Mathematical

Surveys

and Monographs

Volume 72

Jacobi Operators and Completely Integrable Nonlinear Lattices

Gerald Teschl

University of Vienna

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American Mathematical Society

Editorial Board

Georgia M. Benkart  Michael Loss
Peter Landweber  Tudor Stefan Ratiu, Chair

1991 Mathematics subject classification. 39Axx, 47B39, 58F07

Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.

Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.

Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.

Corrections and complements to this book are available from:

http://www.mat.univie.ac.at/˜ gerald/ftp/book-jac/

Typeset by AMS-L^{A}TEX, PicTEX, and Makeindex. Version: August 24, 2017.

Library of Congress Cataloging-in-Publication Data

Teschl, Gerald, 1970–

Jacobi Operators and Completely Integrable Nonlinear Lattices / Gerald Teschl. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; V. 72)

Includes bibliographical references and index.

ISBN 0-8218-1940-2

1. Jacobi operators. 2. Differentiable dynamical systems. I. Title. II. Series: Mathemtical surveys and monographs ; no. 72. QA329.2.T47 1999 99-39165 515’.7242–dc21 CIP

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permissions should be addressed to the Assistant to the Publisher, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission@ams.org.

\begin{matrix}c \\ ◯ \end{matrix} 2000 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted too the United States Government.

To Susanne

Contents

Preface  xiii

Part 1. Jacobi Operators

Chapter 1. Jacobi operators  3
1.1. General properties  3
1.2. Jacobi operators  13
1.3. A simple example  19
1.4. General second order difference expressions  20
1.5. The infinite harmonic crystal in one dimension  22
Chapter 2. Foundations of spectral theory for Jacobi operators  27
2.1. Weyl m-functions  27
2.2. Properties of solutions  29
2.3. Positive solutions  34
2.4. Weyl circles  37
2.5. Canonical forms of Jacobi operators and the moment problem  40
2.6. Some remarks on unbounded operators  47
2.7. Inverse spectral theory  53
Chapter 3. Qualitative theory of spectra  59
3.1. Locating the spectrum and spectral multiplicities  59
3.2. Locating the essential spectrum  62
3.3. Locating the absolutely continuous spectrum  65
3.4. A bound on the number of eigenvalues  72
Chapter 4. Oscillation theory  75
4.1. Prüfer variables and Sturm’s separation theorem  75

4.2. Classical oscillation theory  80
4.3. Renormalized oscillation theory  83
Chapter 5. Random Jacobi operators  87
5.1. Random Jacobi operators  87
5.2. The Lyapunov exponent and the density of states  91
5.3. Almost periodic Jacobi operators  100
Chapter 6. Trace formulas  105
6.1. Asymptotic expansions  105
6.2. General trace formulas and xi functions  109
Chapter 7. Jacobi operators with periodic coefficients  115
7.1. Floquet theory  115
7.2. Connections with the spectra of finite Jacobi operators  119
7.3. Polynomial identities  123
7.4. Two examples: period one and two  124
7.5. Perturbations of periodic operators  126
Chapter 8. Reflectionless Jacobi operators  133
8.1. Spectral analysis and trace formulas  133
8.2. Isospectral operators  140
8.3. The finite-gap case  142
8.4. Further spectral interpretation  150
Chapter 9. Quasi-periodic Jacobi operators and Riemann theta functions 153
9.1. Riemann surfaces  153
9.2. Solutions in terms of theta functions  155
9.3. The elliptic case, genus one  163
9.4. Some illustrations of the Riemann-Roch theorem  165
Chapter 10. Scattering theory  167
10.1. Transformation operators  167
10.2. The scattering matrix  171
10.3. The Gel’fand-Levitan-Marchenko equations  175
10.4. Inverse scattering theory  180
Chapter 11. Spectral deformations – Commutation methods  187
11.1. Commuting first order difference expressions  187
11.2. The single commutation method  189
11.3. Iteration of the single commutation method  193
11.4. Application of the single commutation method  196
11.5. A formal second commutation  198

11.6. The double commutation method  200
11.7. Iteration of the double commutation method  206
11.8. The Dirichlet deformation method  209
Notes on literature  217

Part 2. Completely Integrable Nonlinear Lattices

Chapter 12. The Toda system  223
12.1. The Toda lattice  223
12.2. Lax pairs, the Toda hierarchy, and hyperelliptic curves  226
12.3. Stationary solutions  233
12.4. Time evolution of associated quantities  236
Chapter 13. The initial value problem for the Toda system  241
13.1. Finite-gap solutions of the Toda hierarchy  241
13.2. Quasi-periodic finite-gap solutions and the time-dependent Baker-Akhiezer function247
13.3. A simple example – continued  250
13.4. Inverse scattering transform  251
13.5. Some additions in case of the Toda lattice  254
13.6. The elliptic case – continued  256
Chapter 14. The Kac-van Moerbeke system  257
14.1. The Kac-van Moerbeke hierarchy and its relation to the Toda hierarchy257
14.2. Kac and van Moerbeke’s original equations  263
14.3. Spectral theory for supersymmetric Dirac-type difference operators 264
14.4. Associated solutions  265
14.5. N-soliton solutions on arbitrary background  267
Notes on literature  273
Appendix A. Compact Riemann surfaces – a review  275
A.1. Basic notation  275
A.2. Abelian differentials  277
A.3. Divisors and the Riemann-Roch theorem  279
A.4. Jacobian variety and Abel’s map  283
A.5. Riemann’s theta function  286
A.6. The zeros of the Riemann theta function  288
A.7. Hyperelliptic Riemann surfaces  291
Appendix B. Herglotz functions  299
Appendix C. Jacobi Difference Equations with Mathematica  309
C.1. The package DiffEqs and first order difference equations  309

C.2. The package JacDEqs and Jacobi difference equations  313
C.3. Simple properties of Jacobi difference equations  315
C.4. Orthogonal Polynomials  319
C.5. Recursions  321
C.6. Commutation methods  322
C.7. Toda lattice  328
C.8. Kac-van Moerbeke lattice  331
Bibliography  335
Glossary of notations  347
Index  353

Preface

The name of the game

Jacobi operators appear in a variety of applications. They can be viewed as the discrete analogue of Sturm-Liouville operators and their investigation has many similarities with Sturm-Liouville theory. Spectral and inverse spectral theory for Jacobi operators plays a fundamental role in the investigation of completely integrable nonlinear lattices, in particular the Toda lattice and its modified counterpart, the Kac-van Moerbeke lattice.

Why I have written this book

Whereas numerous books about Sturm-Liouville operators have been written, only few on Jacobi operators exist. In particular, there is currently no monograph available which covers all basic topics (like spectral and inverse spectral theory, scattering theory, oscillation theory and positive solutions, (quasi-)periodic operators, spectral deformations, etc.) typically found in textbooks on Sturm-Liouville operators.

In the case of the Toda lattice a textbook by M. Toda [230] exists, but none of the recent advances in the theory of nonlinear lattices are covered there.

Audience and prerequisites

As audience I had researchers in mind. This book can be used to get acquainted with selected topics as well as to look up specific results. Nevertheless, no previous knowledge on difference equations is assumed and all results are derived in a selfcontained manner. Hence the present book is accessible to graduate students as well. Previous experience with Sturm-Liouville operators might be helpful but is not necessary. Still, a solid working knowledge from other branches of mathematics is needed. In particular, I have assumed that the reader is familiar with the theory of (linear) self-adjoint operators in Hilbert spaces which can be found in (e.g.)

[192] or [241]. This theory is heavily used in the first part. In addition, the reader might have to review material from complex analysis (see Appendix A and B) and differential equations on Banach manifolds (second part only) if (s)he feels (s)he does not have the necessary background. However, this knowledge is mainly needed for understanding proofs rather than the results themselves.

The style of this book

The style of this monograph is strongly influenced by my personal bias. I have striven to present an intuitive approach to each subject and to include the simplest possible proof for every result. Most proofs are rather sketchy in character, so that the main idea becomes clear instead of being drowned by technicalities. Nevertheless, I have always tried to include enough information for the reader to fill in the remaining details (her)himself if desired. To help researchers, using this monograph as a reference, to quickly spot the result they are looking for, most information is found in display style formulas.

The entire treatment is supposed to be mathematically rigorous. I have tried to prove every statement I make and, in particular, these little obvious things, which turn out less obvious once one tries to prove them. In this respect I had Marchenko’s monograph on Sturm-Liouville operators [167] and the one by Weidmann [241] on functional analysis in mind.

Literature

The first two chapters are of an introductory nature and collect some wellknown “folklore”, the successive more advanced chapters are a synthesis of results from research papers. In most cases I have rearranged the material, streamlined proofs, and added further facts which are not published elsewhere. All results appear without special attribution to who first obtained them but there is a section entitled “Notes on literature” in each part which contains references to the literature plus hints for additional reading. The bibliography is selective and far from being complete. It contains mainly references I (am aware of and which I) have actually used in the process of writing this book.

Terminology and notation

For the most part, the terminology used agrees with generally accepted usage. Whenever possible, I have tried to preserve original notation. Unfortunately I had to break with this policy at various points, since I have given higher priority to a consistent (and self-explaining) notation throughout the entire monograph. A glossary of notation can be found towards the end.

Contents

For convenience of the reader, I have split the material into two parts; one on Jacobi operators and one on completely integrable lattices. In particular, the second part is to a wide extent independent of the first one and anybody interested only in completely integrable lattices can move directly to the second part (after browsing Chapter 1 to get acquainted with the notation).

Part I

Chapter 1 gives an introduction to the theory of second order difference equations and bounded Jacobi operators. All basic notations and properties are presented here. In addition, this chapter provides several easy but extremely helpful gadgets. We investigate the case of constant coefficients and, as a motivation for the reader, the infinite harmonic crystal in one dimension is discussed.

Chapter 2 establishes the pillars of spectral and inverse spectral theory for Jacobi operators. Here we develop what is known as discrete Weyl-Titchmarsh-Kodaira theory. Basic things like eigenfunction expansions, connections with the moment problem, and important properties of solutions of the Jacobi equation are shown in this chapter.

Chapter 3 considers qualitative theory of spectra. It is shown how the essential, absolutely continuous, and point spectrum of specific Jacobi operators can be located in some cases. The connection between existence of α-subordinate solutions and α-continuity of spectral measures is discussed. In addition, we investigate under which conditions the number of discrete eigenvalues is finite.

Chapter 4 covers discrete Sturm-Liouville theory. Both classical oscillation and renormalized oscillation theory are developed.

Chapter 5 gives an introduction to the theory of random Jacobi operators. Since there are monographs (e.g., [40]) devoted entirely to this topic, only basic results on the spectra and some applications to almost periodic operators are presented.

Chapter 6 deals with trace formulas and asymptotic expansions which play a fundamental role in inverse spectral theory. In some sense this can be viewed as an application of Krein’s spectral shift theory to Jacobi operators. In particular, the tools developed here will lead to a powerful reconstruction procedure from spectral data for reflectionless (e.g., periodic) operators in Chapter 8.

Chapter 7 considers the special class of operators with periodic coefficients. This class is of particular interest in the one-dimensional crystal model and several profound results are obtained using Floquet theory. In addition, the case of impurities in one-dimensional crystals (i.e., perturbation of periodic operators) is studied.

Chapter 8 again considers a special class of Jacobi operators, namely reflectionless ones, which exhibit an algebraic structure similar to periodic operators. Moreover, this class will show up again in Chapter 10 as the stationary solutions of the Toda equations.

Chapter 9 shows how reflectionless operators with no eigenvalues (which turn out to be associated with quasi-periodic coefficients) can be expressed in terms of Riemann theta functions. These results will be used in Chapter 13 to compute explicit formulas for solutions of the Toda equations in terms of Riemann theta functions.

Chapter 10 provides a comprehensive treatment of (inverse) scattering theory for Jacobi operators with constant background. All important objects like reflection/transmission coefficients, Jost solutions and Gel’fand-Levitan-Marchenko equations are considered. Again this applies to impurities in one-dimensional crystals. Furthermore, this chapter forms the main ingredient of the inverse scattering transform for the Toda equations.

Chapter 11 tries to deform the spectra of Jacobi operators in certain ways. We compute transformations which are isospectral and such which insert a finite number of eigenvalues. The standard transformations like single, double, or Dirichlet commutation methods are developed. These transformations can be used as powerful tools in inverse spectral theory and they allow us to compute new solutions from old solutions of the Toda and Kac-van Moerbeke equations in Chapter 14.

Part II

Chapter 12 is the first chapter on integrable lattices and introduces the Toda system as hierarchy of evolution equations associated with the Jacobi operator via the standard Lax approach. Moreover, the basic (global) existence and uniqueness theorem for solutions of the initial value problem is proven. Finally, the stationary hierarchy is investigated and the Burchnall-Chaundy polynomial computed.

Chapter 13 studies various aspects of the initial value problem. Explicit formulas in case of reflectionless (e.g., (quasi-)periodic) initial conditions are given in terms of polynomials and Riemann theta functions. Moreover, the inverse scattering transform is established.

The final Chapter 14 introduces the Kac van-Moerbeke hierarchy as modified counterpart of the Toda hierarchy. Again the Lax approach is used to establish the basic (global) existence and uniqueness theorem for solutions of the initial value problem. Finally, its connection with the Toda hierarchy via a Miura-type transformation is studied and used to compute N-soliton solutions on arbitrary background.

Appendix

Appendix A reviews the theory of Riemann surfaces as needed in this monograph. While most of us will know Riemann surfaces from a basic course on complex analysis or algebraic geometry, this will be mainly from an abstract viewpoint like in [86] or [129], respectively. Here we will need a more “computational” approach and I hope that the reader can extract this knowledge from Appendix A.

Appendix B compiles some relevant results from the theory of Herglotz functions and Borel measures. Since not everybody is familiar with them, they are included for easy reference.

Appendix C shows how a program for symbolic computation, Mathematica^{®}, can be used to do some of the computations encountered during the main bulk. While I don’t believe that programs for symbolic computations are an indispensable tool for doing research on Jacobi operators (or completely integrable lattices), they are at least useful for checking formulas. Further information and Mathematica^{®}notebooks can be found at

          http://www.mat.univie.ac.at/~gerald/ftp/book-jac/

Acknowledgments

This book has greatly profited from collaborations and discussions with W. Bulla, F. Gesztesy, H. Holden, M. Krishna, and B. Simon. In addition, many people generously devoted considerable time and effort to reading earlier versions of the manuscript and making many corrections. In particular, I wish to thank D. Damanik, H. Hanßmann, A. von der Heyden, R. Killip, T. Sørensen, S. Teschl (Timischl), K. Unterkofler, and H. Widom. Next, I am happy to express my gratitude to P. Deift, J. Geronimo, and E. Lieb for helpful suggestions and advise. I also like to thank the staff at the American Mathematical Society for the fast and professional production of this book.

Partly supported by the Austrian Science Fund (http://www.fwf.ac.at/) under Grant No. P12864-MAT.

Finally, no book is free of errors. So if you find one, or if you have comments or suggestions, please let me know. I will make all corrections and complements available at the URL above.

Gerald Teschl

Vienna, Austria May, 1999

Gerald Teschl Institut für Mathematik Universität Wien Strudlhofgasse 4 1090 Wien, Austria

E-mail address: gerald@mat.univie.ac.at URL: http://www.mat.univie.ac.at/˜ gerald/

Part 1

Chapter 1

Jacobi Operators Jacobi operators

This chapter introduces to the theory of second order difference equations and Jacobi operators. All the basic notation and properties are presented here. In addition, it provides several easy but extremely helpful gadgets. We investigate the case of constant coefficients and, as an application, discuss the infinite harmonic crystal in one dimension.

1.1. General properties

The issue of this section is mainly to fix notation and to establish all for us relevant properties of symmetric three-term recurrence relations in a self-contained manner.

We start with some preliminary notation. For I ⊆ ℤ and M a set we denote by ℓ(I, M) the set of M-valued sequences (f(n))_{n∈I}. Following common usage we will frequently identify the sequence f = f(.) = (f(n))_{n∈I} with f(n) whenever it is clear that n is the index (I being understood). We will only deal with the cases M = ℝ, ℝ^{2}, ℂ, and ℂ^{2}. Since most of the time we will have M = ℂ, we omit M in this case, that is, ℓ(I) = ℓ(I, ℂ). For N_{1}, N_{2} ∈ ℤ we abbreviate ℓ(N_{1}, N_{2}) = ℓ({n ∈ ℤ|N_{1} < n < N_{2}}), ℓ(N_{1}, ∞) = ℓ({n ∈ ℤ|N_{1} < n}), and ℓ(−∞, N_{2}) = ℓ({n ∈ ℤ|n < N_{2}}) (sometimes we will also write ℓ(N_{2}, −∞) instead of ℓ(−∞, N_{2}) for notational convenience). If M is a Banach space with norm |.|, we define

ℓ^{p}(I, M) = {f ∈ ℓ(I, M)| ∑_{n∈I} |f(n)|^{p} < ∞}, 1 ≤ p < ∞,
(1.1)  ℓ^{∞}(I, M) = {f ∈ ℓ(I, M)| sup_{n∈I} |f(n)| < ∞}.

Introducing the following norms

(1.2)  ∥f∥_{p} = (∑_{n∈I} |f(n)|^{p})^{1/p} , 1 ≤ p < ∞,  ∥f∥_{∞} = sup_{n∈I} |f(n)|,

makes ℓ^{p}(I, M), 1 ≤ p ≤ ∞, a Banach space as well.

Furthermore, ℓ_{0}(I, M) denotes the set of sequences with only finitely many values being nonzero and ℓ_{±}^{p}(ℤ, M) denotes the set of sequences in ℓ(ℤ, M) which are ℓ^{p} near ±∞, respectively (i.e., sequences whose restriction to ℓ(±ℕ, M) belongs to ℓ^{p}(±ℕ, M). Here ℕ denotes the set of positive integers). Note that, according to our definition, we have

(1.3)  ℓ_{0}(I, M) ⊆ ℓ^{p}(I, M) ⊆ ℓ^{q}(I, M) ⊆ ℓ^{∞}(I, M), p < q,

with equality holding if and only if I is finite (assuming dim M > 0).

In addition, if M is a (separable) Hilbert space with scalar product ⟨., ..⟩_{M}, then the same is true for ℓ^{2}(I, M) with scalar product and norm defined by

⟨f, g⟩ = ∑_{n∈I}⟨f(n), g(n)⟩_{M},
(1.4)  ∥f∥ = ∥f∥_{2} = \sqrt{⟨f, f⟩}, f, g ∈ ℓ^{2}(I, M).

For what follows we will choose I = ℤ for simplicity. However, straightforward modifications can be made to accommodate the general case I ⊂ ℤ.

During most of our investigations we will be concerned with difference expressions, that is, endomorphisms of ℓ(ℤ);

(1.5)  \begin{aligned}R : & ℓ(ℤ) & → & ℓ(ℤ) \\ & f & ↦ & R f \end{aligned}

(we reserve the name difference operator for difference expressions defined on a subset of ℓ^{2}(ℤ)). Any difference expression R is uniquely determined by its corresponding matrix representation

(1.6)  (R(m, n))_{m,n∈ℤ}, R(m, n) = (Rδ_{n})(m) = ⟨δ_{m}, R δ_{n}⟩,

where

(1.7)  δ_{n}(m) = δ_{m,n} = \{ \begin{matrix}0 & n ≠ m \\ 1 & n = m \end{matrix}

is the canonical basis of ℓ(ℤ). The order of R is the smallest nonnegative integer N = N_{+}+N_{−} such that R(m, n) = 0 for all m, n with n−m > N_{+} and m−n > N_{−}. If no such number exists, the order is infinite.

We call R symmetric (resp. skew-symmetric) if R(m, n) = R(n, m) (resp. R(m, n) = −R(n, m)).

Maybe the simplest examples for a difference expression are the shift expressions

(1.8)  (S^{±}f)(n) = f(n ± 1).

They are of particular importance due to the fact that their powers form a basis for the space of all difference expressions (viewed as a module over the ring ℓ(ℤ)). Indeed, we have

(1.9)  R = ∑_{k∈ℤ} R(., . + k)(S^{+})^{k},  (S^{±})^{−1} = S^{∓}.

Here R(., . + k) denotes the multiplication expression with the sequence (R(n, n + k))_{n∈ℤ}, that is, R(., . + k) : f(n) ↦ R(n, n + k)f(n). In order to simplify notation we agree to use the short cuts

(1.10)  f^{±} = S^{±}f, f^{++} = S^{+}S^{+}f, etc.,

whenever convenient. In connection with the difference expression (1.5) we also define the diagonal, upper, and lower triangular parts of R as follows

(1.11)  [R]_{0} = R(., .), [R]_{±} = ∑_{k∈ℕ} R(., . ± k)(S^{±})^{k},

implying R = [R]_{+} + [R]_{0} + [R]_{−}.

Having these preparations out of the way, we are ready to start our investigation of second order symmetric difference expressions. To set the stage, let a, b ∈ ℓ(ℤ, ℝ) be two real-valued sequences satisfying

(1.12)  a(n) ∈ ℝ\{0}, b(n) ∈ ℝ,  n ∈ ℤ,

and introduce the corresponding second order, symmetric difference expression

(1.13)  \begin{aligned}τ : & ℓ(ℤ) & → & ℓ(ℤ) \\ & f(n) & ↦ & a(n)f(n + 1) + a(n − 1)f(n − 1) + b(n)f(n) \end{aligned} .

It is associated with the tridiagonal matrix

(1.14)  ( \begin{matrix}... & ... & ... \\ a(n − 2) & b(n − 1) & a(n − 1) \\ & a(n − 1) & b(n) & a(n) \\ & & a(n) & b(n + 1) & a(n + 1) \\ & & ... & ... & ... \end{matrix} )

and will be our main object for the rest of this section and the tools derived here – even though simple – will be indispensable for us.

Before going any further, I want to point out that there is a close connection between second order, symmetric difference expressions and second order, symmetric differential expressions. This connection becomes more apparent if we use the difference expressions

(1.15)  (∂f)(n) = f(n + 1) − f(n),  (∂^{∗}f)(n) = f(n − 1) − f(n),

(note that ∂, ∂^{∗} are formally adjoint) to rewrite τ in the following way

(τf)(n) = −(∂^{∗}a∂f)(n) + (a(n − 1) + a(n) + b(n))f(n)
(1.16)  = −(∂a^{−}∂^{∗}f)(n) + (a(n − 1) + a(n) + b(n))f(n).

This form resembles very much the Sturm-Liouville differential expression, wellknown in the theory of ordinary differential equations.

In fact, the reader will soon realize that there are a whole lot more similarities between differentials, integrals and their discrete counterparts differences and sums. Two of these similarities are the product rules

(∂fg)(n) = f(n)(∂g)(n) + g(n + 1)(∂f)(n),
(1.17)  (∂^{∗}fg)(n) = f(n)(∂^{∗}g)(n) + g(n − 1)(∂^{∗}f)(n)

and the summation by parts formula (also known as Abel transform)

(1.18) ∑_{j=m}^{n} g(j)(∂f)(j) = g(n)f(n + 1) − g(m − 1)f(m) + ∑_{j=m}^{n}(∂^{∗}g)(j)f(j).

Both are readily verified. Nevertheless, let me remark that ∂, ∂^{∗} are no derivations since they do not satisfy Leibnitz rule. This very often makes the discrete case different (and sometimes also harder) from the continuous one. In particular, many calculations become much messier and formulas longer.

There is much more to say about relations for the difference expressions (1.15) analogous to the ones for differentiation. We refer the reader to, for instance, [4], [87], or [147] and return to (1.13).

Associated with τ is the eigenvalue problem τu = zu. The appropriate setting for this eigenvalue problem is the Hilbert space ℓ^{2}(ℤ). However, before we can pursue the investigation of the eigenvalue problem in ℓ^{2}(ℤ), we need to consider the Jacobi difference equation

(1.19)  τu = z u,  u ∈ ℓ(ℤ), z ∈ ℂ.

Using a(n) ≠ 0 we see that a solution u is uniquely determined by the values u(n_{0}) and u(n_{0} + 1) at two consecutive points n_{0}, n_{0} + 1 (you have to work much harder to obtain the corresponding result for differential equations). It follows, that there are exactly two linearly independent solutions.

Combining (1.16) and the summation by parts formula yields Green’s formula

(1.20)  ∑_{j=m}^{n} (f(τg) − (τf)g)(j) = W_{n}(f, g) − W_{m−1}(f, g)

for f, g ∈ ℓ(ℤ), where we have introduced the (modified) Wronskian

(1.21)  W_{n}(f, g) = a(n)(f(n)g(n + 1) − g(n)f(n + 1)).

Green’s formula will be the key to self-adjointness of the operator associated with τ in the Hilbert space ℓ^{2}(ℤ) (cf. Theorem 1.5) and the Wronskian is much more than a suitable abbreviation as we will show next.

Evaluating (1.20) in the special case where f and g both solve (1.19) (with the same parameter z) shows that the Wronskian is constant (i.e., does not depend on n) in this case. (The index n will be omitted in this case.) Moreover, it is nonzero if and only if f and g are linearly independent.

Since the (linear) space of solutions is two dimensional (as observed above) we can pick two linearly independent solutions c, s of (1.19) and write any solution u of (1.19) as a linear combination of these two solutions

(1.22)  u(n) = \frac{W (u, s)}{W (c, s)}c(n) − \frac{W (u, c)}{W (c, s)}s(n).

For this purpose it is convenient to introduce the following fundamental solutions c, s ∈ ℓ(ℤ)

(1.23)  τc(z, ., n_{0}) = z c(z, ., n_{0}),  τs(z, ., n_{0}) = z s(z, ., n_{0}),

fulfilling the initial conditions

(1.24)  \begin{matrix}c(z, n_{0}, n_{0}) = 1, & c(z, n_{0} + 1, n_{0}) = 0, \\ s(z, n_{0}, n_{0}) = 0, & s(z, n_{0} + 1, n_{0}) = 1. \end{matrix}

Most of the time the base point n_{0} will be unessential and we will choose n_{0} = 0 for simplicity. In particular, we agree to omit n_{0} whenever it is 0, that is,

(1.25)  c(z, n) = c(z, n, 0),  s(z, n) = s(z, n, 0).

Since the Wronskian of c(z, ., n_{0}) and s(z, ., n_{0}) does not depend on n we can evaluate it at n_{0}

(1.26)  W (c(z, ., n_{0}), s(z, ., n_{0})) = a(n_{0})

and consequently equation (1.22) simplifies to

(1.27)  u(n) = u(n_{0})c(z, n, n_{0}) + u(n_{0} + 1)s(z, n, n_{0}).

Sometimes a lot of things get more transparent if (1.19) is regarded from the viewpoint of dynamical systems. If we introduce \underline{u} = (u, u^{+}) ∈ ℓ(ℤ, ℂ^{2}), then (1.19) is equivalent to

(1.28)  \underline{u}(n + 1) = U(z, n + 1)\underline{u}(n),  \underline{u}(n − 1) = U(z, n)^{−1}\underline{u}(n),

where U(z, .) is given by

U(z, n) = \frac{1}{a(n)} ( \begin{matrix}0 & a(n) \\ −a(n − 1) & z − b(n) \end{matrix} ) ,
(1.29)  U^{−1}(z, n) = \frac{1}{a(n − 1)} ( \begin{matrix}z − b(n) & −a(n) \\ a(n − 1) & 0 \end{matrix} ) .

The matrix U(z, n) is often referred to as transfer matrix. The corresponding (non-autonomous) flow on ℓ(ℤ, ℂ^{2}) is given by the fundamental matrix

Φ(z, n, n_{0}) = ( \begin{matrix}c(z, n, n_{0}) & s(z, n, n_{0}) \\ c(z, n + 1, n_{0}) & s(z, n + 1, n_{0}) \end{matrix} )

(1.30)  = \{ \begin{matrix}U(z, n) ⋯ U(z, n_{0} + 1) & n > n_{0} \\ 1l & n = n_{0} \\ U^{−1}(z, n + 1) ⋯ U^{−1}(z, n_{0}) & n < n_{0} \end{matrix} .

More explicitly, equation (1.27) is now equivalent to

(1.31)  \binom{u(n)}{u(n + 1)} = Φ(z, n, n_{0})\binom{u(n_{0})}{u(n_{0} + 1)}.

Using (1.31) we learn that Φ(z, n, n_{0}) satisfies the usual group law

(1.32)  Φ(z, n, n_{0}) = Φ(z, n, n_{1})Φ(z, n_{1}, n_{0}),  Φ(z, n_{0}, n_{0}) = 1l

and constancy of the Wronskian (1.26) implies

(1.33)  det Φ(z, n, n_{0}) = \frac{a(n_{0})}{a(n)}.

Let us use Φ(z, n) = Φ(z, n, 0) and define the upper, lower Lyapunov exponents

\overline{γ}^{±}(z) = lim sup_{n→±∞} \frac{1}{|n|} ln ∥Φ(z, n, n_{0})∥ = lim sup_{n→±∞} \frac{1}{|n|} ln ∥Φ(z, n)∥,
(1.34) \underline{γ}^{±}(z) = lim inf_{n→±∞} \frac{1}{|n|} ln ∥Φ(z, n, n_{0})∥ = lim inf_{n→±∞} \frac{1}{|n|} ln ∥Φ(z, n)∥.

Here

(1.35)  ∥Φ∥ = sup_{u∈ℂ^{2}\\{0\}} \frac{∥Φu∥_{ℂ^{2}}}{∥u∥_{ℂ^{2}}}

denotes the operator norm of Φ. By virtue of (use (1.32))

(1.36)  ∥Φ(z, n_{0})∥^{−1}∥Φ(z, n)∥ ≤ ∥Φ(z, n, n_{0})∥ ≤ ∥Φ(z, n_{0})^{−1}∥∥Φ(z, n)∥

the definition of \overline{γ}^{±}(z), \underline{γ}^{±}(z) is indeed independent of n_{0}. Moreover, \underline{γ}^{±}(z) ≥ 0 if a(n) is bounded. In fact, since \underline{γ}^{±}(z) < 0 would imply lim_{j→±∞} ∥Φ(z, n_{j}, n_{0})∥ = 0 for some subsequence n_{j} contradicting (1.33).

If \underline{γ}^{±}(z) = \overline{γ}^{±}(z) we will omit the bars. A number λ ∈ ℝ is said to be hyperbolic at ±∞ if \underline{γ}^{±}(λ) = \overline{γ}^{±}(λ) > 0, respectively. The set of all hyperbolic numbers is denoted by Hyp_{±}(Φ). For λ ∈ Hyp_{±}(Φ) one has existence of corresponding stable and unstable manifolds V^{±}(λ).

Lemma 1.1. Suppose that |a(n)| does not grow or decrease exponentially and that |b(n)| does not grow exponentially, that is,

(1.37)  lim_{n→±∞} \frac{1}{|n|} ln |a(n)| = 0,  lim_{n→±∞} \frac{1}{|n|} ln(1 + |b(n)|) = 0.

If λ ∈ Hyp_{±}(Φ), then there exist one-dimensional linear subspaces V^{±}(λ) ⊆ ℝ^{2} such that

\underline{v} ∈ V^{±}(λ) ⇔ lim_{n→±∞} \frac{1}{|n|} ln ∥Φ(λ, n)\underline{v}∥ = −γ^{±}(λ),
(1.38)  \underline{v} ∉ V^{±}(λ) ⇔ lim_{n→±∞} \frac{1}{|n|} ln ∥Φ(λ, n)\underline{v}∥ = γ^{±}(λ),

respectively.

Proof. Set

(1.39)  A(n) = ( \begin{matrix}1 & 0 \\ 0 & a(n) \end{matrix} )

and abbreviate

Ũ(z, n) = A(n)U(z, n)A(n − 1)^{−1} = \frac{1}{a(n − 1)} ( \begin{matrix}0 & 1 \\ −a(n − 1)^{2} & z − b(n) \end{matrix} ) ,

Φ̃(z, n) = A(n)Φ(z, n)A(0)^{−1}. (1.40)

Then (1.28) translates into

(1.41)  \underline{ũ}(n + 1) = Ũ(z, n + 1)\underline{ũ}(n),  \underline{ũ}(n − 1) = Ũ(z, n)^{−1}\underline{ũ}(n),

where \underline{ũ} = A\underline{u} = (u, au^{+}), and we have

(1.42)  det Ũ(z, n) = 1 and lim_{n→∞} \frac{1}{|n|} ln ∥Ũ(z, n)∥ = 0

due to our assumption (1.37). Moreover,

(1.43)  \frac{min(1, a(n))}{max(1, a(0))} ≤ \frac{∥Φ̃(z, n)∥}{∥Φ(z, n)∥} ≤ \frac{max(1, a(n))}{min(1, a(0))}

and hence lim_{n→±∞} |n|^{−1} ln ∥Φ̃(z, n)∥ = lim_{n→±∞} |n|^{−1} ln ∥Φ(z, n)∥ whenever one of the limits exists. The same is true for the limits of |n|^{−1} ln ∥Φ̃(z, n)v∥ and |n|^{−1} ln ∥Φ(z, n)v∥. Hence it suffices to prove the result for matrices Φ̃ satisfying (1.42). But this is precisely the (deterministic) multiplicative ergodic theorem of Osceledec (see [201]). □

Observe that by looking at the Wronskian of two solutions u ∈ V^{±}(λ), v ∉ V^{±}(λ) it is not hard to see that the lemma becomes false if a(n) is exponentially decreasing.

For later use observe that

(1.44)  Φ(z, n, m)^{−1} = \frac{a(n)}{a(m)}J Φ(z, n, m)^{⊤}J^{−1},  J = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ,

(where Φ^{⊤} denotes the transposed matrix of Φ) and hence

(1.45)  |a(m)|∥Φ(z, n, m)^{−1}∥ = |a(n)|∥Φ(z, n, m)∥.

We will exploit this notation later in this monograph but for the moment we return to our original point of view.

The equation

(1.46)  (τ − z)f = g

for fixed z ∈ ℂ, g ∈ ℓ(ℤ), is referred to as inhomogeneous Jacobi equation. Its solution can can be completely reduced to the solution of the corresponding homogeneous Jacobi equation (1.19) as follows. Introduce

(1.47)  K(z, n, m) = \frac{s(z, n, m)}{a(m)}.

Then the sequence

f(n) = f_{0} c(z, n, n_{0}) + f_{1}s(z, n, n_{0})

+\begin{matrix}n \\ ∑ & ∗ \\ m=n_{0}+1 \end{matrix} K(z, n, m)g(m), (1.48)

where

(1.49)  \begin{matrix}n−1 \\ ∑ & ∗ \\ j=n_{0} \end{matrix} f(j) = \{ \begin{matrix}∑_{j=n_{0}}^{n−1} f(j) for & n > n_{0} \\ 0  for & n = n_{0} \\ − ∑_{j=n}^{n_{0}−1} f(j) for & n < n_{0} \end{matrix} ,

satisfies (1.46) and the initial conditions f(n_{0}) = f_{0}, f(n_{0} + 1) = f_{1} as can be checked directly. The summation kernel K(z, n, m) has the following properties: K(z, n, n) = 0, K(z, n + 1, n) = a(n)^{−1}, K(z, n, m) = −K(z, m, n), and

(1.50)  K(z, n, m) = \frac{u(z, m)v(z, n) − u(z, n)v(z, m)}{W (u(z), v(z))}

for any pair u(z), v(z) of linearly independent solutions of τu = zu.

Another useful result is the variation of constants formula. It says that if one solution u of (1.19) with u(n) ≠ 0 for all n ∈ ℤ is known, then a second (linearly independent, W (u, v) = 1) solution of (1.19) is given by

(1.51)  v(n) = u(n) \begin{matrix}n−1 \\ ∑ & ∗ \\ j=n_{0} \end{matrix} \frac{1}{a(j)u(j)u(j + 1)}.

It can be verified directly as well.

Sometimes transformations can help to simplify a problem. The following two are of particular interest to us. If u fulfills (1.19) and u(n) ≠ 0, then the sequence ϕ(n) = u(n + 1)/u(n) satisfies the (discrete) Riccati equation

(1.52)  a(n)ϕ(n) + \frac{a(n − 1)}{ϕ(n − 1)} = z − b(n).

Conversely, if ϕ fulfills (1.52), then the sequence

(1.53)  u(n) = \begin{matrix}n−1 \\ ∏ & ∗ \\ j=n_{0} \end{matrix} ϕ(j) = \{ \begin{matrix}∏_{j=n_{0}}^{n−1} ϕ(j) for & n > n_{0} \\ 1  for & n = n_{0} \\ ∏_{j=n}^{n_{0}−1} ϕ(j)^{−1} for & n < n_{0} \end{matrix} ,

fulfills (1.19) and is normalized such that u(n_{0}) = 1. In addition, we remark that the sequence ϕ(n) might be written as finite continued fraction,

(1.54) a(n)ϕ(n) = z − b(n) − \frac{a(n − 1)^{2}}{z − b(n − 1) − _{^{...} − \frac{a(n_{0} + 1)^{2}}{z − b(n_{0} + 1) − \frac{a(n_{0})}{ϕ(n_{0})}}}}

for n > n_{0} and

(1.55)  a(n)ϕ(n) = \frac{a(n)^{2}}{z − b(n + 1) − _{^{...} − \frac{a(n_{0} − 1)^{2}}{z − b(n_{0}) − a(n_{0})ϕ(n_{0})}}}

for n < n_{0}.

If ã is a sequence with ã(n) ≠ 0 and u fulfills (1.19), then the sequence

(1.56)  ũ(n) = u(n)\begin{matrix}n−1 \\ ∏ & ∗ \\ j=n_{0} \end{matrix} ã(j),

fulfills

(1.57)  \frac{a(n)}{ã(n)}ũ(n + 1) + a(n − 1)ã(n − 1)ũ(n − 1) + b(n)ũ(n) = z ũ(n).

Especially, taking ã(n) = sgn(a(n)) (resp. ã(n) = −sgn(a(n))), we see that it is no restriction to assume a(n) > 0 (resp. a(n) < 0) (compare also Lemma 1.6 below).

We conclude this section with a detailed investigation of the fundamental solutions c(z, n, n_{0}) and s(z, n, n_{0}). To begin with, we note (use induction) that both c(z, n ± k, n) and s(z, n ± k, n), k ≥ 0, are polynomials of degree at most k with respect to z. Hence we may set

(1.58)  s(z, n ± k, n) = ∑_{j=0}^{k} s_{j,±k}(n)z^{j}, c(z, n ± k, n) = ∑_{j=0}^{k} c_{j,±k}(n)z^{j}.

Using the coefficients s_{j,±k}(n) and c_{j,±k}(n) we can derive a neat expansion for arbitrary difference expressions. By (1.9) it suffices to consider (S^{±})^{k}.

Lemma 1.2. Any difference expression R of order at most 2k + 1 can be expressed as

(1.59)  R = ∑_{j=0}^{k} (c_{j} + s_{j}S^{+})τ^{j},  c_{j}, s_{j} ∈ ℓ(ℤ), k ∈ ℕ_{0},

with c_{j} = s_{j} = 0 if and only if R = 0. In other words, the set {τ^{j}, S^{+}τ^{j}}_{j∈ℕ_{0}} forms a basis for the space of all difference expressions.

We have

(1.60)  (S^{±})^{k} = ∑_{j=0}^{k} (c_{j,±k} + s_{j,±k}S^{+})τ^{j},

where s_{j,±k}(n) and c_{j,±k}(n) are defined in (1.58).

Proof. We first prove (1.59) by induction on k. The case k = 0 is trivial. Since the matrix element τ^{k}(n, n ± k) = ∏_{j=0}^{k−1} a(n ± j − {0 \atop 1}) ≠ 0 is nonzero we can choose s_{k}(n) = R(n, n + k + 1)/τ^{k}(n − 1, n + k − 1), c_{k}(n) = R(n, n − k)/τ^{k}(n, n − k) and apply the induction hypothesis to R − (c_{k} − s_{k}S^{+})τ^{k}. This proves (1.59). The rest is immediate from

(1.61)  (R(s(z, ., n))(n) = ∑_{j=0}^{k} s_{j}(n)z^{j}, (R(c(z, ., n))(n) = ∑_{j=0}^{k} c_{j}(n)z^{j}.
□

As a consequence of (1.61) we note

Corollary 1.3. Suppose R is a difference expression of order k. Then R = 0 if and only if R|_{Ker(τ−z)} = 0 for k + 1 values of z ∈ ℂ. (Here R|_{Ker(τ−z)} = 0 says that Ru = 0 for any solution u of τu = zu.)

Next, Φ(z, n_{0}, n_{1}) = Φ(z, n_{1}, n_{0})^{−1} provides the useful relations
( \begin{matrix}c(z, n_{0}, n_{1}) & s(z, n_{0}, n_{1}) \\ c(z, n_{0} + 1, n_{1}) & s(z, n_{0} + 1, n_{1}) \end{matrix} )

= \frac{a(n_{1})}{a(n_{0})} ( \begin{matrix}s(z, n_{1} + 1, n_{0}) & −s(z, n_{1}, n_{0}) \\ −c(z, n_{1} + 1, n_{0}) & c(z, n_{1}, n_{0}) \end{matrix} ) (1.62)

and a straightforward calculation (using (1.27)) yields

s(z, n, n_{0} + 1) = −\frac{a(n_{0} + 1)}{a(n_{0})}c(z, n, n_{0}),
s(z, n, n_{0} − 1) = c(z, n, n_{0}) + \frac{z − b(n_{0})}{a(n_{0})}s(z, n, n_{0}),
c(z, n, n_{0} + 1) = \frac{z − b(n_{0} + 1)}{a(n_{0})}c(z, n, n_{0}) + s(z, n, n_{0}),
(1.63)  c(z, n, n_{0} − 1) = −\frac{a(n_{0} − 1)}{a(n_{0})}s(z, n, n_{0}).

Our next task will be expansions of c(z, n, n_{0}), s(z, n, n_{0}) for large z. Let J_{n_{1},n_{2}} be the Jacobi matrix

(1.64)  J_{n_{1},n_{2}} = ( \begin{matrix}b(n_{1} + 1) & a(n_{1} + 1) \\ a(n_{1} + 1) & b(n_{1} + 2) & ... \\ & ... & ... & ... \\ & & ... & b(n_{2} − 2) & a(n_{2} − 2) \\ & & & a(n_{2} − 2) & b(n_{2} − 1) \end{matrix} ) .

Then we have the following expansion for s(z, n, n_{0}), n > n_{0},

(1.65)  s(z, n, n_{0}) = \frac{det(z − J_{n_{0},n})}{∏_{j=n_{0}+1}^{n−1} a(j)} = \frac{z^{k} − ∑_{j=1}^{k} p_{n_{0},n}(j)z^{k−j}}{∏_{j=1}^{k} a(n_{0} + j)} ,

where k = n − n_{0} − 1 ≥ 0 and

(1.66)  p_{n_{0},n}(j) = \frac{tr(J_{n_{0},n}^{j}) − ∑_{ℓ=1}^{j−1} p_{n_{0},n}(ℓ)tr(J_{n_{0},n}^{j−ℓ})}{j}, 1 ≤ j ≤ k.

To verify the first equation, use that if z is a zero of s(., n, n_{0}), then (s(z, n_{0} + 1, n_{0}), …, s(z, n−1, n_{0})) is an eigenvector of (1.64) corresponding to the eigenvalue z. Since the converse statement is also true, the polynomials (in z) s(z, n, n_{0}) and det(z − J_{n_{0},n}) only differ by a constant which can be deduced from (1.30). The second is a well-known property of characteristic polynomials (cf., e.g., [91]).

The first few traces are given by

tr(J_{n_{0},n_{0}+k+1}) = ∑_{j=n_{0}+1}^{n_{0}+k} b(j),

tr(J_{n_{0},n_{0}+k+1}^{2}) = ∑_{j=n_{0}+1}^{n_{0}+k} b(j)^{2} + 2 ∑_{j=n_{0}+1}^{n_{0}+k−1} a(j)^{2},

tr(J_{n_{0},n_{0}+k+1}^{3}) = ∑_{j=n_{0}+1}^{n_{0}+k} b(j)^{3} + 3 ∑_{j=n_{0}+1}^{n_{0}+k−1} a(j)^{2}(b(j) + b(j + 1)),

tr(J_{n_{0},n_{0}+k+1}^{4}) = ∑_{j=n_{0}+1}^{n_{0}+k} b(j)^{4} − 4 ∑_{j=n_{0}+1}^{n_{0}+k−1} a(j)^{2}(b(j)^{2} + b(j + 1)b(j)

+b(j + 1)^{2} + \frac{a(j)^{2}}{2} ) + 4 ∑_{j=n_{0}+1}^{n_{0}+k−2} a(j + 1)^{2}. (1.67)

An explicit calculation yields for n > n_{0} + 1

c(z, n, n_{0}) = \frac{−a(n_{0})z^{n−n_{0}−2}}{∏_{j=n_{0}+1}^{n−1} a(j)} (1 − \frac{1}{z} ∑_{j=n_{0}+2}^{n−1} b(j) + O(\frac{1}{z^{2}} )) ,

(1.68)  s(z, n, n_{0}) = \frac{z^{n−n_{0}−1}}{∏_{j=n_{0}+1}^{n−1} a(j)} (1 − \frac{1}{z} ∑_{j=n_{0}+1}^{n−1} b(j) + O(\frac{1}{z^{2}} )) ,

and (using (1.63) and (1.62)) for n < n_{0}

c(z, n, n_{0}) = \frac{z^{n_{0}−n}}{∏_{j=n}^{n_{0}−1} a(j)} (1 − \frac{1}{z} ∑_{j=n+1}^{n_{0}} b(j) + O(\frac{1}{z^{2}} )) ,

(1.69)  s(z, n, n_{0}) = \frac{−a(n_{0})z^{n_{0}−n−1}}{∏_{j=n}^{n_{0}−1} a(j)} (1 − \frac{1}{z} ∑_{j=n+1}^{n_{0}−1} b(j) + O(\frac{1}{z^{2}} )) .

1.2. Jacobi operators

In this section we scrutinize the eigenvalue problem associated with (1.19) in the Hilbert space ℓ^{2}(ℤ).

Recall that the scalar product and norm is given by

(1.70)  ⟨f, g⟩ = ∑_{n∈ℤ} \overline{f(n)}g(n), ∥f∥ = \sqrt{⟨f, f⟩}, f, g ∈ ℓ^{2}(ℤ),

where the bar denotes complex conjugation.

For simplicity we assume from now on (and for the rest of this monograph) that a,b are bounded sequences.

Hypothesis H. 1.4. Suppose

(1.71)  a, b ∈ ℓ^{∞}(ℤ, ℝ),  a(n) ≠ 0.

Associated with a, b is the Jacobi operator

(1.72)  \begin{aligned}H : & ℓ^{2}(ℤ) & → & ℓ^{2}(ℤ) \\ & f & ↦ & τf \end{aligned} ,

whose basic properties are summarized in our first theorem.

Theorem 1.5. Assume (H.1.4). Then H is a bounded self-adjoint operator. Moreover, a,b bounded is equivalent to H bounded since we have ∥a∥_{∞} ≤ ∥H∥, ∥b∥_{∞} ≤ ∥H∥ and

(1.73)  ∥H∥ ≤ 2∥a∥_{∞} + ∥b∥_{∞},

where ∥H∥ denotes the operator norm of H.

Proof. The fact that lim_{n→±∞} W_{n}(f, g) = 0, f, g ∈ ℓ^{2}(ℤ), together with Green’s formula (1.20) shows that H is self-adjoint, that is,

(1.74)  ⟨f, Hg⟩ = ⟨Hf, g⟩,  f, g ∈ ℓ^{2}(ℤ).

For the rest consider a(n)^{2} + a(n − 1)^{2} + b(n)^{2} = ∥Hδ_{n}∥^{2} ≤ ∥H∥^{2} and

(1.75)  |⟨f, Hf⟩| ≤ (2∥a∥_{∞} + ∥b∥_{∞})∥f∥^{2}.
□

Before we pursue our investigation of Jacobi operators H, let us have a closer look at Hypothesis (H.1.4).

The previous theorem shows that the boundedness of H is due to the boundedness of a and b. This restriction on a, b is by no means necessary. However, it significantly simplifies the functional analysis involved and is satisfied in most cases of practical interest. You can find out how to avoid this restriction in Section 2.6.

The assumption a(n) ≠ 0 is also not really necessary. In fact, we have not even used it in the proof of Theorem 1.5. If a(n_{0}) = 0, this implies that H can be decomposed into the direct sum H_{n_{0}+1,−} ⊕H_{n_{0},+} on ℓ^{2}(−∞, n_{0} +1)⊕ℓ^{2}(n_{0}, ∞) (cf. (1.90) for notation). Nevertheless I want to emphasize that a(n) ≠ 0 was crucial in the previous section and is connected with the existence of (precisely) two linearly independent solutions, which again is related to the fact that the spectrum of H has multiplicity at most two (cf. Section 2.5).

Hence the analysis of H in the case a(n_{0}) = 0 can be reduced to the analysis of restrictions of H which will be covered later in this section. In addition, the following lemma shows that the case a(n) ≠ 0 can be reduced to the case a(n) > 0 or a(n) < 0.

Lemma 1.6. Assume (H.1.4) and pick ε ∈ ℓ(ℤ, {−1, +1}). Introduce a_{ε}, b_{ε} by

(1.76)  a_{ε}(n) = ε(n)a(n), b_{ε}(n) = b(n), n ∈ ℤ,

and the unitary involution U_{ε} by

(1.77)  U_{ε} = U_{ε}^{−1} = ( \begin{matrix}n−1 \\ ∏ & ∗ \\ j=0 \end{matrix} ε(j) δ_{m,n})_{m,n∈ℤ}.

Let H be a Jacobi operator associated with the difference expression (1.13). Then H_{ε} defined as

(1.78)  H_{ε} = U_{ε}HU_{ε}^{−1}

is associated with the difference expression

(1.79)  (τ_{ε}f)(n) = a_{ε}(n)f(n + 1) + a_{ε}(n − 1)f(n − 1) + b_{ε}(n)f(n)

and is unitarily equivalent to H.

Proof. Straightforward. □

The next transformation is equally useful and will be referred to as reflection at n_{0}. It shows how information obtained near one endpoint, say +∞, can be transformed into information near the other, −∞.

Lemma 1.7. Fix n_{0} ∈ ℤ and consider the unitary involution

(1.80)  (U_{R}f)(n) = (U_{R}^{−1}f)(n) = f(2n_{0} − n)

or equivalently (U_{R}f)(n_{0} + k) = f(n_{0} − k). Then the operator

(1.81)  H_{R} = U_{R}HU_{R}^{−1},

is associated with the sequences

(1.82)  a_{R}(n_{0} − k − 1) = a(n_{0} + k), b_{R}(n_{0} − k) = b(n_{0} + k), k ∈ ℤ,

or equivalently a_{R}(n) = a(2n_{0} − n − 1), b_{R}(n) = b(2n_{0} − n).

Proof. Again straightforward. □

Associated with U_{R} are the two orthogonal projections

(1.83)  P_{R}^{±} = \frac{1}{2}(1l ± U_{R}),  P_{R}^{−} + P_{R}^{+} = 1l, P_{R}^{−}P_{R}^{+} = P_{R}^{+}P_{R}^{−} = 0

and a corresponding splitting of H into two parts H = H_{R}^{+} ⊕ H_{R}^{−}, where

(1.84)  H_{R}^{±} = P_{R}^{+}HP_{R}^{±} + P_{R}^{−}HP_{R}^{∓} = \frac{1}{2}(H ± H_{R}).

The symmetric part H_{R}^{+} (resp. antisymmetric part H_{R}^{−}) commutes (resp. anticommutes) with U_{R}, that is, [U_{R}, H_{R}^{+}] = U_{R}H_{R}^{+} − H_{R}^{+}U_{R} = 0 (resp. {U_{R}, H_{R}^{−}} = U_{R}H_{R}^{−} + H_{R}^{−}U_{R} = 0). If H = H_{R}^{−} we enter the realm of supersymmetric quantum mechanics (cf., e.g., [225] and Section 14.3).

After these two transformations we will say a little more about the spectrum σ(H) of H. More precisely, we will estimate the location of σ(H).

Lemma 1.8. Let

(1.85)  c_{±}(n) = b(n) ± (|a(n)| + |a(n − 1)|).

Then we have

(1.86)  σ(H) ⊆ [ inf_{n∈ℤ} c_{−}(n), sup_{n∈ℤ} c_{+}(n)].

Proof. We will first show that H is semi-bounded from above by sup c_{+}. From (1.16) we infer

⟨f, Hf⟩ = ∑_{n∈ℤ} ( − a(n)|f(n + 1) − f(n)|^{2}

+ (a(n − 1) + a(n) + b(n))|f(n)|^{2}). (1.87)

By Lemma 1.6 we can first choose a(n) > 0 to obtain

(1.88)  ⟨f, Hf⟩ ≤ sup_{n∈ℤ} c_{+}(n) ∥f∥^{2}.

Similarly, choosing a(n) < 0 we see that H is semibounded from below by inf c_{−},

(1.89)  ⟨f, Hf⟩ ≥ inf_{n∈ℤ} c_{−}(n) ∥f∥^{2},

completing the proof. □

We remark that these bounds are optimal in the sense that equality is attained for (e.g.) a(n) = 1/2, b(n) = 0 (cf. Section 1.3).

We will not only consider H but also restrictions of H; partly because they are of interest on their own, partly because their investigation gives information about H.

To begin with, we define the following restrictions H_{±,n_{0}} of H to the subspaces ℓ^{2}(n_{0}, ±∞),

H_{+,n_{0}}f(n) = \{ \begin{matrix}a(n_{0} + 1)f(n_{0} + 2) + b(n_{0} + 1)f(n_{0} + 1), n = n_{0} + 1 \\ (τf)(n), n > n_{0} + 1 \end{matrix} ,
(1.90) H_{−,n_{0}}f(n) = \{ \begin{matrix}a(n_{0} − 2)f(n_{0} − 2) + b(n_{0} − 1)f(n_{0} − 1), n = n_{0} − 1 \\ (τf)(n), n < n_{0} − 1 \end{matrix} .

In addition, we also define for β ∈ ℝ ∪ {∞}

H_{+,n_{0}}^{0} = H_{+,n_{0}+1}, H_{+,n_{0}}^{β} = H_{+,n_{0}} − a(n_{0})β^{−1}⟨δ_{n_{0}+1}, .⟩δ_{n_{0}+1}, β ≠ 0,

H_{−,n_{0}}^{∞} = H_{−,n_{0}}, H_{−,n_{0}}^{β} = H_{−,n_{0}+1} − a(n_{0})β⟨δ_{n_{0}}, .⟩δ_{n_{0}}, β ≠ ∞. (1.91)

All operators H_{±,n_{0}}^{β} are bounded and self-adjoint.

Last, we define the following finite restriction H_{n_{1},n_{2}} to the subspaces ℓ^{2}(n_{1}, n_{2})

(1.92) H_{n_{1},n_{2}}f(n) = \{ \begin{matrix}a(n_{1} + 1)f(n_{1} + 2) + b(n_{1} + 1)f(n_{1} + 1), n = n_{1} + 1 \\ (τf)(n), n_{1} + 1 < n < n_{2} − 1 \\ a(n_{2} − 2)f(n_{2} − 2) + b(n_{2} − 1)f(n_{2} − 1), n = n_{2} − 1 \end{matrix} .

The operator H_{n_{1},n_{2}} is clearly associated with the Jacobi matrix J_{n_{1},n_{2}} (cf. (1.64)). Moreover, we set H_{n_{1},n_{2}}^{∞,∞} = H_{n_{1},n_{2}}, H_{n_{1},n_{2}}^{0,β_{2}} = H_{n_{1}+1,n_{2}}^{∞,β_{2}}, and

H_{n_{1},n_{2}}^{β_{1},β_{2}} = H_{n_{1},n_{2}}^{∞,β_{2}} − a(n_{1})β_{1}^{−1}⟨δ_{n_{1}+1}, .⟩δ_{n_{1}+1}, β_{1} ≠ 0,
(1.93)  H_{n_{1},n_{2}}^{β_{1},β_{2}} = H_{n_{1},n_{2}+1}^{β_{1},∞} − a(n_{2})β_{2}⟨δ_{n_{2}}, .⟩δ_{n_{2}}, β_{2} ≠ ∞.

Remark 1.9. H_{+,n_{0}}^{β} can be associated with the following domain

(1.94)  𝔇(H_{+,n_{0}}^{β}) = {f ∈ ℓ^{2}(n_{0}, ∞)| cos(α)f(n_{0}) + sin(α)f(n_{0} + 1) = 0},

β = cot(α) ≠ 0, if one agrees that only points with n > n_{0} are of significance and that the last point is only added as a dummy variable so that one does not have to specify an extra expression for (τf)(n_{0} + 1). In particular, the case β = ∞ (i.e., corresponding to the boundary condition f(n_{0}) = 0) will be referred to as Dirichlet boundary condition at n_{0}. Analogously for H_{−,n_{0}}^{β}, H_{n_{1},n_{2}}^{β_{1},β_{2}}.

One of the most important objects in spectral theory is the resolvent (H−z)^{−1}, z ∈ ρ(H), of H. Here ρ(H) = ℂ\σ(H) denotes the resolvent set of H. The matrix elements of (H − z)^{−1} are called Green function

(1.95)  G(z, n, m) = ⟨δ_{n}, (H − z)^{−1}δ_{m}⟩,  z ∈ ρ(H).

Clearly,

(1.96)  (τ − z)G(z, ., m) = δ_{m}(.),  G(z, m, n) = G(z, n, m)

and

(1.97)  ((H − z)^{−1}f)(n) = ∑_{m∈ℤ} G(z, n, m)f(m), f ∈ ℓ^{2}(ℤ), z ∈ ρ(H).

We will derive an explicit formula for G(z, n, m) in a moment. Before that we need to construct solutions u_{±}(z) of (1.19) being square summable near ±∞.

Set

(1.98)  u(z, .) = (H − z)^{−1}δ_{0}(.) = G(z, ., 0),  z ∈ ρ(H).

By construction u fulfills (1.19) only for n > 0 and n < 0. But if we take u(z, −2), u(z, −1) as initial condition we can obtain a solution u_{−}(z, n) of τu = zu on the whole of ℓ(ℤ) which coincides with u(z, n) for n < 0. Hence u_{−}(z) satisfies u_{−}(z) ∈ ℓ_{−}^{2}(ℤ) as desired. A solution u_{+}(z) ∈ ℓ_{+}^{2}(ℤ) is constructed similarly.

As anticipated, these solutions allow us to write down the Green function in a somewhat more explicit way

(1.99)  G(z, n, m) = \frac{1}{W (u_{−}(z), u_{+}(z))} \{ \begin{matrix}u_{+}(z, n)u_{−}(z, m) for & m ≤ n \\ u_{+}(z, m)u_{−}(z, n) for & n ≤ m \end{matrix} ,

z ∈ ρ(H). Indeed, since the right hand side of (1.99) satisfies (1.96) and is square summable with respect to n, it must be the Green function of H.

For later use we also introduce the convenient abbreviations

g(z, n) = G(z, n, n) = \frac{u_{+}(z, n)u_{−}(z, n)}{W (u_{−}(z), u_{+}(z))},
h(z, n) = 2a(n)G(z, n, n + 1) − 1
(1.100)  = \frac{a(n)(u_{+}(z, n)u_{−}(z, n + 1) + u_{+}(z, n)u_{−}(z, n + 1))}{W (u_{−}(z), u_{+}(z))}.

Note that for n ≤ m we have

G(z, n, m) = g(z, n_{0})c(z, n, n_{0})c(z, m, n_{0})

+ g(z, n_{0} + 1)s(z, n, n_{0})s(z, m, n_{0})

+ h(z, n_{0})\frac{c(z, n, n_{0})s(z, m, n_{0}) + c(z, m, n_{0})s(z, n, n_{0})}{2a(n_{0})}

− \frac{c(z, n, n_{0})s(z, m, n_{0}) − c(z, m, n_{0})s(z, n, n_{0})}{2a(n_{0})}. (1.101)

Similar results hold for the restrictions: Let

(1.102)  s_{β}(z, n, n_{0}) = sin(α)c(z, n, n_{0}) − cos(α)s(z, n, n_{0})

with β = cot(α) (i.e., the sequence s_{β}(z, n, n_{0}) fulfills the boundary condition cos(α)s_{β}(z, n_{0}, n_{0}) + sin(α)s_{β}(z, n_{0} + 1, n_{0}) = 0). Then we obtain for the resolvent of H_{±,n_{0}}^{β}

(1.103)  ((H_{±,n_{0}}^{β} − z)^{−1}u)(n) = ∑_{m {> \atop <} n_{0}} G_{±,n_{0}}^{β}(z, m, n)u(m), z ∈ ρ(H_{±,n_{0}}^{β}),

where

(1.104) G_{±,n_{0}}^{β}(z, m, n) = \frac{±1}{W (s_{β}(z), u_{±}(z))} \{ \begin{matrix}s_{β}(z, n, n_{0})u_{±}(z, m) for & m{≥ \atop ≤}n \\ s_{β}(z, m, n_{0})u_{±}(z, n) for & n{≥ \atop ≤}m \end{matrix}

(use (H_{±,n_{0}}^{β} − z)^{−1} to show the existence of u_{±}(z, .) for z ∈ ρ(H_{±,n_{0}}^{β})).

Remark 1.10. The solutions being square summable near ±∞ (resp. satisfying the boundary condition cos(α)f(n_{0}) + sin(α)f(n_{0} + 1) = 0) are unique up to constant multiples since the Wronskian of two such solutions vanishes (evaluate it at ±∞ (resp. n_{0})). This implies that the point spectrum of H, H_{±,n_{0}}^{β} is always simple.

In addition to H_{±,n_{0}}^{β} we will be interested in the following direct sums of these operators

(1.105)  H_{n_{0}}^{β} = H_{−,n_{0}}^{β} ⊕ H_{+,n_{0}}^{β},

in the Hilbert space {f ∈ ℓ^{2}(ℤ)| cos(α)f(n_{0}) + sin(α)f(n_{0} + 1) = 0}. The reason why H_{n_{0}}^{β} is of interest to us follows from the close spectral relation to H as can be seen from their resolvents (resp. Green functions)

G_{n_{0}}^{∞}(z, n, m) = G(z, n, m) − \frac{G(z, n, n_{0})G(z, n_{0}, m)}{G(z, n_{0}, n_{0})},
G_{n_{0}}^{β}(z, n, m) = G(z, n, m) − γ^{β}(z, n_{0})^{−1}(G(z, n, n_{0} + 1) + βG(z, n, n_{0}))

×(G(z, n_{0} + 1, m) + βG(z, n_{0}, m)), β ∈ ℝ, (1.106)

where

γ^{β}(z, n) = \frac{(u_{+}(z, n + 1) + βu_{+}(z, n))(u_{−}(z, n + 1) + βu_{−}(z, n))}{W (u_{−}(z), u_{+}(z))}
(1.107)  = g(z, n + 1) + \frac{β}{a(n)}h(z, n) + β^{2}g(z, n).

Remark 1.11. The operator H_{n_{0}}^{β} is equivalently given by

(1.108)  H_{n_{0}}^{β} = (1l − P_{n_{0}}^{β})H(1l − P_{n_{0}}^{β})

in the Hilbert space (1l − P_{n_{0}}^{β})ℓ^{2}(ℤ) = {f ∈ ℓ^{2}(ℤ)|⟨δ_{n_{0}}^{β}, f⟩ = 0}, where P_{n_{0}}^{β} denotes the orthogonal projection onto the one-dimensional subspace spanned by δ_{n_{0}}^{β} = cos(α)δ_{n_{0}} + sin(α)δ_{n_{0}+1}, β = cot(α), α ∈ [0, π) in ℓ^{2}(ℤ).

Finally, we derive some interesting difference equations for g(z, n) to be used in Section 6.1.

Lemma 1.12. Let u, v be two solutions of (1.19). Then g(n) = u(n)v(n) satisfies

(1.109)  \frac{(a^{+})^{2}g^{++} − a^{2}g}{z − b^{+}} + \frac{a^{2}g^{+} − (a^{−})^{2}g^{−}}{z − b} = (z − b^{+})g^{+} − (z − b)g,

and

(1.110)  (a^{2}g^{+} − (a^{−})^{2}g^{−} + (z − b)^{2}g)^{2} = (z − b)^{2}(W (u, v)^{2} + 4a^{2}gg^{+}).

Proof. First we calculate (using (1.19))

(1.111)  a^{2}g^{+} − (a^{−})^{2}g^{−} = −(z − b)^{2}g + a(z − b)(uv^{+} + u^{+}v).

Adding (z − b)^{2}g and taking squares yields the second equation. Dividing both sides by z − b and adding the equations corresponding to n and n + 1 yields the first. □

Remark 1.13. There exists a similar equation for γ^{β}(z, n). Since it is quite complicated, it seems less useful. Set γ^{β}(n) = (u(n + 1) + βu(n))(v(n + 1) + βv(n)), then we have

((a^{+}A^{−})^{2}(γ^{β})^{+} − (a^{−}A)^{2}(γ^{β})^{−} + B^{2}γ^{β})^{2}
(1.112)  = (A^{−}B)^{2}((\frac{A}{a}W (u, v))^{2} + 4(a^{+})^{2}γ^{β}(γ^{β})^{+}),

with

A = a + β(z − b^{+}) + β^{2}a^{+},
B = a^{−}(z − b^{+}) + β((z − b^{+})(z − b) + a^{+}a^{−} − a^{2})

+ β^{2}a^{+}(z − b). (1.113)

It can be verified by a long and tedious (but straightforward) calculation.

1.3. A simple example

We have been talking about Jacobi operators for quite some time now, but we have not seen a single example yet. Well, here is one, the free Jacobi operator H_{0} associated with constant sequences a(n) = a, b(n) = b. The transformation z → 2az + b reduces this problem to the one with a_{0}(n) = 1/2, b_{0}(n) = 0. Thus we will consider the equation

(1.114)  \frac{1}{2}(u(n + 1) + u(n − 1)) = zu(n).

Without restriction we choose n_{0} = 0 throughout this section (note that we have s(z, n, n_{0}) = s(z, n − n_{0}), etc.) and omit n_{0} in all formulas. By inspection (try the ansatz u(n) = k^{n}) u_{±}(z, .) are given by

(1.115)  u_{±}(z, n) = (z ± R_{2}^{1/2}(z))^{n},

where R_{2}^{1/2}(z) = −\sqrt{z − 1}\sqrt{z + 1}. Here and in the sequel \sqrt{.} always denotes the standard branch of the square root, that is,

(1.116)  \sqrt{z} = |\sqrt{z}| exp(i arg(z)/2), arg(z) ∈ (−π, π], z ∈ ℂ.

Since W (u_{−}, u_{+}) = R_{2}^{1/2}(z) we need a second solution for z^{2} = 1, which is given by s(±1, n) = (±1)^{n+1}n. For the fundamental solutions we obtain

s(z, n) = \frac{(z + R_{2}^{1/2}(z))^{n} − (z − R_{2}^{1/2}(z))^{n}}{2R_{2}^{1/2}(z)} ,

(1.117)  c(z, n) = \frac{s(z, n − 1)}{s(z, −1)} = −s(z, n − 1).

Notice that s(−z, n) = (−1)^{n+1}s(z, n). For n > 0 we have the following expansion

s(z, n) = ∑_{j=0}^{[[n/2]]} \binom{n}{2j + 1}(z^{2} − 1)^{j}z^{n−2j−1}

(1.118)  = ∑_{k=0}^{[[n/2]]} ((−1)^{k} ∑_{j=k}^{[[n/2]]} \binom{n}{2j + 1}\binom{j}{k}) z^{n−2k−1},

where [[x]] = sup{n ∈ ℤ | n < x}. It is easily seen that we have ∥H_{0}∥ = 1 and further that

(1.119)  σ(H_{0}) = [−1, 1].

For example, use unitarity of the Fourier transform

(1.120)  \begin{aligned}U : & ℓ^{2}(ℤ) & → & L^{2}(−π, π) \\ & u(n) & ↦ & ∑_{n∈ℤ} u(n)e^{inx} \end{aligned} .

which maps H_{0} to the multiplication operator by cos(x).

The Green function of H_{0} explicitly reads (z ∈ ℂ\[−1, 1])

(1.121)  G_{0}(z, m, n) = \frac{(z + R_{2}^{1/2}(z))^{|m−n|}}{R_{2}^{1/2}(z)} .

In particular, we have

g_{0}(z, n) = \frac{1}{R_{2}^{1/2}(z)} = −2 ∑_{j=0}^{∞} \binom{2j}{j}\frac{1}{(2z)^{2j+1}}

(1.122)  h_{0}(z, n) = \frac{z}{R_{2}^{1/2}(z)} = − ∑_{j=0}^{∞} \binom{2j}{j}\frac{1}{(2z)^{2j}} .

Note that it is sometimes convenient to set k = z + R_{2}^{1/2}(z) (and conversely z = \frac{1}{2}(k + k^{−1})), or k = exp(iκ) (and conversely z = cos(κ)) implying

(1.123)  u_{±}(z, n) = k^{±n} = e^{±iκn}.

The map z ↦ k = z + R_{2}^{1/2}(z) is a holomorphic mapping from the set Π_{+} ≃ (ℂ ∪ {∞})\[−1, 1] to the unit disk {z ∈ ℂ| |z| < 1}. In addition, viewed as a map on the Riemann surface of R_{2}^{1/2}(z), it provides an explicit isomorphism between the Riemann surface of R_{2}^{1/2}(z) and the Riemann sphere ℂ ∪ {∞}.

1.4. General second order difference expressions

We consider the difference expression

(1.124)  τ̂f(n) = \frac{1}{w(n)}(f(n + 1) + f(n − 1) + d(n)f(n)),

where w(n) > 0, d(n) ∈ ℝ, and (w(n)w(n + 1))^{−1}, w(n)^{−1}d(n) are bounded sequences. It gives rise to an operator Ĥ, called Helmholtz operator, in the weighted Hilbert space ℓ^{2}(ℤ; w) with scalar product

(1.125)  ⟨f, g⟩ = ∑_{n∈ℤ} w(n)\overline{f(n)}g(n),  f, g ∈ ℓ^{2}(ℤ; w).

Green’s formula (1.20) holds with little modifications and Ĥ is easily seen to be bounded and self-adjoint. There is an interesting connection between Jacobi and Helmholtz operators stated in the next theorem.

Theorem 1.14. Let H be the Jacobi operator associated with the sequences a(n) > 0, b(n) and let Ĥ be the Helmholtz operator associated with the sequences w(n) > 0, d(n). If we relate these sequences by

(1.126)  \begin{matrix}w(2m) = w(0) \begin{matrix}m−1 \\ ∏ & ∗ \\ j=0 \end{matrix} \frac{a(2j)^{2}}{a(2j + 1)^{2}} \\ w(2m + 1) = \frac{1}{a(2m)^{2}w(2m)} \end{matrix} ,  d(n) = w(n)b(n),

respectively

(1.127)  a(n) = \frac{1}{\sqrt{w(n)w(n + 1)}},  b(n) = \frac{d(n)}{w(n)},

then the operators H and Ĥ are unitarily equivalent, that is, H = UĤU^{−1}, where U is the unitary transformation

(1.128)  \begin{aligned}U : & ℓ^{2}(ℤ; w) & → & ℓ^{2}(ℤ) \\ & u(n) & ↦ & \sqrt{w(n)}u(n) \end{aligned} .

Proof. Straightforward. □

Remark 1.15. (i). The most general three-term recurrence relation

(1.129)  τ̃f(n) = ã(n)f(n + 1) + b̃(n)f(n) + c̃(n)f(n − 1),

with ã(n)c̃(n+1) > 0, can be transformed to a Jacobi recurrence relation as follows. First we render τ̃ symmetric,

(1.130)  τ̃f(n) = \frac{1}{w(n)}(c(n)f(n + 1) + c(n − 1)f(n − 1) + d(n)f(n)),

where

w(n) = \begin{matrix}n−1 \\ ∏ & ∗ \\ j=n_{0} \end{matrix} \frac{ã(j)}{c̃(j + 1)},

(1.131)  c(n) = w(n)ã(n) = w(n + 1)c̃(n + 1), d(n) = w(n)b̃(n).

Let H̃ be the self-adjoint operator associated with τ̃ in ℓ^{2}(ℤ; w). Then the unitary operator

(1.132)  \begin{aligned}U : & ℓ^{2}(ℤ; w) & → & ℓ^{2}(ℤ) \\ & u(n) & ↦ & \sqrt{w(n)}u(n) \end{aligned}

transforms H̃ into a Jacobi operator H = UH̃U^{−1} in ℓ^{2}(ℤ) associated with the sequences

a(n) = \frac{c(n)}{\sqrt{w(n)w(n + 1)}} = sgn(ã(n))\sqrt{ã(n)c̃(n + 1)},
(1.133)  b(n) = \frac{d(n)}{w(n)} = b̃(n).

In addition, the Wronskians are related by

c(n)(f(n)g(n + 1) − f(n + 1)g(n)) =

a(n)((Uf)(n)(Ug)(n + 1) − (Uf)(n + 1)(Ug)(n)). (1.134)

(ii). Let c(n) > 0 be given. Defining

(1.135)  \begin{matrix}w(2m) = \begin{matrix}m−1 \\ ∏ & ∗ \\ j=0 \end{matrix} (\frac{a(2j)c(2j + 1)}{c(2j)a(2j + 1)})^{2} \\ w(2m + 1) = \frac{c(2m)^{2}}{a(2m)^{2}w(2m)} \end{matrix} ,  d(n) = w(n)b(n),

the transformation U maps H to an operator H̃ = UHU^{−1} associated with the difference expression (1.130).

1.5. The infinite harmonic crystal in one dimension

Finally, I want to say something about how Jacobi operators arise in applications. Despite the variety of possible applications of difference equations we will focus on only one model from solid state physics: the infinite harmonic crystal in one dimension. Hence we consider a linear chain of particles with harmonic nearest neighbor interaction. If x(n, t) denotes the deviation of the n-th particle from its equilibrium position, the equations of motion read

m(n)\frac{d^{2}}{dt^{2}} x(n, t) = k(n)(x(n + 1, t) − x(n, t)) + k(n − 1)(x(n − 1, t) − x(n, t))
(1.136)  = −(∂^{∗}k∂x)(n, t),

where m(n) > 0 is the mass of the n-th particle and k(n) is the force constant between the n-th and (n + 1)-th particle.

[figure: m(n−1); k(n−1); m(n); k(n); m(n+1); x(n,t)]

This model is only valid as long as the relative displacement is not too large (i.e., at least smaller than the distance of the particles in the equilibrium position). Moreover, from a physical viewpoint it is natural to assume k, m, m^{−1} ∈ ℓ^{∞}(ℤ, ℝ) and k(n) ≠ 0. Introducing conjugate coordinates

(1.137)  p(n, t) = m(n)\frac{d}{dt}x(n, t), q(n, t) = x(n, t),

the system (1.136) can be written as Hamiltonian system with Hamiltonian given by

(1.138)  H(p, q) = ∑_{n∈ℤ} (\frac{p(n)^{2}}{2m(n)} + \frac{k(n)}{2}(q(n + 1) − q(n))^{2}).

Since the total energy of the system is supposed to be finite, a natural phase space for this system is (p, q) ∈ ℓ^{2}(ℤ, ℝ^{2}) with symplectic form

(1.139)  ω((p_{1}, q_{1}), (p_{2}, q_{2})) = ∑_{n∈ℤ} (p_{1}(n)q_{2}(n) − p_{2}(n)q_{1}(n)).

Using the symplectic transform

(1.140)  (p, q) → (v, u) = (\frac{p}{\sqrt{m}}, \sqrt{m} q)

we get a new Hamiltonian

(1.141)  H̃(v, u) = \frac{1}{2} ∑_{n∈ℤ} (v(n)^{2} + 2a(n)u(n)u(n + 1) + b(n)u(n)^{2}),

where

(1.142)  a(n) = \frac{−k(n)}{\sqrt{m(n)m(n + 1)}}, b(n) = \frac{k(n) + k(n − 1)}{m(n)}.

The corresponding equations of evolution read

\frac{d}{dt}u(n, t) = \frac{∂H̃(v, u)}{∂v(n, t)} = v(n, t),
(1.143)  \frac{d}{dt}v(n, t) = −\frac{∂H̃(v, u)}{∂u(n, t)} = −Hu(n, t),

where H is our usual Jacobi operator associated with the sequences (1.142). Equivalently we have

(1.144)  \frac{d^{2}}{dt^{2}} u(n, t) = −Hu(n, t).

Since this system is linear, standard theory implies

u(n, t) = cos(t\sqrt{H})u(n, 0) + \frac{sin(t\sqrt{H})}{\sqrt{H}}v(n, 0),
(1.145)  v(n, t) = cos(t\sqrt{H})v(n, 0) − \frac{sin(t\sqrt{H})}{\sqrt{H}}Hu(n, 0),

where

(1.146)  cos(t\sqrt{H}) = ∑_{ℓ=0}^{∞} \frac{(−1)^{ℓ}t^{2ℓ}}{(2ℓ)!} H^{ℓ}, \frac{sin(t\sqrt{H})}{\sqrt{H}} = ∑_{ℓ=0}^{∞} \frac{(−1)^{ℓ}t^{2ℓ+1}}{(2ℓ + 1)!} H^{ℓ}.

In particular, introducing

(1.147)  \underline{S}(λ, n) = (c(λ, n), s(λ, n))

and expanding the initial conditions in terms of eigenfunctions (cf. Section 2.5, equation (2.133))

(1.148)  \underline{u}(λ) = ∑_{n∈ℤ} u(n, 0)\underline{S}(λ, n), \underline{v}(λ) = ∑_{n∈ℤ} v(n, 0)\underline{S}(λ, n)

we infer

u(n, t) = ∫_{σ(H)} (\underline{u}(λ) cos(t\sqrt{λ}) + \underline{v}(λ)\frac{sin(t\sqrt{λ})}{\sqrt{λ}})\underline{S}(λ, n)dρ(λ),

(1.149) v(n, t) = ∫_{σ(H)} (\underline{v}(λ) cos(t\sqrt{λ}) − \underline{u}(λ)\frac{sin(t\sqrt{λ})}{\sqrt{λ}}λ)\underline{S}(λ, n)dρ(λ),

where ρ(λ) is the spectral matrix of H.

This shows that in order to understand the dynamics of (1.144) one needs to understand the spectrum of H.

For example, in the case where H ≥ 0 (or if \underline{u}(λ), \underline{u}(λ) = 0 for λ ≤ 0) we clearly infer

∥u(t)∥ ≤ ∥u(0)∥ + t∥v(0)∥,
(1.150)  ∥v(t)∥ ≤ ∥v(0)∥ + ∥\sqrt{H}v(0)∥,

since | cos(t\sqrt{λ})| ≤ 1, |\frac{sin(t\sqrt{λ})}{t\sqrt{λ}}| ≤ 1 for t ∈ ℝ, λ ≥ 0.

Given this setting two questions naturally arise. Firstly, given m(n), k(n) what can be said about the characteristic frequencies of the crystal? Spectral theory for H deals with this problem. Secondly, given the set of characteristic frequencies (i.e., the spectrum of H), is it possible to reconstruct m(n), k(n)? This question is equivalent to inverse spectral theory for H once we establish how to reconstruct k, m from a, b. This will be done next.

Note that u(n) = \sqrt{m(n)} > 0 solves τu = 0. Hence, if we assume k > 0, then H ≥ 0 by Corollary 11.2. In particular,

(1.151)  σ(H) ⊆ [0, 2(∥k∥_{∞}∥m^{−1}∥_{∞})]

and 0 is in the essential spectrum of H by Lemma 3.8. Moreover, since

(1.152)  ∑_{n∈±ℕ} \frac{1}{−a(n)u(n)u(n + 1)} = ∑_{n∈±ℕ} \frac{1}{k(n)} = ∞

(recall k ∈ ℓ^{∞}(ℤ)), H is critical (cf. Section 2.3). Thus we can recover k, m via

(1.153)  k(n) = −m(0)a(n)u(n)u(n + 1), m(n) = m(0)u(n)^{2},

where u(n) is the unique positive solution of τu = 0 satisfying u(0) = \sqrt{m(0)}. That m, k can only be recovered up to a constant multiple (i.e., m(0)) is not surprising due to the corresponding scaling invariance of (1.136). If the positive solution of τu = 0 is not unique, we get a second free parameter. However, note that for each choice the corresponding Helmholtz operator Ĥ = m^{−1}∂k∂^{∗} is unitary equivalent to H (and so are Ĥ_{±} and H_{±}).

From a physical viewpoint the case of a crystal with N atoms in the base cell is of particular interest, that is, k and m are assumed to be periodic with period N. In this case, the same is true for a, b and we are led to the study of periodic Jacobi operators. The next step is to consider impurities in this crystal. If such impurities appear only local, we are led to scattering theory with periodic background (to be dealt with in Chapter 7). If they are randomly distributed over the crystal we are led to random operators (to be dealt with in Chapter 5).

But for the moment let us say a little more about the simplest case where m and k are both constants (i.e., N = 1). After the transformation t ↦ \sqrt{\frac{m}{2k}} t we can even assume m = 2k = 1, that is,

(1.154)  \frac{d^{2}}{dt^{2}} x(n, t) = \frac{1}{2}(x(n + 1, t) + x(n − 1, t)) − x(n, t).

The so-called plane wave solutions are given by

(1.155)  u^{±}(n, t) = ei(κn ± ν(κ)t),

where the wavelength κ^{−1} and the frequency ν are connected by the dispersion relation

(1.156)  ν(κ) = \sqrt{1 − cos(κ)} = \sqrt{2}| sin(\frac{κ}{2})|.

Since u^{±}(n, t) is only meaningful for n ∈ ℤ, we can restrict κ to values in the first Brillouin zone, that is, κ ∈ (−π, π].

These solutions correspond to infinite total energy of the crystal. However, one can remedy this problem by taking (continuous) superpositions of these plane waves. Introducing

(1.157)  u(κ) = ∑_{n∈ℤ} u(n, 0)eiκn, v(κ) = ∑_{n∈ℤ} v(n, 0)eiκn

we obtain

(1.158)  u(n, t) = \frac{1}{2π} ∫_{−π}^{π} (u(κ) cos(ν(κ)t) + v(κ)\frac{sin(ν(κ)t)}{ν(κ)})e−iκndκ.

Or, equivalently,

(1.159)  u(n, t) = ∑_{m∈ℤ} c_{n−m}(t)u(m, 0) + s_{n−m}(t)v(m, 0),

where

c_{n}(t) = \frac{1}{2π} ∫_{−π}^{π} cos(ν(κ)t)eiκndκ = J_{2|n|}(\sqrt{2}t),
s_{n}(t) = \frac{1}{2π} ∫_{−π}^{π} \frac{sin(ν(κ)t)}{ν(κ)}eiκndκ = ∫_{0}^{t} c_{n}(s)ds
(1.160)  = \frac{t^{2|n|+1}}{2^{|n|}(|n| + 1)!} _{1}F_{2}(\frac{2|n| + 1}{2}; (\frac{2|n| + 3}{2}, 2|n| + 1); −\frac{t^{2}}{2}).

Here J_{n}(x), _{p}F_{q}(\underline{u}; \underline{v}; x) denote the Bessel and generalized hypergeometric functions, respectively. From this form one can deduce that a localized wave (say compactly supported at t = 0) will spread as t increases (cf. [194], Corollary to Theorem XI.14). This phenomenon is due to the fact that different plane waves travel with different speed and is called dispersion.

You might want to observe the following fact: if the coupling constant k(n_{0}) between the n_{0}-th and (n_{0} + 1)-th particle is zero (i.e., no interaction between these particles), then the chain separates into two parts (cf. the discussion after Theorem 1.5 and note that k(n_{0}) = 0 implies a(n_{0}) = 0 for the corresponding Jacobi operator).

We will encounter chains of particles again in Section 12.1. However, there we will consider a certain nonlinear interaction.

Foundations of spectral theory for Jacobi operators

The theory presented in this chapter is the discrete analog of what is known as Weyl-Titchmarsh-Kodaira theory for Sturm-Liouville operators. The discrete version has the advantage of being less technical and more transparent.

Again, the present chapter is of fundamental importance and the tools developed here are the pillars of spectral theory for Jacobi operators.

2.1. Weyl m-functions

In this section we will introduce and investigate Weyl m-functions. Rather than the classical approach of Weyl (cf. Section 2.4) we advocate a different one which is more natural in the discrete case.

As in the previous chapter, u_{±}(z, .) denote the solutions of (1.19) in ℓ(ℤ) which are square summable near ±∞, respectively.

We start by defining the Weyl m-functions

(2.1)  m_{±}(z, n_{0}) = ⟨δ_{n_{0}±1}, (H_{±,n_{0}} − z)^{−1}δ_{n_{0}±1}⟩ = G_{±,n_{0}}(z, n_{0} ± 1, n_{0} ± 1).

By virtue of (1.104) we also have the more explicit form

(2.2) m_{+}(z, n_{0}) = −\frac{u_{+}(z, n_{0} + 1)}{a(n_{0})u_{+}(z, n_{0})},  m_{−}(z, n_{0}) = −\frac{u_{−}(z, n_{0} − 1)}{a(n_{0} − 1)u_{−}(z, n_{0})}.

The base point n_{0} is of no importance in what follows and we will only consider m_{±}(z) = m_{±}(z, 0) most of the time. Moreover, all results for m_{−}(z) can be obtained from the corresponding results for m_{+}(z) using reflection (cf. Lemma 1.7).

The definition (2.1) implies that the function m_{±}(z) is holomorphic in ℂ\σ(H_{±}) and that it satisfies

(2.3)  m_{±}(\overline{z}) = \overline{m_{±}(z)},  |m_{±}(z)| ≤ ∥(H_{±} − z)^{−1}∥ ≤ \frac{1}{|Im(z)|}.

Moreover, m_{±}(z) is a Herglotz function (i.e., it maps the upper half plane into itself, cf. Appendix B). In fact, this is a simple consequence of the first resolvent identity

Im(m_{±}(z)) = Im(z)⟨δ_{±1}, (H_{±} − \overline{z})^{−1}(H_{±} − z)^{−1}δ_{±1}⟩
(2.4)  = Im(z)∥(H_{±} − z)^{−1}δ_{±1}∥^{2}.

Hence by Theorem B.2, m_{±}(z) has the following representation

(2.5)  m_{±}(z) = ∫_{ℝ} \frac{dρ_{±}(λ)}{λ − z},  z ∈ ℂ\ℝ,

where ρ_{±} = ∫_{(−∞,λ]} dρ_{±} is a nondecreasing bounded function which is given by Stieltjes inversion formula (cf. Theorem B.2)

(2.6)  ρ_{±}(λ) = \frac{1}{π} lim_{δ↓0} lim_{ε↓0} ∫_{−∞}^{λ+δ}Im(m_{±}(x + iε))dx.

Here we have normalized ρ_{±} such that it is right continuous and obeys ρ_{±}(λ) = 0, λ < σ(H_{±}).

Let P_{Λ}(H_{±}), Λ ⊆ ℝ, denote the family of spectral projections corresponding to H_{±} (spectral resolution of the identity). Then dρ_{±} can be identified using the spectral theorem,

(2.7)  m_{±}(z) = ⟨δ_{±1}, (H_{±} − z)^{−1}δ_{±1}⟩ = ∫_{ℝ} \frac{d⟨δ_{±1}, P_{(−∞,λ]}(H_{±})δ_{±1}⟩}{λ − z}.

Thus we see that dρ_{±} = d⟨δ_{±1}, P_{(−∞,λ]}(H_{±})δ_{±1}⟩ is the spectral measure of H_{±} associated to the sequence δ_{±1}.

Remark 2.1. (i). Clearly, similar considerations hold for arbitrary expectations of resolvents of self-adjoint operators (cf. Lemma 6.1). (ii). Let me note at this point that the fact which makes discrete life so much easier than continuous life is, that δ_{1} is an element of our Hilbert space. In contradistinction, the continuous analog of δ_{n}, the delta distribution δ_{x}, is not a square integrable function. (However, if one considers non-negative Sturm-Liouville operators H, then δ_{x} lies in the scale of spaces associated to H. See [206] for further details.)

It follows in addition that all moments m_{±,ℓ} of dρ_{±} are finite and given by

(2.8)  m_{±,ℓ} = ∫_{ℝ} λ^{ℓ}dρ_{±}(λ) = ⟨δ_{±1}, (H_{+})^{ℓ}δ_{±1}⟩.

Moreover, there is a close connection between the so-called moment problem (i.e., determining dρ_{±} from all its moments m_{±,ℓ}) and the reconstruction of H_{±} from dρ_{±}. Indeed, since m_{±,0} = 1, m_{±,1} = b(±1), m_{±,2} = a(±1 − {0 \atop 1})^{2} + b(±1)^{2}, etc., we infer

(2.9)  b(±1) = m_{±,1}, a(±1 − {0 \atop 1})^{2} = m_{±,2} − (m_{±,1})^{2}, etc. .

We will consider this topic in Section 2.5.

You might have noticed, that m_{±}(z, n) has (up to the factor −a(n − {0 \atop 1})) the same structure as the function ϕ(n) used when deriving the Riccati equation (1.52).

Comparison with the formulas for ϕ(n) shows that

(2.10)  u_{±}(z, n) = u_{±}(z, n_{0})\begin{matrix}n−1 \\ ∏ & ∗ \\ j=n_{0} \end{matrix} (−a(j − {0 \atop 1})m_{±}(z, j))^{±1}

and that m_{+}(z, n) satisfies the following recurrence relation

(2.11)  a(n − {0 \atop 1})^{2}m_{±}(z, n) + \frac{1}{m_{±}(z, n ∓ 1)} + z − b(n) = 0.

The functions m_{±}(z, n) are the correct objects from the viewpoint of spectral theory. However, when it comes to calculations, the following pair of Weyl mfunctions

(2.12)  m̃_{±}(z, n) = ∓\frac{u_{±}(z, n + 1)}{a(n)u_{±}(z, n)}, m̃_{±}(z) = m̃_{±}(z, 0)

is often more convenient than the original one. The connection is given by

(2.13)  m_{+}(z, n) = m̃_{+}(z, n),  m_{−}(z, n) = \frac{a(n)^{2}m̃_{−}(z, n) − z + b(n)}{a(n − 1)^{2}}

(note also m_{−}(z, n + 1)^{−1} = −a(n)^{2}m̃_{−}(z, n)) and the corresponding spectral measures (for n = 0) are related by

(2.14)  dρ_{+} = dρ̃_{+},  dρ_{−} = \frac{a(0)^{2}}{a(−1)^{2}} dρ̃_{−}.

You might want to note that m̃_{−}(z) does not tend to 0 as Im(z) → ∞ since the linear part is present in its Herglotz representation

(2.15)  m̃_{−}(z) = \frac{z − b(n)}{a(0)^{2}} + ∫_{ℝ} \frac{dρ̃_{−}(λ)}{λ − z}.

Finally, we introduce the Weyl m-functions m_{±}^{β}(z, n) associated with H_{±,n}^{β}. They are defined analogously to m_{±}(z, n). Moreover, the definition of H_{+,n}^{β} in terms of H_{+,n} suggests that m_{+}^{β}(z, n), β ≠ 0, can be expressed in terms of m_{+}(z, n). Using (1.91) and the second resolvent identity we obtain

(2.16)  m_{+}^{β}(z, n) = ⟨δ_{n+1}, (H_{+,n}^{β} − z)^{−1}δ_{n+1}⟩ = \frac{βm_{+}(z, n)}{β − a(n)m_{+}(z, n)}.

Similarly, the Weyl m-functions m_{−}^{β}(z, n) associated with H_{−,n}^{β}, β ≠ ∞, can be expressed in terms of m_{−}(z, n + 1),

(2.17)  m_{−}^{β}(z, n) = ⟨δ_{n}, (H_{−,n}^{β} − z)^{−1}δ_{n}⟩ = \frac{m_{−}(z, n + 1)}{1 − βa(n)m_{−}(z, n + 1)}.

2.2. Properties of solutions

The aim of the present section is to establish some fundamental properties of special solutions of (1.19) which will be indispensable later on.

As an application of Weyl m-functions we first derive some additional properties of the solutions u_{±}(z, .). By (1.27) we have

(2.18)  u_{±}(z, n) = a(0)u_{±}(z, 0)(a(0)^{−1}c(z, n) ∓ m̃_{±}(z)s(z, n)),

where the constant (with respect to n) a(0)u_{±}(z, 0) is at our disposal. If we choose a(0)u_{±}(z, 0) = 1 (bearing in mind that c(z, n), s(z, n) are polynomials with respect to z), we infer for instance (using (2.4))

(2.19)  \overline{u_{±}(z, n)} = u_{±}(\overline{z}, n).

Moreover, u_{±}(z, n) are holomorphic with respect to z ∈ ℂ\σ(H_{±}). But we can do even better. If μ is an isolated eigenvalue of H_{±}, then m̃_{±}(z) has a simple pole at z = μ (since it is Herglotz; see also (2.36)). By choosing a(0)u_{±}(z, 0) = (z − μ) we can remove the singularity at z = μ. In summary,

Lemma 2.2. The solution u_{±}(z, n) of (1.19) which is square summable near ±∞ exist for z ∈ ℂ\σ_{ess}(H_{±}), respectively. If we choose

(2.20)  u_{±}(z, n) = \frac{c(z, n)}{a(0)} ∓ m̃_{±}(z)s(z, n),

then u_{±}(z, n) is holomorphic for z ∈ ℂ\σ(H_{±}), u_{±}(z, .) ≢ 0, and \overline{u_{±}(z, .)} = u_{±}(\overline{z}, .). In addition, we can include a finite number of isolated eigenvalues in the domain of holomorphy of u_{±}(z, n).

Moreover, the sums

(2.21)  ∑_{j=n}^{∞} u_{+}(z_{1}, j)u_{+}(z_{2}, j), ∑_{j=−∞}^{n} u_{−}(z_{1}, j)u_{−}(z_{2}, j)

are holomorphic with respect to z_{1} (resp. z_{2}) provided u_{±}(z_{1}, .) (resp. u_{±}(z_{2}, .)) are.

Proof. Only the last assertion has not been shown yet. It suffices to prove the claim for one n, say n = 1. Without restriction we suppose u_{+}(z, 0) = −a(0)^{−1} (if u_{+}(z, 0) = 0, shift everything by one). Then u_{+}(z, n) = (H_{+} − z)^{−1}δ_{1}(n), n ≥ 1, and hence

(2.22)  ∑_{j=1}^{∞} u_{+}(z_{1}, j)u_{+}(z_{2}, j) = ⟨δ_{1}, (H_{+} − z_{1})^{−1}(H_{+} − z_{2})^{−1}δ_{1}⟩_{ℓ^{2}(ℕ)}.

The remaining case is similar. □

Remark 2.3. If (λ_{0}, λ_{1}) ⊂ ρ(H), we can define u_{±}(λ, n), λ ∈ [λ_{0}, λ_{1}]. Indeed, by (2.20) it suffices to prove that m_{±}(λ) tends to a limit (in ℝ ∪ {∞}) as λ ↓ λ_{0} or λ ↑ λ_{1}. This follows from monotonicity of m_{±}(λ),

(2.23)  m′_{±}(λ) = −⟨δ_{±1}, (H_{±} − λ)^{−2}δ_{±1}⟩ < 0, λ ∈ (λ_{0}, λ_{1})

(compare also equation (B.18)). Here the prime denotes the derivative with respect to λ. However, u_{±}(λ_{0,1}, n) might not be square summable near ±∞ in general.

In addition, the discrete eigenvalues of H_{+}^{β} are the zeros of \frac{β}{a(0)} − m_{+}(λ) (see (2.16)) and hence decreasing as a function of β. Similarly, the discrete eigenvalues of H_{−}^{β} are the zeros of \frac{1}{a(0)β} −m_{−}(λ) (see (2.17)) and hence increasing as a function of β.

Let u(z) be a solution of (1.19) with z ∈ ℂ\ℝ. If we choose f = u(z), g = \overline{u(z)} in (1.20), we obtain

(2.24)  [u(z)]_{n} = [u(z)]_{m−1} − \begin{matrix}n \\ ∑ & ∗ \\ j=m \end{matrix} |u(z, j)|^{2},

where [.]_{n} denotes the Weyl bracket,

(2.25)  [u(z)]_{n} = \frac{W_{n}(u(z), \overline{u(z)})}{2iIm(z)} = a(n)\frac{Im(u(z, n)\overline{u(z, n + 1)})}{Im(z)}.

Especially for u_{±}(z, n) as in Lemma 2.2 we get

(2.26)  [u_{±}(z)]_{n} = \{ \begin{matrix}∑_{j=n+1}^{∞} |u_{+}(z, j)|^{2} \\ − ∑_{j=−∞}^{n} |u_{−}(z, j)|^{2} \end{matrix}

and for s(z, n)

(2.27)  [s(z)]_{n} = \{ \begin{matrix}− ∑_{j=0}^{n} |s(z, j)|^{2}, & n ≥ 0 \\ ∑_{j=n+1}^{0} |s(z, j)|^{2}, & n < 0 \end{matrix} .

Moreover, let u(z, n), v(z, n) be solutions of (1.19). If u(z, n) is differentiable with respect to z, we obtain

(2.28)  (τ − z)u′(z, n) = u(z, n)

(here the prime denotes the derivative with respect to z) and using Green’s formula (1.20) we infer

(2.29)  ∑_{j=m}^{n} v(z, j)u(z, j) = W_{n}(v(z), u′(z)) − W_{m−1}(v(z), u′(z)).

Even more interesting is the following result.

Lemma 2.4. Let u_{±}(z, n) be solutions of (1.19) as in Lemma 2.2. Then we have

(2.30)  W_{n}(u_{±}(z), \frac{d}{dz} u_{±}(z)) = \{ \begin{matrix}− ∑_{j=n+1}^{∞} u_{+}(z, j)^{2} \\ ∑_{j=−∞}^{n} u_{−}(z, j)^{2} \end{matrix} .

Proof. Green’s formula (1.20) implies

(2.31)  W_{n}(u_{+}(z), u_{+}(z̃)) = (z − z̃) ∑_{j=n+1}^{∞} u_{+}(z, j)u_{+}(z̃, j)

and furthermore,

W_{n}(u_{+}(z), u′_{+}(z)) = lim_{z̃→z} W_{n}(u_{+}(z), \frac{u_{+}(z) − u_{+}(z̃)}{z − z̃})
(2.32)  = ∑_{j=n+1}^{∞} u_{+}(z, j)^{2}.
□

As a first application of this lemma, let us investigate the isolated poles of G(z, n, m). If z = λ_{0} is such an isolated pole (i.e., an isolated eigenvalue of H), then W (u_{−}(λ_{0}), u_{+}(λ_{0})) = 0 and hence u_{±}(λ_{0}, n) differ only by a (nonzero) constant multiple. Moreover,

\frac{d}{dz} W (u_{−}(z), u_{+}(z))|_{z=λ_{0}} = W_{n}(u_{−}(λ_{0}), u′_{+}(λ_{0})) + W_{n}(u′_{−}(λ_{0}), u_{+}(λ_{0}))
(2.33)  = − ∑_{j∈ℤ} u_{−}(λ_{0}, j)u_{+}(λ_{0}, j)

by the lemma and hence

(2.34)  G(z, n, m) = −\frac{P (λ_{0}, n, m)}{z − λ_{0}} + O(z − λ_{0})^{0},

where

(2.35)  P (λ_{0}, n, m) = \frac{u_{±}(λ_{0}, n)u_{±}(λ_{0}, m)}{∑_{j∈ℤ} u_{±}(λ_{0}, j)^{2}}.

Similarly, for H_{±} we obtain

(2.36)  lim_{z→μ}(z − μ)G_{±}(z, n, m) = −\frac{s(μ, n)s(μ, m)}{∑_{j∈±ℕ} s(μ, j)^{2}}.

Thus the poles of the kernel of the resolvent at isolated eigenvalues are simple. Moreover, the negative residue equals the kernel of the projector onto the corresponding eigenspace.

As a second application we show monotonicity of G(z, n, n) with respect to z in a spectral gap. Differentiating (1.99), a straightforward calculation shows

(2.37) G′(z, n, n) = \frac{u_{+}(z, n)^{2}W_{n}(u_{−}(z), u̇_{−}(z)) − u_{−}(z, n)^{2}W_{n}(u_{+}(z), u̇_{+}(z))}{W (u_{−}(z), u_{+}(z))^{2}},

which is positive for z ∈ ℝ\σ(H) by Lemma 2.4. The same result also follows from

G′(z, n, m) = \frac{d}{dz} ⟨δ_{n}, (H − z)^{−1}δ_{m}⟩ = ⟨δ_{n}, (H − z)^{−2}δ_{m}⟩
(2.38)  = ∑_{j∈ℤ} G(z, n, j)G(z, j, m).

Finally, let us investigate the qualitative behavior of the solutions u_{±}(z) more closely.

Lemma 2.5. Suppose z ∈ ℂ\σ_{ess}(H_{±}). Then we can find C_{±}, γ^{±} > 0 such that

(2.39)  |u_{±}(z, n)| ≤ C_{±} exp(−γ^{±}n), ±n ∈ ℕ.

For γ^{±} we can choose

(2.40)  γ^{±} = ln (1 + (1 − ε)\frac{sup_{β∈ℝ∪\{∞\}} dist(σ(H_{±}^{β}), z)}{2 sup_{n∈ℕ} |a(±n)|}), ε > 0

(the corresponding constant C_{±} will depend on ε).

Proof. By the definition of H_{±}^{β} it suffices to consider the case β = ∞, δ = dist(σ(H_{+}), z) > 0 (otherwise alter b(1) or shift the interval). Recall G_{+}(z, 1, n) = u_{+}(z, n)/(−a(0)u_{+}(z, 0)) and consider (γ > 0)

e^{γ(n−1)}G_{+}(z, 1, n) = ⟨δ_{1}, P_{−γ}(H_{+} − z)^{−1}P_{γ}δ_{n}⟩ = ⟨δ_{1}, (P_{−γ}H_{+}P_{γ} − z)^{−1}δ_{n}⟩
(2.41)  = ⟨δ_{1}, (H_{+} − z + Q_{γ})^{−1}δ_{n}⟩,

where

(2.42)  (P_{γ}u)(n) = e^{γn}u(n)

and

(2.43)  (Q_{γ}u)(n) = a(n)(e^{γ} − 1)u(n + 1) + a(n − 1)(e^{−γ} − 1)u(n − 1).

Moreover, choosing γ = ln(1 + (1 − ε)δ/(2 sup_{n∈ℕ} |a(±n)|)) we have

(2.44)  ∥Q_{γ}∥ ≤ 2 sup_{n∈ℕ} |a(±n)|(e^{γ} − 1) = (1 − ε)δ.

Using the first resolvent equation

∥(H_{+} − z + Q_{γ})^{−1}∥ ≤ ∥(H_{+} − z)^{−1}∥ + ∥(H_{+} − z)^{−1}∥

×∥Q_{γ}∥∥(H_{+} − z + Q_{γ})^{−1}∥ (2.45)

and ∥(H_{+} − z)^{−1}∥ = δ^{−1} implies the desired result

(2.46)  |e^{γ(n−1)}G_{+}(z, 1, n)| ≤ ∥(H_{+} − z + Q_{γ})^{−1}∥ ≤ \frac{1}{δε}.
□

This result also shows that for any solution u(z, n) which is not a multiple of u_{±}(z, .), we have

(2.47)  \sqrt{|u(z, n)|^{2} + |u(z, n + 1)|^{2}} ≥ \frac{const}{|a(n)|}e^{γ^{±}n}

since otherwise the Wronskian of u and u_{±}(z) would vanish as n → ±∞. Hence, (cf. (1.34))

(2.48)  \{λ ∈ ℝ|\underline{γ}^{±}(λ) = 0\} ⊆ σ_{ess}(H_{±})

since \underline{γ}^{±}(λ) > γ^{±}, λ ∉ σ_{ess}(H_{±}).

2.3. Positive solutions

In this section we want to investigate solutions of τu = λu with λ ≤ σ(H). We will assume

(2.49)  a(n) < 0

throughout this section. As a warm-up we consider the case λ < σ(H).

Lemma 2.6. Suppose a(n) < 0 and let λ < σ(H). Then we can assume

(2.50)  u_{±}(λ, n) > 0, n ∈ ℤ,

and we have

(2.51)  (n − n_{0})s(λ, n, n_{0}) > 0, n ∈ ℤ\{n_{0}}.

Proof. From (H − λ) > 0 one infers (H_{+,n} − λ) > 0 and hence

(2.52)  0 < ⟨δ_{n+1}, (H_{+,n} − λ)^{−1}δ_{n+1}⟩ = \frac{u_{+}(λ, n + 1)}{−a(n)u_{+}(λ, n)},

showing that u_{+}(λ) can be chosen to be positive. Furthermore, for n > n_{0} we obtain

(2.53)  0 < ⟨δ_{n}, (H_{+,n_{0}} − λ)^{−1}δ_{n}⟩ = \frac{u_{+}(λ, n)s(λ, n, n_{0})}{−a(n_{0})u_{+}(λ, n_{0})},

implying s(λ, n, n_{0}) > 0 for n > n_{0}. The remaining case is similar. □

The general case λ ≤ σ(H) requires an additional investigation. We first prove that (2.51) also holds for λ ≤ σ(H) with the aid of the following lemma.

Lemma 2.7. Suppose a(n) < 0 and let u ≢ 0 solve τu = u. Then u(n) ≥ 0, n ≥ n_{0} (resp. n ≤ n_{0}), implies u(n) > 0, n > n_{0} (resp. n < n_{0}). Similarly, u(n) ≥ 0, n ∈ ℤ, implies u > 0, n ∈ ℤ.

Proof. Fix n > n_{0} (resp. n < n_{0}). Then, u(n) = 0 implies u(n ± 1) > 0 (since u cannot vanish at two consecutive points) contradicting 0 = (b(n) − λ)u(n) = −a(n)u(n + 1) − a(n − 1)u(n − 1) > 0. □

The desired result now follows from

(2.54)  s(λ, n, n_{0}) = lim_{ε↓0} s(λ − ε, n, n_{0}) ≥ 0, n > n_{0},

and the lemma. In addition, we note

(2.55)  b(n) − λ = −a(n)s(λ, n + 1, n − 1) > 0, λ ≤ σ(H).

The following corollary is simple but powerful.

Corollary 2.8. Suppose u_{j}(λ, n), j = 1, 2, are two solutions of τu = λu, λ ≤ σ(H), with u_{1}(λ, n_{0}) = u_{2}(λ, n_{0}) for some n_{0} ∈ ℤ. Then if

(2.56)  (n − n_{0})(u_{1}(λ, n) − u_{2}(λ, n)) > 0 (resp. < 0)

holds for one n ∈ ℤ\{n_{0}}, then it holds for all. If u_{1}(λ, n) = u_{2}(λ, n) for one n ∈ ℤ\{n_{0}}, then u_{1} and u_{2} are equal.

Proof. Use u_{1}(λ, n) − u_{2}(λ, n) = c s(λ, n, n_{0}) for some c ∈ ℝ. □

In particular, this says that for λ ≤ σ(H) solutions u(λ, n) can change sign (resp. vanish) at most once (since s(z, n, n_{0}) does).

For the sequence

(2.57)  u_{m}(λ, n) = \frac{s(λ, n, m)}{s(λ, 0, m)}, m ∈ ℕ,

this corollary implies that

(2.58)  ϕ_{m}(λ) = u_{m}(λ, 1) = \frac{s(λ, 1, m)}{s(λ, 0, m)}

is increasing with m since we have u_{m+1}(λ, m) > 0 = u_{m}(λ, m). Next, since a(0)s(λ, 1, m) + a(−1)s(λ, −1, m) = (λ − b(0))s(λ, 0, m) implies

(2.59)  ϕ_{m}(λ) < \frac{λ − b(0)}{a(0)},

we can define (2.60)

ϕ_{+}(λ) = lim_{m→∞} ϕ_{m}(λ), u_{+}(λ, n) = lim_{m→∞} u_{m}(λ, n) = c(λ, n) + ϕ_{+}(λ)s(λ, n).

By construction we have u_{+}(λ, n) > u_{m}(λ, n), n ∈ ℕ, implying u_{+}(λ, n) > 0, n ∈ ℕ. For n < 0 we have at least u_{+}(λ, n) ≥ 0 since u_{m}(λ, n) > 0 and thus u_{+}(λ, n) > 0, n ∈ ℤ, by Lemma 2.7.

Let u(λ, n) be a solution with u(λ, 0) = 1, u(λ, 1) = ϕ(λ). Then, by the above analysis, we infer that u(λ, n) > 0, n ∈ ℕ, is equivalent to ϕ(λ) ≥ ϕ_{+}(λ) and hence u(λ, n) ≥ u_{+}(λ, n), n ∈ ℕ (with equality holding if and only if u(λ) = u_{+}(λ)). In this sense u_{+}(λ) is the minimal positive solution (also principal or recessive solution) near +∞. In particular, since every solution can change sign at most once, we infer that if there is a square summable solution near +∞, then it must be equal to u_{+}(λ, n) (up to a constant multiple), justifying our notation.

Moreover, if u(λ) is different from u_{+}(λ), then constancy of the Wronskian

(2.61)  \frac{u_{+}(λ, n)}{u(λ, n)} − \frac{u_{+}(λ, n + 1)}{u(λ, n + 1)} = \frac{W (u(λ), u_{+}(λ))}{−a(n)u(λ, n)u(λ, n + 1)}

together with W (u(λ), u_{+}(λ)) = a(0)(ϕ_{+}(λ) − ϕ(λ)) > 0 shows that the sequence u_{+}(λ, n)/u(λ, n) is decreasing for all u(λ) ≠ u_{+}(λ) if and only if u_{+}(λ) is minimal. Moreover, we claim

(2.62)  lim_{n→∞} \frac{u_{+}(λ, n)}{u(λ, n)} = 0.

In fact, suppose lim_{n→∞} u_{+}(λ, n)/u(λ, n) = ε > 0. Then u_{+}(λ, n) > εu(λ, n), n ∈ ℕ, and hence u_{ε} = (u_{+}(λ, n)−εu(λ, n))/(1−ε) > 0, n ∈ ℕ. But u_{+}(λ, n) < u_{ε}(λ, n) implies u(λ, n) < u_{+}(λ, n), a contradiction.

Conversely, (2.62) for one u(λ) uniquely characterizes u_{+}(λ, n) since (2.62) remains valid if we replace u(λ) by any positive linear combination of u(λ) and u_{+}(λ).

Summing up (2.61) shows that

(2.63)  ∑_{j=0}^{∞} \frac{1}{−a(j)u(λ, j)u(λ, j + 1)} = \frac{1}{a(0)(ϕ_{+}(λ) − ϕ(λ))} < ∞.

Moreover, the sequence

(2.64)  v(λ, n) = u(λ, n) ∑_{j=n}^{∞} \frac{1}{−a(j)u(λ, j)u(λ, j + 1)} > 0

solves τu = λu (cf. (1.51)) and equals u_{+}(λ, n) up to a constant multiple since lim_{n→∞} v(λ, n)/u(λ, n) = 0.

By reflection we obtain a corresponding minimal positive solution u_{−}(λ, n) near −∞. Let us summarize some of the results obtained thus far.

Lemma 2.9. Suppose a(n) < 0, λ ≤ σ(H) and let u(λ, n) be a solution with u(λ, n) > 0, ±n ≥ 0. Then the following conditions are equivalent.

(i). u(λ, n) is minimal near ±∞.

(ii). We have

(2.65)  \frac{u(λ, n)}{v(λ, n)} ≤ \frac{v(λ, 0)}{u(λ, 0)}, ±n ≥ 0,
for any solution v(λ, n) with v(λ, n) > 0, ±n ≥ 0.

(iii). We have

(2.66)  lim_{n→±∞} \frac{u(λ, n)}{v(λ, n)} = 0.
for one solution v(λ, n) with v(λ, n) > 0, ±n ≥ 0.

(iv). We have

(2.67)  ∑_{j∈±ℕ} \frac{1}{−a(j)u(j)u(j + 1)} = ∞.

Recall that minimality says that for a solution u(λ, n) with u(λ, 0) = 1, u(λ, 1) = ϕ(λ) to be positive on ℕ, we need ϕ(λ) ≥ ϕ_{+}(λ). Similarly, for u(λ, n) to be positive on −ℕ we need ϕ(λ) ≤ ϕ_{−}(λ). In summary, u(n) > 0, n ∈ ℤ, if and only if ϕ_{+}(λ) ≤ ϕ(λ) ≤ ϕ_{−}(λ) and thus any positive solution can (up to constant multiples) be written as

(2.68)  u(λ, n) = \frac{1 − σ}{2} u_{−}(λ, n) + \frac{1 + σ}{2} u_{+}(λ, n), σ ∈ [−1, 1].

Two cases may occur

(i). u_{−}(λ, n), u_{+}(λ, n) are linearly dependent (i.e., ϕ_{+}(λ) = ϕ_{−}(λ)) and there is only one (up to constant multiples) positive solution. In this case H −λ is called critical.

(ii). u_{−}(λ, n), u_{+}(λ, n) are linearly independent and

(2.69)  u_{σ}(λ, n) = \frac{1 + σ}{2} u_{+}(λ, n) + \frac{1 − σ}{2} u_{−}(λ, n),

is positive if and only if σ ∈ [−1, 1]. In this case H−λ is called subcritical .

If H −λ > σ(H), then H −λ is always subcritical by Lemma 2.6. To emphasize this situation one sometimes calls H − λ supercritical if H − λ > σ(H).

In case (ii) it is not hard to show using (2.62) that for two positive solutions u_{j}(λ, n), j = 1, 2, we have

(2.70)  u_{σ}(λ, n) = \frac{1 + σ}{2} u_{1}(λ, n) + \frac{1 − σ}{2} u_{2}(λ, n) > 0 ⇔ σ ∈ [−1, 1],

if and only if the solutions u_{1,2} equal u_{±} up to constant multiples.

Remark 2.10. Assuming a(n) < 0, the requirement H − λ ≥ 0 is also necessary for a positive solution to exist. In fact, any positive solution can be used to factor H − λ = A^{∗}A ≥ 0 (cf. Corollary 11.2).

Similarly, if a(n) > 0, the requirement H − λ ≤ 0 is sufficient and necessary for a positive solution to exist (consider −(H − λ) ≥ 0). If a(n) is not of a fixed sign, no general statement is possible.

2.4. Weyl circles

In this section we will advocate a different approach for Weyl m-functions. There are two main reasons for doing this. First of all this approach is similar to Weyl’s original one for differential equations and second it will provide an alternative characterization of Weyl m-functions as limits needed later.

Let s_{β}(z, .), c_{β}(z, .) be the fundamental system of (1.19) corresponding to the initial conditions

(2.71)

s_{β}(z, 0) = − sin(α),  s_{β}(z, 1) = cos(α),
c_{β}(z, 0) = cos(α),  c_{β}(z, 1) = sin(α),

where β = cot(α). Clearly,

W (c_{β}(z), s_{β}(z)) = a(0). (2.72)

The general idea is to approximate H_{±}^{β} by finite matrices. We will choose the boundary conditions associated with β, β_{N} at n = 0, N, respectively. The corresponding matrix H_{0,N}^{β,β_{N}} will have eigenvalues λ_{j}(N), 1 ≤ j ≤ Ñ, and corresponding eigenvectors s_{β}(λ_{j}(N), n) (since s_{β}(z, n) fulfills the boundary condition at 0). We note that λ_{j}(N), 1 ≤ j ≤ Ñ, depend also on β.

If we set

(2.73)  m̃_{N}^{β}(z) = sgn(N)\frac{W_{N}(s_{β}(λ_{j}(N)), c_{β}(z))}{a(0)W_{N}(s_{β}(λ_{j}(N)), s_{β}(z))},  N ∈ ℤ\\{0\}

(independent of the eigenvalue chosen), then the solution

(2.74)  u_{N}(z, n) = a(0)^{−1}c_{β}(z, n) − sgn(N)m̃_{N}^{β}(z)s_{β}(z, n)

satisfies the boundary condition at N, that is, W_{N}(s_{β}(λ_{1}), u_{N}(z)) = 0. The function m̃_{N}^{β}(z) is rational (w.r.t. z) and has poles at the eigenvalues λ_{j}(N). In particular, they are the analogs of the Weyl m̃-functions for finite Jacobi operators. Hence it suggests itself to consider the limit N → ±∞, where our hope is to obtain the Weyl m̃-functions for the Jacobi operators H_{±}^{β}.

We fix λ_{0} ∈ ℝ and set

(2.75)  β_{N} = −\frac{s_{β}(λ_{0}, N + 1)}{s_{β}(λ_{0}, N)}

implying λ_{0} = λ_{j}(N) for one j. It will turn out that the above choice for β_{N} is not really necessary since m̃_{N}^{β}(z) will, as a consequence of boundedness of H, converge for arbitrary sequences β_{N}. However, the above choice also works in the case of unbounded operators.

Before we turn to the limit, let us derive some additional properties for finite N. The limits

lim_{z→λ_{j}(N)} W_{N}(s_{β}(λ_{j}(N)), c_{β}(z)) = −a(0),
(2.76) lim_{z→λ_{j}(N)} \frac{W_{N}(s_{β}(λ_{j}(N)), s_{β}(z))}{z − λ_{j}(N)} = W_{N}(s_{β}(λ_{j}(N)), s′_{β}(λ_{j}(N)))

imply that all poles of m_{N}(z, β) are simple. Using (2.29) to evaluate (2.76) one infers that the negative residue at λ_{j}(N) is given by

(2.77)  γ_{N}^{β}(λ_{j}(N)) = ( ∑_{n= {1 \atop N+1}}^{{N \atop 0}} s_{β}(λ_{j}(N), n)^{2})^{−1}, N > < 0.

The numbers γ_{N}^{β}(λ_{j}(N)) are called norming constants. Hence one gets

(2.78)  m̃_{N}^{β}(z) = − ∑_{j=1}^{Ñ} \frac{γ_{N}^{β}(λ_{j}(N))}{z − λ_{j}(N)} + \{ \begin{matrix}\frac{β^{∓1}}{a(0)}, & β^{∓1} ≠ ∞ \\ \frac{z−b( {1 \atop 0} )}{a(0)^{2}}, & β^{∓1} = ∞ \end{matrix} , N > < 0,

and with the help of (2.24) we obtain

(2.79)  ∑_{n= {1 \atop N+1}}^{{N \atop 0}} |u_{N}(z, n)|^{2} = \frac{Im(m̃_{N}^{β}(z))}{Im(z)}, N > < 0,

that is, m_{N}^{β}(z) are Herglotz functions.

Now we turn to the limits N → ±∞. Fix z ∈ ℂ\ℝ. Observe that if λ_{1} varies in ℝ, then β_{N} takes all values in ℝ ∪ {∞} Ñ times. Rewriting (2.73) shows that

(2.80)  m̃_{N}^{β}(z) = \frac{sgn(N)}{a(0)} \frac{c_{β}(z, N)β_{N} + c_{β}(z, N + 1)}{s_{β}(z, N)β_{N} + s_{β}(z, N + 1)}

is a Möbius transformation and hence the values of m̃_{N}^{β}(z) for different β_{N} ∈ ℝ ∪ {∞} lie on a circle (also called Weyl circle) in the complex plane (note that z ∈ ℝ would correspond to the degenerate circle ℝ ∪ {∞}). The center of this circle is

(2.81)  c_{N} = sgn(N)\frac{W_{N}(c_{β}(z), \overline{s_{β}(z)})}{2a(0)Im(z)[s_{β}(z)]_{N}}

and the radius is

(2.82)  r_{N} = |\frac{W (c_{β}(z), s_{β}(z))}{2a(0)Im(z)[s_{β}(z)]_{N}} | = \frac{1}{2|Im(z)[s_{β}(z)]_{N}|}.

Using

(2.83)  [a(0)^{−1}c_{β}(z) − sgn(N)m̃s_{β}(z)]_{N} = [s_{β}(z)]_{N}(|m̃ − c_{N}|^{2} − r_{N}^{2})

we see that this circle is equivalently characterized by the equation

(2.84)  {m̃ ∈ ℂ|[a(0)^{−1}c_{β}(z) + sgn(N)m̃s_{β}(z)]_{N} = 0}.

Since [.]_{N}, N > 0, is decreasing with respect to N, the circle corresponding to N +1 lies inside the circle corresponding to N. Hence these circles tend to a limit point for any sequence (β_{N} ∈ ℝ ∪ {∞})_{N∈ℕ} since

(2.85)  lim_{N→∞} −[s_{β}(z)]_{N} = ∑_{n=0}^{∞} |s_{β}(z, n)|^{2} = ∞

(otherwise H_{+}^{β} would have a non-real eigenvalue). Similarly for N < 0. Thus the pointwise convergence of m̃_{N}^{β}(z) is clear and we may define

(2.86)  m̃_{±}^{β}(z) = lim_{N→±∞} m̃_{N}^{β}(z).

Moreover, the above sequences are locally bounded in z (fix N and take all circles corresponding to a (sufficiently small) neighborhood of any point z and note that all following circles lie inside the ones corresponding to N) and by Vitali’s theorem ([229], p. 168) they converge uniformly on every compact set in ℂ_{±} = {z ∈ ℂ| ± Im(z) > 0}, implying that m̃_{±}^{β}(z) are again holomorphic on ℂ_{±}.

Remark 2.11. (i). Since m̃_{N}^{β}(z) converges for arbitrary choice of the sequence β_{N} we even have

(2.87)  m̃_{±}^{β}(z) = lim_{N→±∞} \frac{c_{β}(z, N)}{a(0)s_{β}(z, N)}.

Moreover, this approach is related to (2.60). Using (1.62), (1.63) shows ϕ_{m}(λ) = −c(λ, m)/s(λ, m) and establishes the equivalence. (ii). That the Weyl circles converge to a point is a consequence of the boundedness of a(n), b(n). In the general case the limit could be a circle or a point (independent of z ∈ ℂ\ℝ). Accordingly one says that τ is limit circle or limit point at ±∞. (See Section 2.6 for further information on unbounded operators.)

As anticipated by our notation, m̃_{±}^{β}(z) are closely related to m_{±}^{β}(z). This will be shown next. We claim that

(2.88)  u_{±}(z, n) = a(0)^{−1}c_{β}(z, n) ∓ m̃_{±}^{β}(z)s_{β}(z, n)

is square summable near ±∞. Since m̃_{+}^{β} lies in the interior of all Weyl circles the limit lim_{N→∞}[u_{+}(z)]_{N} ≥ 0 must exist and hence u_{+} ∈ ℓ_{+}^{2}(ℤ) by (2.24). Moreover, u_{+} ∈ ℓ_{+}^{2}(ℤ) implies [u_{+}]_{∞} = 0. Similarly for u_{−}(z). In addition, (cf. (2.4))

(2.89)  ∑_{n= {1 \atop −∞}}^{{∞ \atop 0}} |u_{±}(z, n)|^{2} = \frac{Im(m̃_{±}^{β}(z))}{Im(z)},

implies that m̃_{±}^{β}(z) are Herglotz functions (note that u_{±}(z, n) depends on β because of the normalization u_{±}(z, 0) = a(0)^{−1} cos(α) ± m̃_{±}^{β}(z) sin(α)). In particular, their Herglotz representation reads

(2.90)  m̃_{±}^{β}(z) = \frac{β^{∓1}}{a(0)} + ∫_{ℝ} \frac{dρ̃_{±}^{β}(λ)}{λ − z},  β ≠ {0 \atop ∞}.

This finally establishes the connection

(2.91)  m̃_{±}^{∞}(z) = m̃_{±}(z) = \frac{∓u_{±}(z, 1)}{a(0)u_{±}(z, 0)}

as expected. Furthermore, m̃_{±}^{β_{1}}(z) can be expressed in terms of m̃_{±}^{β_{2}}(z) (use that u_{±} is unique up to a constant) by

(2.92)  m̃_{±}^{β_{1}}(z) = ±\frac{1}{a(0)} \frac{a(0) cos(α_{2} − α_{1})m̃_{±}^{β_{2}}(z) ∓ sin(α_{2} − α_{1})}{a(0) sin(α_{2} − α_{1})m̃_{±}^{β_{2}}(z) ± cos(α_{2} − α_{1})},

where β_{1,2} = cot(α_{1,2}). Specializing to the case β_{1} = β, β_{2} = ∞ we infer

(2.93)  m̃_{+}^{β}(z, n) = \frac{βm_{+}(z) + a(0)^{−1}}{β − a(0)m_{+}(z)} , m̃_{−}^{β}(z, n) = \frac{m_{−}(z, 1) + a(0)^{−1}β}{1 − βa(0)m_{−}(z, 1)}

which should be compared with (2.16), (2.17), respectively.

2.5. Canonical forms of Jacobi operators and the moment problem

The aim of this section is to derive canonical forms for H, H_{±} and to relate the spectra of these operators to the corresponding measures encountered in the previous sections.

Since s(z, n) is a polynomial in z we infer by induction (cf. Lemma 1.2)

(2.94)  s(H_{+}, n)δ_{1} = ∑_{j=0}^{n} s_{j,n}(0)H_{+}^{j}δ_{1} = δ_{n},

implying that δ_{1} is a cyclic vector for H_{+}. We recall the measure

dρ_{+}(λ) = d⟨δ_{1}, P_{(−∞,λ]}(H_{+})δ_{1}⟩ (2.95)

and consider the Hilbert space L^{2}(ℝ, dρ_{+}). Since dρ_{+} is supported on σ(H_{+}) this space is the same as the space L^{2}(σ(H_{+}), dρ_{+}). The scalar product is given by

(2.96)  ⟨f, g⟩_{L^{2}} = ∫_{ℝ} \overline{f(λ)}g(λ) dρ_{+}(λ).

If f, g are polynomials we can evaluate their scalar product without even knowing dρ_{+}(λ) since

(2.97)  ⟨f, g⟩_{L^{2}} = ⟨f(H_{+})δ_{1}, g(H_{+})δ_{1}⟩.

Applying this relation in the special case f(λ) = s(λ, m), g(λ) = s(λ, n), we obtain from equation (2.94) that the polynomials s(z, n), n ∈ ℕ, are orthogonal with respect to this scalar product, that is,

(2.98)  ⟨s(λ, m), s(λ, n)⟩_{L^{2}} = ∫_{ℝ} s(λ, m)s(λ, n) dρ_{+}(λ) = δ_{m,n}.

We will see in Theorem 4.5 that s(λ, n) has n − 1 distinct real roots which interlace the roots of s(λ, n + 1).

Now consider the following transformation U from the set ℓ_{0}(ℕ) onto the set of all polynomials (eigenfunction expansion)

(Uf)(λ) = ∑_{n=1}^{∞} f(n)s(λ, n),

(2.99)  (U^{−1}F)(n) = ∫_{ℝ} s(λ, n)F (λ)dρ_{+}(λ).

A simple calculation for F (λ) = (Uf)(λ) shows that U is unitary,

(2.100)  ∑_{n=1}^{∞} |f(n)|^{2} = ∫_{ℝ} |F (λ)|^{2}dρ_{+}(λ).

This leads us to the following result.

Theorem 2.12. The unitary transformation

(2.101)  Ũ : \begin{matrix}ℓ^{2}(ℕ) & → & L^{2}(ℝ, dρ_{+}) \\ f(n) & ↦ & ∑_{n=1}^{∞} f(n)s(λ, n) \end{matrix}

(where the sum is to be understood as norm limit) maps the operator H_{+} to the multiplication operator by λ. More explicitly,

(2.102)  H_{+} = Ũ^{−1}H̃Ũ,

where

(2.103)  \begin{aligned}H̃ : & L^{2}(ℝ, dρ_{+}) & → & L^{2}(ℝ, dρ_{+}) \\ & F (λ) & ↦ & λF (λ) \end{aligned} .

Proof. Since dρ_{+} is compactly supported the set of all polynomials is dense in L^{2}(ℝ, dρ_{+}) (Lemma B.1) and U extends to the unitary transformation Ũ. The rest follows from

H̃F (λ) = ŨH_{+}Ũ^{−1}F (λ) = ŨH_{+} ∫_{ℝ} s(λ, n)F (λ)dρ_{+}(λ)
(2.104)  = Ũ ∫_{ℝ} λs(λ, n)F (λ)dρ_{+}(λ) = λF (λ).
□

This implies that the spectrum of H_{+} can be characterized as follows (see Lemma B.5). Let the Lebesgue decomposition of dρ_{+} be given by

(2.105)  dρ_{+} = dρ_{+,pp} + dρ_{+,ac} + dρ_{+,sc},

where pp, ac, and sc refer to the pure point, absolutely continuous, and singularly continuous part of the measure ρ_{+} (with respect to Lebesgue measure), respectively.

Then the pure point, absolutely continuous, and singular continuous spectra of H_{+} are given by (see Lemma B.5)

σ(H_{+}) = {λ ∈ ℝ|λ is a growth point of ρ_{+}},
σ_{pp}(H_{+}) = {λ ∈ ℝ|λ is a growth point of ρ_{+,pp}},
σ_{ac}(H_{+}) = {λ ∈ ℝ|λ is a growth point of ρ_{+,ac}},
(2.106)  σ_{sc}(H_{+}) = {λ ∈ ℝ|λ is a growth point of ρ_{+,sc}}.

Recall that σ_{pp}(H_{+}) is in general not equal to the point spectrum σ_{p}(H_{+}) (i.e., the set of eigenvalues of H_{+}). However, we have at least

(2.107)  σ_{pp}(H_{+}) = \overline{σ_{p}(H_{+})},

where the bar denotes closure.

An additional decomposition in continuous and singular part with respect to the Hausdorff measure dh^{α} (see Appendix B) will be of importance as well,

(2.108)  dρ_{+} = dρ_{+,αc} + dρ_{+,αs}.

The corresponding spectra are defined analogously

σ_{αc}(H_{+}) = {λ ∈ ℝ|λ is a growth point of ρ_{+,αc}},
σ_{αs}(H_{+}) = {λ ∈ ℝ|λ is a growth point of ρ_{+,αs}}.

They will be used in Section 3.3.

Finally, we show how a^{2}, b can be reconstructed from the measure ρ_{+}. In fact, even the moments m_{+,j}, j ∈ ℕ, are sufficient for this task. This is generally known as (Hamburger) moment problem.

Suppose we have a given sequence m_{j}, j ∈ ℕ_{0}, such that

(2.109)  C(k) = det ( \begin{matrix}m_{0} & m_{1} & ⋯ & m_{k−1} \\ m_{1} & m_{2} & ⋯ & m_{k} \\ ⋮ & ⋮ & ... & ⋮ \\ m_{k−1} & m_{k} & ⋯ & m_{2k−2} \end{matrix} ) > 0, k ∈ ℕ.

Without restriction we will assume m_{0} = 1. Using this we can define a sesquilinear form on the set of polynomials as follows

(2.110)  ⟨P (λ), Q(λ)⟩_{L^{2}} = ∑_{j,k=0}^{∞} m_{j+k}\overline{p_{j}}q_{k},

where P (z) = ∑_{j=0}^{∞} p_{j}z^{j}, Q(z) = ∑_{j=0}^{∞} q_{j}z^{j} (note that all sums are finite). The polynomials

(2.111) s(z, k) = \frac{1}{\sqrt{C(k − 1)C(k)}} det ( \begin{matrix}m_{0} & m_{1} & ⋯ & m_{k−1} \\ m_{1} & m_{2} & ⋯ & m_{k} \\ ⋮ & ⋮ & ... & ⋮ \\ m_{k−2} & m_{k−1} & ⋯ & m_{2k−3} \\ 1 & z & ⋯ & z^{k−1} \end{matrix} ) , k ∈ ℕ

(set C(0) = 1), form a basis for the set of polynomials which is immediate from

(2.112)  s(z, k) = \sqrt{\frac{C(k − 1)}{C(k)}} (z^{k−1} + \frac{D(k − 1)}{C(k − 1)}z^{k−2} + O(z^{k−3})) ,

where D(0) = 0, D(1) = m_{1}, and

(2.113)  D(k) = det ( \begin{matrix}m_{0} & m_{1} & ⋯ & m_{k−2} & m_{k} \\ m_{1} & m_{2} & ⋯ & m_{k−1} & m_{k+1} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ m_{k−1} & m_{k} & ⋯ & m_{2k−3} & m_{2k−1} \end{matrix} ) , k ∈ ℕ.

Moreover, this basis is orthonormal, that is,

(2.114)  ⟨s(λ, j), s(λ, k)⟩_{L^{2}} = δ_{j,k},

since

⟨s(λ, k), λ^{j}⟩ = \frac{1}{\sqrt{C(k − 1)C(k)}} det ( \begin{matrix}m_{0} & m_{1} & ⋯ & m_{k−1} \\ m_{1} & m_{2} & ⋯ & m_{k} \\ ⋮ & ⋮ & ... & ⋮ \\ m_{k−2} & m_{k−1} & ⋯ & m_{2k−3} \\ m_{j} & m_{j+1} & ⋯ & m_{j+k−1} \end{matrix} )

(2.115)  = \{ \begin{matrix}0, & 0 ≤ j ≤ k − 2 \\ \sqrt{\frac{C(k)}{C(k−1)}}, & j = k − 1 \end{matrix} .

In particular, the sesquilinear form (2.110) is positive definite and hence an inner product (note that C(k) > 0 is also necessary for this).

Expanding the polynomial zs(z, k) in terms of s(z, j), j ∈ ℕ, we infer

zs(z, k) = ∑_{j=0}^{k+1}⟨s(λ, j), λs(λ, k)⟩_{L^{2}}s(z, j)
(2.116)  = a(k)s(z, k + 1) + b(k)s(z, k) + a(k − 1)s(z, k − 1)

(set s(z, 0) = 0) with

(2.117) a(k) = ⟨s(λ, k + 1), λs(λ, k)⟩_{L^{2}}, b(k) = ⟨s(λ, k), λs(λ, k)⟩_{L^{2}}, k ∈ ℕ.

In addition, comparing powers of z in (2.116) shows

(2.118)  a(k) = \frac{\sqrt{C(k − 1)C(k + 1)}}{C(k)}, b(k) = \frac{D(k)}{C(k)} − \frac{D(k − 1)}{C(k − 1)}.

In terms of our original setting this says that given the measure dρ_{+} (or its moments, m_{+,j}, j ∈ ℕ) we can compute s(λ, n), n ∈ ℕ, via orthonormalization of the set λ^{n}, n ∈ ℕ_{0}. This fixes s(λ, n) up to a sign if we require s(λ, n) real-valued. Then we can compute a(n), b(n) as above (up to the sign of a(n) which changes if we change the sign of s(λ, n)). Summarizing, dρ_{+} uniquely determines a(n)^{2} and b(n). Since knowing dρ_{+}(λ) is equivalent to knowing m_{+}(z), the same is true for m_{+}(z) (compare also the proof of Theorem 2.29). In fact, we have an even stronger result.

Theorem 2.13. Suppose that the bounded measure dρ_{+} is not supported on a finite set. Then there exists a unique bounded Jacobi operator H_{+} having dρ_{+} as spectral measure.

Proof. We have already seen that the necessary and sufficient condition for our reconstruction procedure to work is that the sesquilinear form generated by the moments m_{j} of dρ_{+} is positive definite. Pick any nonzero polynomial P (λ). Due to our assumption we can find ε > 0 and an interval I such that ρ_{+}(I) ≠ 0 and P (λ)^{2} ≥ ε, λ ∈ I. Hence ⟨P (λ), P (λ)⟩ ≥ ερ_{+}(I).

As a consequence we can define a(n), b(n), s(λ, n), and the unitary transform Ũ as before. By construction H_{+} = Ũ^{−1}H̃Ũ is a bounded Jacobi operator associated with a(n), b(n). That ρ_{+} is the spectral measure of H_{+} follows from (using (Ũδ_{1})(λ) = 1)

(2.119)  ⟨δ_{1}, P_{Λ}(H_{+})δ_{1}⟩ = ⟨Ũδ_{1}, P_{Λ}(H̃)Ũδ_{1}⟩ = ∫_{ℝ} χ_{Λ}(λ)dρ_{+}(λ)

for any Borel set Λ ⊆ ℝ. □

If dρ_{+} is supported on N points, the reconstruction procedure will break down after N steps (i.e., C(N + 1) = 0) and we get a finite Jacobi matrix with dρ_{+} as spectral measure.

We also remark

(2.120)  c(z, n) = −a(0) ∫_{ℝ} \frac{s(z, n) − s(λ, n)}{z − λ} dρ_{+}(λ), n ∈ ℕ,

since one easily verifies τ_{+}c(z, n) = zc(z, n) − a(0)δ_{0}(n), n ∈ ℕ (use (2.98) with m = 1). Moreover, this implies

(2.121)  u_{+}(z, n) = \frac{c(z, n)}{a(0)} − m_{+}(z)s(z, n) = ∫_{ℝ} \frac{s(λ, n)}{z − λ} dρ_{+}(λ)

and it is not hard to verify

(2.122)  G_{+}(z, n, m) = ∫_{ℝ} \frac{s(λ, n)s(λ, m)}{λ − z}dρ_{+}(λ).

The Jacobi operator H can be treated along the same lines. Since we essentially repeat the analysis of H_{+} we will be more sketchy.

Consider the vector valued polynomials

(2.123)  \underline{S}(z, n) = (c(z, n), s(z, n)).

The analog of (2.94) reads

(2.124)  s(H, n)δ_{1} + c(H, n)δ_{0} = δ_{n}.

This is obvious for n = 0, 1 and the rest follows from induction upon applying H to (2.124). We introduce the spectral measures

(2.125)  dρ_{i,j}(.) = d⟨δ_{i}, P_{(−∞,λ]}(H)δ_{j}⟩,

and the (hermitian) matrix valued measure

(2.126)  dρ = ( \begin{matrix}dρ_{0,0} & dρ_{0,1} \\ dρ_{1,0} & dρ_{1,1} \end{matrix} ) .

The diagonal part consists of positive measures and the off-diagonal part can be written as the difference of two positive measures

(2.127)  dρ_{0,1}(λ) = dρ_{1,0}(λ) = dρ_{0,1,+}(λ) − dρ_{0,1,−}(λ),

where

dρ_{0,1,+}(λ) = \frac{1}{2}d⟨(δ_{0} + δ_{1}), P_{(−∞,λ]}(H)(δ_{0} + δ_{1})⟩,
(2.128) dρ_{0,1,−}(λ) = \frac{1}{2}(d⟨δ_{0}, P_{(−∞,λ]}(H)δ_{0}⟩ + d⟨δ_{1}, P_{(−∞,λ]}(H)δ_{1}⟩).

Moreover, dρ is a positive matrix measure and we have a corresponding Hilbert space L^{2}(ℝ, ℂ^{2}, dρ) with scalar product given by

(2.129)  ⟨\underline{F} , \underline{G}⟩_{L^{2}} = ∑_{i,j=0}^{1} ∫_{ℝ} \overline{F_{i}(λ)}G_{j}(λ)dρ_{i,j}(λ) ≡ ∫_{ℝ} \overline{\underline{F} (λ)}\underline{G}(λ)dρ(λ)

and if \underline{F} , \underline{G} are vector valued polynomials, then

(2.130)  ⟨\underline{F} , \underline{G}⟩_{L^{2}} = ⟨F_{0}(H)δ_{0} + F_{1}(H)δ_{1}, G_{0}(H)δ_{0} + G_{1}(H)δ_{1}⟩.

By (2.124) the vector valued polynomials \underline{S}(λ, n) are orthogonal with respect to dρ,

(2.131)  ⟨\underline{S}(., m), \underline{S}(., n)⟩_{L^{2}} = δ_{m,n}.

The formulas analogous to (2.117) then read

(2.132) a(n) = ⟨\underline{S}(λ, n + 1), λ\underline{S}(λ, n)⟩_{L^{2}}, b(n) = ⟨\underline{S}(λ, n), λ\underline{S}(λ, n)⟩_{L^{2}}, n ∈ ℤ.

Next, we consider the following transformation U from the set ℓ_{0}(ℤ) onto the set of vector-valued polynomials (eigenfunction expansion)

(Uf)(λ) = ∑_{n∈ℤ} f(n)\underline{S}(λ, n),

(2.133)  (U^{−1}\underline{F})(n) = ∫_{ℝ} \underline{S}(λ, n)\underline{F} (λ)dρ(λ).

Again a simple calculation for \underline{F} (λ) = (Uf)(λ) shows that U is unitary,

(2.134)  ∑_{n∈ℤ} |f(n)|^{2} = ∫_{ℝ} \overline{\underline{F} (λ)}\underline{F} (λ)dρ(λ).

Extending U to a unitary transformation Ũ we obtain as in the case of H_{+} the following

Theorem 2.14. The unitary transformation

(2.135)  Ũ : \begin{matrix}ℓ^{2}(ℤ) & → & L^{2}(ℝ, ℂ^{2}, dρ) \\ f(n) & ↦ & ∑_{n=−∞}^{∞} f(n)\underline{S}(λ, n) \end{matrix}

(where the sum is to be understood as norm limit) maps the operator H to the multiplication operator by λ, that is,

(2.136)  H̃ = ŨHŨ^{−1},

where

(2.137)  \begin{aligned}H̃ : & L^{2}(ℝ, ℂ^{2}, dρ) & → & L^{2}(ℝ, ℂ^{2}, dρ) \\ & \underline{F} (λ) & ↦ & λ\underline{F} (λ) \end{aligned} .

For the Green function of H we obtain

(2.138)  G(z, n, m) = ∫_{ℝ} \frac{\underline{S}(λ, n)\underline{S}(λ, m)}{λ − z}dρ(λ).

By Lemma B.13, in order to characterize the spectrum of H one only needs to consider the trace dρ^{tr} of dρ given by

(2.139)  dρ^{tr} = dρ_{0,0} + dρ_{1,1}.

Let the Lebesgue decomposition (cf. (2.105)) of dρ^{tr} be given by

(2.140)  dρ^{tr} = dρ_{pp}^{tr} + dρ_{ac}^{tr} + dρ_{sc}^{tr},

then the pure point, absolutely continuous, and singular continuous spectra of H are given by

σ(H) = {λ ∈ ℝ|λ is a growth point of ρ^{tr}},
σ_{pp}(H) = {λ ∈ ℝ|λ is a growth point of ρ_{pp}^{tr}},
σ_{ac}(H) = {λ ∈ ℝ|λ is a growth point of ρ_{ac}^{tr}},
(2.141)  σ_{sc}(H) = {λ ∈ ℝ|λ is a growth point of ρ_{sc}^{tr}}.

The Weyl matrix M(z) is defined as

M(z) = ∫_{−∞}^{∞} \frac{dρ(λ)}{λ − z} − \frac{1}{2a(0)} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

(2.142)  = ( \begin{matrix}g(z, 0) & \frac{h(z,0)}{2a(0)} \\ \frac{h(z,0)}{2a(0)} & g(z, 1) \end{matrix} ) ,  z ∈ ℂ\σ(H).

Explicit evaluation yields

(2.143)  M(z) = \frac{1}{m̃_{+}(z) + m̃_{−}(z)} ( \begin{matrix}−\frac{1}{a(0)^{2}} & \frac{m̃_{+}(z)−m̃_{−}(z)}{2a(0)} \\ \frac{m̃_{+}(z)−m̃_{−}(z)}{2a(0)} & m̃_{+}(z)m̃_{−}(z) \end{matrix} ) ,

and the determinant reads

(2.144)  det M(z) = −\frac{1}{4a(0)^{2}} .

In terms of the original Weyl m-functions we obtain

g(z, 0) = \frac{−1}{z − b(0) − a(0)^{2}m_{+}(z) − a(−1)^{2}m_{−}(z)},
(2.145)  h(z, 0) = (z − b(0) + a(0)^{2}m_{+}(z) − a(−1)^{2}m_{−}(z))g(z, 0).

Finally, notice that we can replace c(z, n), s(z, n) by any other pair of linearly independent solutions. For example, we could use c_{β}(z, n), s_{β}(z, n). As in (2.123) we define

(2.146)  \underline{S}_{β}(z, n) = (c_{β}(z, n), s_{β}(z, n)) = U_{α}\underline{S}(z, n),

where U_{α} is rotation by the angle α (β = cot α), that is,

(2.147)  U_{α} = ( \begin{matrix}cos α & sin α \\ − sin α & cos α \end{matrix} ) .

Hence all objects need to be rotated by the angle α. For instance, introducing M^{β}(z) = U_{α}M(z)U_{α}^{−1} we infer

(2.148) M^{β}(z) = \frac{1}{m̃_{+}^{β}(z) + m̃_{−}^{β}(z)} ( \begin{matrix}−\frac{1}{a(0)^{2}} & \frac{m̃_{+}^{β}(z)−m̃_{−}^{β}(z)}{2a(0)} \\ \frac{m̃_{+}^{β}(z)−m̃_{−}^{β}(z)}{2a(0)} & m̃_{+}^{β}(z)m̃_{−}^{β}(z) \end{matrix} ) .

Note also

(2.149)  \frac{−1}{a(0)^{2}(m̃_{+}^{β}(z) + m̃_{−}^{β}(z))} = sin^{2}(α) γ^{β}(z, 0).

If we restrict ourselves to the absolutely continuous part of the spectrum, we can do even more. We abbreviate m̃_{±}(λ) = lim_{ε↓0} m̃_{±}(λ + iε), λ ∈ ℝ, implying (cf. Appendix B)

dρ_{0,0,ac}(λ) = \frac{Im(m̃_{+}(λ) + m̃_{−}(λ))}{πa(0)^{2}|m̃_{+}(λ) + m̃_{−}(λ)|^{2}} dλ,
dρ_{0,1,ac}(λ) = \frac{Im(m̃_{+}(λ))Re(m̃_{−}(λ)) − Re(m̃_{+}(λ))Im(m̃_{−}(λ))}{πa(0)|m̃_{+}(λ) + m̃_{−}(λ)|^{2}}dλ,
(2.150) dρ_{1,1,ac}(λ) = \frac{Im(|m̃_{−}(λ)|^{2}m̃_{+}(λ) + |m̃_{−}(λ)|^{2}m̃_{+}(λ))}{π|m̃_{+}(λ) + m̃_{−}(λ)|^{2}}dλ.

Choosing the new basis

(2.151)  ( \begin{matrix}u_{+}(λ, n) \\ u_{−}(λ, n) \end{matrix} ) = V (λ) ( \begin{matrix}c(λ, n) \\ s(λ, n) \end{matrix} )

with

(2.152)  V (λ) = \frac{1}{m̃_{+}(λ) + m̃_{−}(λ)} ( \begin{matrix}\frac{1}{a(0)} & −m̃_{+}(λ) \\ \frac{1}{a(0)} & m̃_{−}(λ) \end{matrix} )

we get

(2.153)  (V^{−1}(λ))^{∗} dρ_{ac}(λ) V^{−1}(λ) = \frac{1}{π} ( \begin{matrix}Im(m̃_{−}(λ)) & 0 \\ 0 & Im(m̃_{+}(λ)) \end{matrix} ) dλ.

Note that V (λ) is not unitary. We will show how to diagonalize all of dρ in Section 3.1.

2.6. Some remarks on unbounded operators

In this section we temporarily drop the boundedness assumption on the coefficients a, b. This renders H unbounded and implies that we are no longer able to define H on all of ℓ^{2}(ℤ). Nevertheless we can define the minimal and maximal operator allied with τ as follows

(2.154)  \begin{aligned}H_{min} : & 𝔇(H_{min}) & → & ℓ^{2}(ℤ) \\ & f & ↦ & τf \end{aligned} , \begin{aligned}H_{max} : & 𝔇(H_{max}) & → & ℓ^{2}(ℤ) \\ & f & ↦ & τf \end{aligned} ,

with

(2.155)  𝔇(H_{min}) = ℓ_{0}(ℤ),  𝔇(H_{max}) = {f ∈ ℓ^{2}(ℤ)|τf ∈ ℓ^{2}(ℤ)}.

By Green’s formula (1.20) we have H_{min}^{∗} = H_{max} and

(2.156)  \begin{aligned}H_{max}^{∗} = \overline{H_{min}} : & 𝔇(H_{max}^{∗}) & → & ℓ^{2}(ℤ) \\ & f & ↦ & τf \end{aligned}  ,

with

(2.157)  𝔇(H_{max}^{∗}) = \{f ∈ 𝔇(H_{max})| lim_{n→±∞} W_{n}(\overline{f}, g) = 0, g ∈ 𝔇(H_{max})\}.

Here H^{∗}, \overline{H} denote the adjoint, closure of an operator H, respectively. We also remark that lim_{n→±∞} W_{n}(\overline{f}, g) exists for f, g ∈ 𝔇(H_{max}) as can be easily shown using (1.20). Similar definitions apply to H_{±}.

Since we might have H_{max}^{∗} ≠ H_{max}, H_{max} might not be self-adjoint in general. The key to this problem will be the limit point (l.p.), limit circle (l.c.) classification alluded to in Remark 2.11 (ii). To make things precise, we call τ l.p. at ±∞ if s(z_{0}, .) ∉ ℓ^{2}(±ℕ) for some z_{0} ∈ ℂ\ℝ. Otherwise τ is called l.c. at ±∞.

In order to draw some first consequences from this definition we note that all considerations of Section 2.4 do not use boundedness of a, b except for (2.85). However, if ∑_{n∈ℕ} s(z_{0}, n) < ∞ (considering only β = ∞ for simplicity), then the circle corresponding to m̃_{N}(z_{0}) converges to a circle instead of a point as N → ∞. If m̃_{±}(z_{0}) is defined to be any point on this limiting circle, everything else remains unchanged. In particular, u_{+}(z_{0}, .) ∈ ℓ^{2}(ℕ) and s(z_{0}, .) ∈ ℓ^{2}(ℕ) shows that every solution of τu = z_{0}u is in ℓ^{2}(ℕ) if τ is l.c. at +∞.

This enables us to reveal the connections between the l.p. / l.c. characterization and the self-adjointness of unbounded Jacobi operators. We first consider H_{±}.

We recall that H_{min,+} is symmetric and that (H_{min,+})^{∗} = H_{max,+}. We need to investigate the deficiency indices d_{±} = dim Ker(H_{max,+} − z_{±}), z_{±} ∈ ℂ_{±}, of H_{min,+}. They are independent of z_{±} ∈ ℂ_{±} and equal (i.e., d_{−} = d_{+}) since H_{min,+} is real (cf. [192], [241], Chapter 8). Thus, to compute d_{±} it suffices to consider Ker(H_{max,+} − z_{0}). Since any element of Ker(H_{max,+} − z_{0}) is a multiple of s(z_{0}) we infer d_{−} = d_{+} = 0 if s(z_{0}) ∉ ℓ^{2}(ℕ) and d_{−} = d_{+} = 1 if s(z_{0}) ∈ ℓ^{2}(ℕ). This shows that H_{max,+} is self-adjoint if and only if τ is l.p. at +∞. Moreover, s(z_{0}) ∈ ℓ^{2}(ℕ) implies s(z) ∈ ℓ^{2}(ℕ) for all z ∈ ℂ\ℝ since d_{±} = 1 independent of z ∈ ℂ\ℝ. Or, put differently, the l.p. / l.c. definition is independent of z ∈ ℂ\ℝ.

If τ is l.c. at ±∞, then our considerations imply that all solutions of (1.19) for all z ∈ ℂ\ℝ are square summable near ±∞, respectively. This is even true for all z ∈ ℂ.

Lemma 2.15. Suppose that all solutions of τu = zu are square summable near ±∞ for one value z = z_{0} ∈ ℂ. Then this is true for all z ∈ ℂ.

Proof. If u fulfills (1.19), we may apply (1.48) to (τ − z_{0})u = (z − z_{0})u,

u(n) = u(n_{0})c(z_{0}, n, n_{0}) + u(n_{0} + 1)s(z_{0}, n, n_{0})

−\frac{z − z_{0}}{a(0)}\begin{matrix}n \\ ∑ & ∗ \\ j=n_{0}+1 \end{matrix} (c(z_{0}, n)s(z_{0}, j) − c(z_{0}, j)s(z_{0}, n))u(j). (2.158)

By assumption, there exists a constant M ≥ 0 such that

(2.159)  ∑_{j=n_{0}+1}^{∞} |c(z_{0}, j)|^{2} ≤ M,  ∑_{j=n_{0}+1}^{∞} |s(z_{0}, j)|^{2} ≤ M.

Invoking the Cauchy-Schwarz inequality we obtain the estimate (n > n_{0})

| ∑_{j=n_{0}+1}^{n}(c(z_{0}, n)s(z_{0}, j) − c(z_{0}, j)s(z_{0}, n))u(j)|^{2}

≤ ∑_{j=n_{0}+1}^{n} |c(z_{0}, n)s(z_{0}, j) − c(z_{0}, j)s(z_{0}, n)|^{2} ∑_{j=n_{0}+1}^{n} |u(j)|^{2}

≤ M(|c(z_{0}, n)|^{2} + |s(z_{0}, n)|^{2}) ∑_{j=n_{0}+1}^{n} |u(j)|^{2}. (2.160)

Since n_{0} is arbitrary we may choose n_{0} in (2.159) so large, that we have 4a(0)^{−1}|z − z_{0}|M^{2} ≤ 1. Again using Cauchy-Schwarz

∑_{j=n_{0}+1}^{n} |u(j)|^{2} ≤ (|u(n_{0})|^{2} + |u(n_{0} + 1)|^{2})M̃ + \frac{2|z − z_{0}|}{a(0)}M^{2} ∑_{j=n_{0}+1}^{n} |u(j)|^{2}

(2.161)  ≤ (|u(n_{0})|^{2} + |u(n_{0} + 1)|^{2})M̃ + \frac{1}{2} ∑_{j=n_{0}+1}^{n} |u(j)|^{2},

where M̃ is chosen such that ∑_{j=n_{0}+1}^{∞} |c(z_{0}, j, n_{0})|^{2} ≤ M̃, ∑_{j=n_{0}+1}^{∞} |s(z_{0}, j, n_{0})|^{2} ≤ M̃ holds. Solving for the left hand side finishes the proof,

(2.162)  ∑_{j=n_{0}+1}^{n} |u(j)|^{2} ≤ 2(|u(n_{0})|^{2} + |u(n_{0} + 1)|^{2})M̃.
□

In summary, we have the following lemma.

Lemma 2.16. The operator H_{max,±} is self-adjoint if and only if one of the following statements holds.

(i) τ is l.p. at ±∞.

(ii) There is a solution of (1.19) for some z ∈ ℂ (and hence for all) which is

not square summable near ±∞. (iii) W_{±∞}(f, g) = 0, for all f, g ∈ 𝔇(H_{max,±}).

To simplify notation, we will only consider the endpoint +∞ in the following. The necessary modifications for −∞ are straightforward.

Next, let us show a simple but useful criterion for τ being l.p. at +∞. If τ is l.c. at +∞, we can use the Wronskian

(2.163)  \frac{a(0)}{a(n)} = c(z, n)s(z, n + 1) − c(z, n + 1)s(z, n),

to get (using the Cauchy-Schwarz inequality)

(2.164)  ∑_{n∈ℕ} \frac{1}{|a(n)|} ≤ \frac{2}{|a(0)|}\sqrt{∑_{n∈ℕ} |c(z, n)|^{2} ∑_{n∈ℕ} |s(z, n)|^{2}}.

This shows that a sufficient condition for τ to be l.p. at +∞ is

(2.165)  ∑_{n∈ℕ} \frac{1}{|a(n)|} = ∞.

The remaining question is: What happens in the l.c. case? Obviously we need some suitable boundary conditions. The boundary condition

(2.166)  BC_{n_{0},α}(f) = cos(α)f(n_{0}) + sin(α)f(n_{0} + 1) = 0

of Remark 1.9 makes no sense if n_{0} = ±∞. However, (2.166) can be written as W_{n_{0}}(v, f) = 0, where v is any sequence satisfying BC_{n_{0},α}(v) = 0 and |v(n_{0})| + |v(n_{0} + 1)| ≠ 0. Moreover, all different boundary conditions can be obtained by picking v such that W_{n_{0}}(\overline{v}, v) = 0 and W_{n_{0}}(v, f) ≠ 0 for some f. This latter characterization of the boundary condition may be generalized.

We define the set of boundary conditions for τ at +∞ by

(2.167) BC_{+}(τ) = \{v ∈ 𝔇(H_{max,+})| \begin{matrix}W_{+∞}(\overline{v}, v) = 0, W_{+∞}(\overline{v}, f) ≠ 0 for some \\ f ∈ 𝔇(H_{max,+}) if τ is l.c. at + ∞\}. \end{matrix}

Observe that the first requirement holds if v is real. The second is void if τ is l.p. (at +∞). Otherwise, if τ is l.c., there is at least one real v for which it holds (if not, (iii) of Lemma 2.16 implies that τ is l.p.). Two sequences v_{1,2} ∈ BC_{+}(τ) are called equivalent if W_{+∞}(v_{1}, v_{2}) = 0.

Lemma 2.17. Let v ∈ BC_{+}(τ) and set

(2.168)  𝔇_{+}(v) = {f ∈ 𝔇(H_{max,+})|W_{+∞}(v, f) = 0}.

Then

(i) W_{+∞}(v, f) = 0 ⇔ W_{+∞}(v, \overline{f}) = 0,
(ii) W_{+∞}(g, f) = 0 for f, g ∈ 𝔇_{+}(v).

Moreover, W_{+∞}(v_{1}, v_{2}) = 0 is equivalent to 𝔇_{+}(v_{1}) = 𝔇_{+}(v_{2}).

Proof. For all f_{1}, …, f_{4} ∈ 𝔇(H_{max}) we can take the limits n → +∞ in the Plücker identity

(2.169) W_{n}(f_{1}, f_{2})W_{n}(f_{3}, f_{4}) + W_{n}(f_{1}, f_{3})W_{n}(f_{4}, f_{2}) + W_{n}(f_{1}, f_{4})W_{n}(f_{2}, f_{3}) = 0.

Now choose f_{1} = v, f_{2} = f̂, f_{3} = \overline{v}, f_{4} = f to conclude W_{+∞}(v, f) = 0 implies W_{+∞}(\overline{v}, f) = 0. Then choose f_{1} = v, f_{2} = f̂, f_{3} = f, f_{4} = \overline{g} to show (ii). The last assertion follows from (ii) upon choosing v = v_{1}, g = v_{2}. □

Combining this lemma with Green’s formula (1.20) shows

Theorem 2.18. Choose v ∈ BC_{+}(τ), then

(2.170)  H_{+} : \begin{matrix}𝔇_{+}(v) & → & ℓ^{2}(ℕ) \\ f & ↦ & τf \end{matrix}

is a self-adjoint extension of H_{min,+}.

In the l.p. case the boundary condition W_{+∞}(v, f) = 0 is of course always fulfilled and thus 𝔇_{+}(v) = 𝔇(H_{max,+}) for any v ∈ BC_{+}(τ).

Clearly, we can also define self-adjoint operators H_{n_{0},+} and H_{n_{0},+}^{β} corresponding to H_{+} as we did in Section 1.2 for the bounded case.

Now, that we have found self-adjoint extensions, let us come back to the Weyl m-functions. We fix v(n) = s_{β}(λ, n), λ ∈ ℝ, for the boundary condition at +∞. Observe that

(2.171)  m̃_{+}^{β}(z) = \frac{1}{a(0)} lim_{n→+∞} \frac{W_{n}(s_{β}(λ), c_{β}(z))}{W_{n}(s_{β}(λ), s_{β}(z))}

lies on the limiting circle. This is clear if τ is l.p. at +∞. Otherwise, τ l.c. at +∞ implies c_{β}(z), s_{β}(z) ∈ 𝔇(H_{max,+}) and both Wronskians converge to a limit as pointed out earlier in this section. Moreover, if W_{n}(s_{β}(λ), s_{β}(z)) = 0, then s_{β}(z) ∈ 𝔇(H_{+}^{β}) and hence z ∈ σ(H_{+}^{β}). In particular, W_{n}(s_{β}(λ), s_{β}(z)) ≠ 0 for z ∈ ℂ\ℝ and we can call m̃_{+}^{β}(z) the Weyl m̃-function of H_{+}^{β}.

In addition, the function

(2.172)  u_{+}(z, n) = \frac{c_{β}(z, n)}{a(0)} ∓ m̃_{+}^{β}(z)s_{β}(z, n)

is in ℓ_{+}^{2}(ℤ) and satisfies the boundary condition

(2.173)  W_{+∞}(s_{β}(λ), u_{±}(z)) = 0.

The boundary condition uniquely characterizes u_{+}(z, n) up to a constant in the l.c. case.

We have seen that the question of self-adjointness is simple if τ is l.p.. One the other hand, the spectrum gets simple if τ is l.c..

Lemma 2.19. If τ is l.c. at +∞, then the resolvent of H_{+} is a Hilbert-Schmidt operator. In particular, this implies that H_{+} has purely discrete spectrum, σ(H_{+}) = σ_{d}(H_{+}), and

(2.174)  ∑_{λ∈σ_{d}(H_{+})} \frac{1}{1 + λ^{2}} < ∞.

Proof. The result is a consequence of the following estimate

∑_{(n,m)∈ℤ^{2}} |G_{+}(z, m, n)|^{2} = \frac{1}{|W |^{2}} ∑_{n∈ℤ} (|u_{+}(z, n)|^{2} ∑_{m<n} |s(z, m)|^{2}

+ |s(z, n)|^{2} ∑_{m≥n} |u_{+}(z, m)|^{2}) ≤ \frac{2}{|W |^{2}} ∥u_{+}(z)∥^{2} ∥s(z)∥^{2},

where W = W (s(z), u_{+}(z)). □

Our next goal is to find a good parameterization of all self-adjoint extensions if τ is l.c. at +∞.

First of all note that any real solution of τu = λu, λ ∈ ℝ, is in BC_{+}(τ) (since W (u, ũ) ≠ 0 for any linearly independent solution of τu = λu). Now fix

(2.175)  v_{α}(n) = cos(α)c(0, n) + sin(α)s(0, n),  α ∈ [0, π),

and note that different α’s imply different extensions since W (v_{α_{1}}, v_{α_{2}}) = sin(α_{2} − α_{1})/a(0).

Lemma 2.20. All self-adjoint extensions of H_{min,+} correspond to some v_{α} with unique α ∈ [0, π).

Proof. Let H_{+} be a self-adjoint extension of H_{min,+} and λ_{0} ∈ σ(H_{+}) be an eigenvalue with corresponding eigenfunction s(λ_{0}, n). Using Green’s formula (1.20) with f = s(λ_{0}), g ∈ 𝔇(H_{+}) we see 𝔇(H_{+}) ⊆ 𝔇_{+}(s(λ_{0})) and hence 𝔇(H_{+}) = 𝔇_{+}(s(λ_{0})) by maximality of self-adjoint operators. Let α ∈ [0, π) be the unique value for which

(2.176) W_{+∞}(v_{α}, s(λ_{0})) = cos(α)W_{+∞}(c(0), s(λ_{0}))+sin(α)W_{+∞}(s(0), s(λ_{0})) = 0.

Then 𝔇_{+}(s(λ_{0})) = 𝔇_{+}(v_{α}). □

Now we turn back again to operators on ℓ^{2}(ℤ). We use that corresponding definitions and results hold for the other endpoint −∞ as well. With this in mind we have

Theorem 2.21. Choose v_{±} ∈ BC_{±}(τ) as above, then the operator H with domain

(2.177)  𝔇(H) = {f ∈ 𝔇(H_{max})|W_{+∞}(v_{+}, f) = W_{−∞}(v_{−}, f) = 0}

is self-adjoint.

Again, if τ is l.p. at ±∞, the corresponding boundary condition is void and can be omitted. We also note that if τ is l.c. at both ±∞, then we have not found all self-adjoint extensions since we only consider separated boundary conditions (i.e., one for each endpoint) and not coupled ones which connect the behavior of functions at −∞ and +∞.

As before we have

Lemma 2.22. If τ is l.c. at both ±∞, then the resolvent of H is a Hilbert-Schmidt operator. In particular, the spectrum of H is purely discrete.

Most results found for bounded Jacobi operators still hold with minor modifications. One result that requires some changes is Theorem 2.13.

Theorem 2.23. A measure dρ_{+} which is not supported on a finite set is the spectral measure of a unique Jacobi operator H_{+} if and only if the set of polynomials is dense in L^{2}(ℝ, dρ_{+}).

Proof. If the set of polynomials is dense in L^{2}(ℝ, dρ_{+}) we can use the same proof as in Theorem 2.13 to show existence of a unique Jacobi operator with dρ_{+} as spectral measure.

Conversely, let H_{+} be given, and let U be a unitary transform mapping H_{+} to multiplication by λ in L^{2}(ℝ, dρ_{+}) (which exists by the spectral theorem). Then |(Uδ_{1})(λ)|^{2} = 1 since

(2.178) ∫_{Λ} dρ_{+} = ⟨δ_{1}, P_{Λ}(H_{+})δ_{1}⟩ = ⟨Uδ_{1}, P_{Λ}(λ)Uδ_{1}⟩ = ∫_{Λ} |(Uδ_{1})(λ)|^{2}dρ_{+}(λ)

for any Borel set Λ ⊆ ℝ. So, by another unitary transformation, we can assume (Uδ_{1})(λ) = 1. And since the span of (H_{+})^{j}δ_{1}, j ∈ ℕ_{0}, is dense in ℓ^{2}(ℤ), so is the span of (U(H_{+})^{j}δ_{1})(λ) = λ^{j} in L^{2}(ℝ, dρ_{+}). □

A measure dρ_{+} which is not supported on a finite set and for which the set of polynomials is dense in L^{2}(ℝ, dρ_{+}) will be called Jacobi measure.

There are some interesting consequences for the moment problem.

Lemma 2.24. A set {m_{+,j}}_{j∈ℕ} forms the moments of a Jacobi measure if and only if (2.109) holds.

Moreover,

(2.179)  supp(ρ_{+}) ⊆ [−R, R] ⇔ |m_{+,j}| ≤ R^{j}, j ∈ ℕ.

Proof. If (2.109) holds we get sequences a(n), b(n) by (2.118). The spectral measure of any self-adjoint extension has m_{+,j} as moments.

If supp(ρ_{+}) ⊆ [−R, R], then |m_{+,j}| ≤ ∫ |λ|^{j}dρ_{+}(λ) ≤ R^{j} ∫ dρ_{+}(λ) = R^{j}. Conversely, if supp(ρ_{+}) ⊈ [−R, R], then there is an ε > 0 such that C_{ε} = ρ_{+}({λ||λ| > R + ε}) > 0 and hence |m_{+,2j}| ≥ ∫_{|λ|>R+ε} λ^{2j}dρ_{+}(λ) ≥ C_{ε}(R + ε)^{2j}. □

Finally, let us look at uniqueness.

Theorem 2.25. A measure is uniquely determined by its moments if and only if the associated Jacobi difference expression τ (defined via (2.118)) is l.p. at +∞.

Proof. Our assumption implies that H_{min,+} is essentially self-adjoint and hence (H_{min,+} − z)𝔇(H_{min,+}) is dense in ℓ^{2}(ℕ) for any z ∈ ℂ_{±}. Denote by ℌ_{0} the set of polynomials on ℝ and by ℌ the closure of ℌ_{0} with respect to the scalar product (2.110). Then (λ − z)ℌ_{0} is dense in ℌ and hence there is a sequence of polynomials P_{z,n}(λ), z ∈ ℂ_{±}, such that (λ − z)P_{z,n}(λ) converges to 1 in ℌ. Let ρ be a measure with correct moments. Then

(2.180)  ∫_{ℝ} |P_{z,n}(λ) − \frac{1}{λ − z} |^{2}dρ(λ) ≤ ∫_{ℝ} \frac{|λ − z|^{2}}{Im(z)^{2}} |P_{z,n}(λ) − \frac{1}{λ − z} |^{2}dρ(λ)

shows that P_{z,n}(λ) converges to (λ − z)^{−1} in L^{2}(ℝ, dρ) and consequently the Borel transform

(2.181)  ∫ \frac{dρ(λ)}{λ − z} = ⟨1, \frac{1}{λ − z} ⟩_{L^{2}} = lim_{n→∞}⟨1, P_{z,n}⟩

is uniquely determined by the moments. Since ρ is uniquely determined by its Borel transform we are done. □

We know that τ is l.p. at +∞ if the moments are polynomially bounded by (2.179). However, a weaker bound on the growth of the moments also suffices to ensure the l.p. case.

Lemma 2.26. Suppose

(2.182)  |m_{+,j}| ≤ CR^{j}j!, j ∈ ℕ,

then τ associated with {m_{+,j}}_{j∈ℕ} is l.p. at +∞.

Proof. Our estimate implies that e^{izλ} ∈ L^{1}(ℝ, dρ) for |Im(z)| < 1/R. Hence the Fourier transform

(2.183)  ∫_{ℝ} e^{izλ}dρ(λ) = ∑_{j=0}^{∞} m_{+,j} \frac{(iz)^{j}}{j!}

is holomorphic in the strip |Im(z)| < 1/R. This shows that the Fourier transform is uniquely determined by the moments and so is the Borel transform and hence the measure (see (B.9)). □

2.7. Inverse spectral theory

In this section we present a simple recursive method of reconstructing the sequences a^{2}, b when the Weyl matrix (cf. (2.142))

(2.184)  M(z, n) = ( \begin{matrix}g(z, n) & \frac{h(z,n)}{2a(n)} \\ \frac{h(z,n)}{2a(n)} & g(z, n + 1) \end{matrix} ) ,  z ∈ ℂ\σ(H),

is known for one fixed n ∈ ℤ. As a consequence, we are led to several uniqueness results.

By virtue of the Neumann series for the resolvent of H we infer (cf. (6.2) below and Section 6.1 for more details)

g(z, n) = −\frac{1}{z} − \frac{b(n)}{z^{2}} + O(\frac{1}{z^{3}} ),
(2.185)  h(z, n) = −1 − \frac{2a(n)^{2}}{z^{2}} + O(\frac{1}{z^{3}} ).

Hence a(n)^{2}, b(n) can be easily recovered as follows

b(n) = − lim_{z→∞} z(1 + zg(z, n)),
(2.186)  a(n)^{2} = −\frac{1}{2} lim_{z→∞} z^{2}(1 + h(z, n)).

Furthermore, we have the useful identities (use (1.100))

(2.187)  4a(n)^{2}g(z, n)g(z, n + 1) = h(z, n)^{2} − 1

and

(2.188)  h(z, n + 1) + h(z, n) = 2(z − b(n + 1))g(z, n + 1),

which show that g(z, n) and h(z, n) together with a(n)^{2} and b(n) can be determined recursively if, say, g(z, n_{0}) and h(z, n_{0}) are given.

In addition, we infer that a(n)^{2}, g(z, n), g(z, n + 1) determine h(z, n) up to one sign,

(2.189)  h(z, n) = (1 + 4a(n)^{2}g(z, n)g(z, n + 1))^{1/2},

since h(z, n) is holomorphic with respect to z ∈ ℂ\σ(H). The remaining sign can be determined from the asymptotic behavior h(z, n) = −1 + O(z^{−2}).

Hence we have proved the important result that M(z, n_{0}) determines the sequences a^{2}, b. In fact, we have proved the slightly stronger result:

Theorem 2.27. One of the following set of data

(i) g(., n_{0}) and h(., n_{0}) (ii) g(., n_{0} + 1) and h(., n_{0}) (iii) g(., n_{0}), g(., n_{0} + 1), and a(n_{0})^{2} for one fixed n_{0} ∈ ℤ uniquely determines the sequences a^{2} and b.

Remark 2.28. (i) Let me emphasize that the two diagonal elements g(z, n_{0}) and g(z, n_{0} + 1) alone plus a(n_{0})^{2} are sufficient to reconstruct a(n)^{2}, b(n). This is in contradistinction to the case of one-dimensional Schrödinger operators, where the diagonal elements of the Weyl matrix determine the potential only up to reflection.

You might wonder how the Weyl matrix of the operator H_{R} associated with the (at n_{0}) reflected coefficients a_{R}, b_{R} (cf. Lemma 1.7) look like. Since reflection at n_{0} exchanges m_{±}(z, n_{0}) (i.e., m_{R,±}(z, n_{0}) = m_{∓}(z, n_{0})) we infer

g_{R}(z, n_{0}) = g(z, n_{0}),
h_{R}(z, n_{0}) = −h(z, n_{0}) + 2(z − b(n_{0}))g(z, n_{0}),
g_{R}(z, n_{0} + 1) = \frac{a(n_{0})^{2}}{a(n_{0} − 1)^{2}} g(z, n_{0} + 1) + \frac{z − b(n_{0})}{a(n_{0} − 1)^{2}} (h(z, n_{0})

+ (z − b(n_{0}))g(z, n_{0})), (2.190)

in obvious notation. (ii) Remark 6.3(ii) will show that the sign of a(n) cannot be determined from either g(z, n_{0}), h(z, n_{0}), or g(z, n_{0} + 1).

The off-diagonal Green function can be recovered as follows

(2.191)  G(z, n + k, n) = g(z, n) ∏_{j=n}^{n+k−1} \frac{1 + h(z, j)}{2a(j)g(z, j)}, k > 0,

and we remark

a(n)^{2}g(z, n + 1) − a(n − 1)^{2}g(z, n − 1) + (z − b(n))^{2}g(z, n)

= (z − b(n))h(z, n). (2.192)

A similar procedure works for H_{+}. The asymptotic expansion

(2.193)  m_{+}(z, n) = −\frac{1}{z} − \frac{b(n + 1)}{z^{2}} − \frac{a(n + 1)^{2} + b(n + 1)^{2}}{z^{3}} + O(z^{−4})

shows that a(n + 1)^{2}, b(n + 1) can be recovered from m_{+}(z, n). In addition, (2.11) shows that m_{+}(z, n_{0}) determines a(n)^{2}, b(n), m_{+}(z, n), n > n_{0}. Similarly, (by reflection) m_{−}(z, n_{0}) determines a(n − 1)^{2}, b(n), m_{−}(z, n − 1), n < n_{0}. Hence both m_{±}(z, n_{0}) determine a(n)^{2}, b(n) except for a(n_{0} − 1)^{2}, a(n_{0})^{2}, b(n_{0}). However, since a(n_{0} − 1)^{2}, a(n_{0})^{2}, b(n_{0}), and m_{−}(z, n_{0}) can be computed from m̃_{−}(z, n_{0}) we conclude:

Theorem 2.29. The quantities m̃_{+}(z, n_{0}) and m̃_{−}(z, n_{0}) uniquely determine a(n)^{2} and b(n) for all n ∈ ℤ.

Next, we recall the function γ^{β}(z, n) introduced in (1.107) with asymptotic expansion

(2.194) γ^{β}(z, n) = −\frac{β}{a(n)} − \frac{1 + β^{2}}{z} − \frac{b(n + 1) + 2βa(n) + β^{2}b(n)}{z^{2}} + O(\frac{1}{z^{3}} ).

Our goal is to prove

Theorem 2.30. Let β_{1}, β_{2} ∈ ℝ ∪ {∞} with β_{1} ≠ β_{2} be given. Then γ^{β_{j}}(., n_{0}), j = 1, 2, for one fixed n_{0} ∈ ℤ uniquely determine a(n)^{2}, b(n) for all n ∈ ℤ (set γ^{∞}(z, n) = g(z, n)) unless (β_{1}, β_{2}) = (0, ∞), (∞, 0). In the latter case a(n_{0})^{2} is needed in addition. More explicitly, we have

g(z, n) = \frac{γ^{β_{1}}(z, n) + γ^{β_{2}}(z, n) + 2R(z)}{(β_{2} − β_{1})^{2}},
g(z, n + 1) = \frac{β_{2}^{2}γ^{β_{1}}(z, n) + β_{1}^{2}γ^{β_{2}}(z, n) + 2β_{1}β_{2}R(z)}{(β_{2} − β_{1})^{2}},
(2.195)  h(z, n) = \frac{β_{2}γ^{β_{1}}(z, n) + β_{1}γ^{β_{2}}(z, n) + (β_{1} + β_{2})R(z)}{(−2a(n))^{−1}(β_{2} − β_{1})^{2}},

where R(z) is the branch of

(2.196)  R(z) = (\frac{(β_{2} − β_{1})^{2}}{4a(n)^{2}} + γ^{β_{1}}(z, n)γ^{β_{2}}(z, n))^{1/2} = \frac{β_{1} + β_{2}}{2a(n)} + O(\frac{1}{z}),

which is holomorphic for z ∈ ℂ\σ(H) and has asymptotic behavior as indicated. If one of the numbers β_{1}, β_{2} equals ∞, one has to replace all formulas by their limit

using g(z, n) = lim_{β→∞} β^{−2}γ^{β}(z, n).

Proof. Clearly, if (β_{1}, β_{2}) ≠ (0, ∞), (∞, 0), we can determine a(n) from equation (2.194). Hence by Theorem 2.27 it suffices to show (2.195). Since the first equation follows from (2.187) and the other two, it remains to establish the last two equations in (2.195). For this we prove that the system

(2.197)  (g^{+})^{2} + 2\frac{β_{j}}{2a(n)}hg^{+} + \frac{β_{j}^{2}}{4a(n)^{2}} (h^{2} − 1) = g^{+}γ^{β_{j}}(z, n), j = 1, 2,

has a unique solution (g^{+}, h) = (g(z, n + 1), h(z, n)) for |z| large enough which is holomorphic with respect to z and satisfies the asymptotic requirements (2.185). We first consider the case β_{j} ≠ 0, ∞. Changing to new variables (x_{1}, x_{2}), x_{j} = (2a(n)/β_{j})g^{+} + h, our system reads

(2.198)  x_{j}^{2} − 1 = \frac{β_{1}β_{2}}{β_{j}^{2}} \frac{2a(n)γ^{β_{j}}(z, n)}{β_{2} − β_{1}}(x_{1} − x_{2}), j = 1, 2.

Picking |z| large enough we can assume γ^{β_{j}}(z, n) ≠ 0 and the solution set of the new system is given by the intersection of two parabolas. In particular, (2.197) has at most four solutions. Two of them are clearly g^{+} = 0, h = ±1. But they do not have the correct asymptotic behavior and hence are of no interest to us. The remaining two solutions are given by the last two equations of (2.195) with the branch of R(z) arbitrarily. However, we only get correct asymptotics (g^{+} = −z^{−1} + O(z^{−2}) resp. h = −1 + O(z^{−2})) if we fix the branch as in (2.196). This shows that g(z, n + 1), h(z, n) can be reconstructed from γ^{β_{1}}, γ^{β_{2}} and we are done. The remaining cases can be treated similarly. □

Finally, we want to give an alternative characterization of the sequences g(z), h(z) respectively m̃_{±}(z). This characterization will come handy in Section 12.2.

Theorem 2.31. Consider the two conditions (i). Suppose the sequences g(z), h(z) are holomorphic near ∞ and satisfy

(2.199)  \begin{aligned}4a^{2}g(z)g^{+}(z) & = & h^{2}(z) − 1, & g(z) & = & −z^{−1} + O(z^{−2}), \\ h^{+}(z) + h(z) & = & 2(z − b^{+})g^{+}(z), & h(z) & = & −1 + O(z^{−1}). \end{aligned}

(ii). Suppose the sequences m̃_{±}(z) are meromorphic near ∞ and satisfy

(2.200)  a^{2}m̃_{±}(z) + \frac{1}{m̃_{±}^{−}(z)} = ∓(z − b), \begin{matrix}m̃_{+}(z) = −z^{−1} + O(z^{−2}) \\ m̃_{−}(z) = a^{2}z + O(z^{0}) \end{matrix} .

Then (i) (resp. (ii)) is necessary and sufficient for g(z), h(z) (resp. m̃_{±}(z)) to be the corresponding coefficients of the Weyl M-matrix (resp. the Weyl m̃-functions) of H.

Moreover, if g(z), h(z) and m̃_{±}(z) are related by

g(z, n) = \frac{−a(n)^{−2}}{m̃_{+}(z, n) + m̃_{−}(z, n)}, g(z, n + 1) = \frac{m̃_{+}(z, n)m̃_{−}(z, n)}{m̃_{+}(z, n) + m̃_{−}(z, n)},
(2.201)  h(z, n) = \frac{m̃_{+}(z, n) − m̃_{−}(z, n)}{m̃_{+}(z, n) + m̃_{−}(z, n)},

respectively

(2.202)  m̃_{±}(z, n) = −\frac{1 ± h(z, n)}{2a(n)^{2}g(z, n)} = \frac{2g(z, n + 1)}{1 ∓ h(z, n)},

then one condition implies the other.

Proof. Necessity has been established earlier in this section. The relation between the two sets of data is straightforward and hence it suffices to consider (e.g.) m̃_{±}(z). Since m̃_{±}(z) are both meromorphic near ∞ they are uniquely determined by their Laurent expansion around ∞. But the coefficients of the Laurent expansion can be determined uniquely using the recursion relations for m̃_{±}(z) (see Lemma 6.7 for more details). □

Qualitative theory of spectra

In the previous chapter we have derived several tools for investigating the spectra of Jacobi operators. However, given a(n), b(n) it will, generally speaking, not be possible to compute m_{±}(z) or G(z, m, n) explicitly and to read off some spectral properties of H from a(n), b(n) directly. In this section we will try to find more explicit criteria. The main idea will be to relate spectral properties of H with properties of solutions of τu = λu. Once this is done, we can relate properties of the solutions with properties of the coefficients a, b. In this way we can read off spectral features of H from a, b without solving the equation τu = λu.

The endeavor of characterizing the spectrum of H is usually split up into three parts:

(1) Locating the essential spectrum.

(2) Determining which parts of the essential spectrum are absolutely continuous.

(3) Trying to give a bound on the number of eigenvalues.

We will freely use the results and notation from Appendix B. The reader might want to review some of this material first.

3.1. Locating the spectrum and spectral multiplicities

We begin by trying to locate the spectrum of H_{+}. By Lemma B.5, λ ∈ σ(H_{+}) if and only if m_{+}(z) is not holomorphic in a neighborhood of λ. Moreover, by (B.27) we even have

(3.1)  σ(H_{+}) = \overline{\{λ ∈ ℝ| lim_{ε↓0} Im(m_{+}(λ + iε)) > 0\}}.

Note that by equation (B.41), the point spectrum of H_{+} is given by

(3.2)  σ_{p}(H_{+}) = \{λ_{0} ∈ ℝ| lim_{ε↓0} \frac{ε}{i}m_{+}(λ_{0} + iε) > 0\}.

Now let us relate the spectrum to the asymptotic behavior of s(λ, n). Let us first show that s(λ, n) does not grow too fast for λ in the spectrum. More precisely, we have

Lemma 3.1. Let f ∈ ℓ^{2}(ℕ), then the set

(3.3)  S_{f,+}(τ) = {λ ∈ ℝ|f(n)s(λ, n) ∈ ℓ^{2}(ℕ)},

is of full H_{+} spectral measure, that is, ρ_{+}(S_{f,+}(τ)) = 1. The same is true for the set

(3.4)  S_{+}(τ) = \{λ ∈ ℝ| lim inf_{n→∞} \frac{1}{n} ∑_{m=1}^{n} |s(λ, m)|^{2} < ∞\}.

Proof. Since the polynomials s(λ, n) are orthonormal, one verifies

(3.5)  ∥f∥^{2} = ∫_{ℝ} ∑_{n∈ℕ} |f(n)s(λ, n)|^{2}dρ_{+}(λ).

Hence ∑_{n∈ℕ} |f(n)s(λ, n)|^{2} < ∞ a.e. with respect to dρ_{+}. Similarly, by Fatou’s lemma we have

(3.6) ∫_{ℝ} lim inf_{n→∞} \frac{1}{n} ∑_{m=1}^{n} |s(λ, m)|^{2}dρ_{+}(λ) < lim inf_{n→∞} ∫_{ℝ} \frac{1}{n} ∑_{m=1}^{n} |s(λ, m)|^{2}dρ_{+}(λ) = 1,

which implies the second claim. □

Choosing f(n) = n^{−1/2−δ}, δ > 0, we see for example
(3.7)  sup_{n∈ℕ} \frac{|s(λ, n)|}{n^{1/2+δ}} < ∞, λ a.e. dρ_{+}.

As a consequence we obtain

Theorem 3.2. Let δ > 0, then

σ(H_{+}) = \overline{S_{+}(τ)} = \overline{\{λ ∈ ℝ| sup_{n∈ℕ} \frac{|s(λ, n)|}{n^{1/2+δ}} < ∞\}}

(3.8)  = \overline{\{λ ∈ ℝ| lim sup_{n→∞} \frac{1}{n} ln(1 + |s(λ, n)|) = 0\}}.

The spectrum of H_{+} is simple.

Proof. Denote the last two sets (without closure) in (3.8) by S_{1}, S_{2}. Then we need to show σ(H_{+}) = \overline{S_{+}} = \overline{S_{1}} = \overline{S_{2}}. By equation (2.47) we have \overline{S_{+}}, \overline{S_{1}} ⊆ \overline{S_{2}} ⊆ σ(H_{+}) and since S_{+,1,2} are all of full spectral measure, the claim follows as in the proof of (B.27).

That the spectrum of H_{+} is simple is immediate from Theorem 2.12. □

Now let us turn to H. Here we have λ ∈ σ(H) if and only if g(z, 0) + g(z, 1) is not holomorphic in z near λ and

(3.9)  σ(H) = \overline{\{λ ∈ ℝ| lim_{ε↓0} Im(g(λ + iε, 0) + g(λ + iε, 1)) > 0\}}.

To obtain again a characterization in terms of solutions, we need the analog of Lemma 3.1.

Lemma 3.3. Let f ∈ ℓ^{2}(ℤ), then the set

(3.10)  S_{f}(τ) = {λ ∈ ℝ|f(n)s_{β}(λ, n) ∈ ℓ^{2}(ℤ) for some β = β(λ)}

is of full H spectral measure, that is, ρ^{tr}(S_{f}(τ)) = 1. The same is true for the set

(3.11)  S(τ) = \{λ ∈ ℝ| lim inf_{n→∞} \frac{1}{n} ∑_{m=1}^{n} |s_{β}(λ, m)|^{2} < ∞ for some β = β(λ)\}.

Proof. We use U(λ) from (B.62) which diagonalizes dρ and change to the new pair of solutions

(3.12)  \underline{U}(λ, n) = U(λ)\underline{S}(λ, n).

Now, proceeding as in the proof of Lemma 3.1, we obtain

(3.13) ∥f∥ = ∫_{ℝ} (r_{1}(λ) ∑_{n∈ℤ} |f(n)U_{1}(z, n)|^{2} + r_{2}(λ) ∑_{n∈ℤ} |f(n)U_{2}(z, n)|^{2}) dρ^{tr}(λ).

Hence for a.e. λ with respect to dρ^{tr} we have ∑_{n∈ℤ} r_{i}(λ)|f(n)U_{i}(λ, n)|^{2} < ∞ for i = 1, 2. Since r_{1}(λ) + r_{2}(λ) = 1, r_{i}(λ)U_{i}(λ, n) is nonzero for one i. The real part is the required solution. The second part follows from Fatou’s lemma as in the proof of the previous lemma. □

As before, choosing f(n) = (1 + |n|)^{−1/2−δ}, δ > 0, we obtain the following result.

Theorem 3.4. Let δ > 0, then

σ(H) = \overline{S(τ)} = \overline{\{λ ∈ ℝ| sup_{n∈ℤ} \frac{|s_{β}(λ, n)|}{n^{1/2+δ}} < ∞ for some β = β(λ)\}}

(3.14)  = \overline{\{λ ∈ ℝ| lim sup_{n→∞} \frac{1}{n} ln(1 + |s_{β}(λ, n)|) = 0 for some β = β(λ)\}}.

The spectral multiplicity of H is at most two.

Before scrutinizing the spectral multiplicity of H, let me remark a few things.

Remark 3.5. (i). The point spectrum of H can be characterized by

(3.15)  σ_{p}(H) = \{λ ∈ ℝ| lim_{ε↓0} \frac{ε}{i}(g(λ + iε, 0) + g(λ + iε, 1)) > 0\}.

At each discrete eigenvalue λ ∈ σ_{d}(H) either m̃_{+}(z, n) and m̃_{−}(z, n) both have a pole or m̃_{−}(z, n) + m̃_{+}(z, n) = 0. (ii). It is interesting to know which parts of the spectrum of H can be read off from dρ_{0,0} alone, that is, from g(z, 0). Since g(z, 0) = −a(0)^{−2}(m̃_{−}(z) + m̃_{+}(z))^{−1}, we know σ_{ess}(H_{−}) ∪ σ_{ess}(H_{+}) and it will follow from Lemma 3.7 that g(z, 0) determines the essential spectrum of H. Moreover, if λ ∈ σ_{d}(H), we either have u_{−}(λ, 0), u_{+}(λ, 0) ≠ 0 and hence m̃_{−}(z, n) + m̃_{+}(z, n) = 0 (implying a pole of g(z, 0) at z = λ) or u_{−}(λ, 0), u_{+}(λ, 0) = 0 and hence poles for both m̃_{±}(z, n) (implying a zero of g(z, 0) at z = λ). Summarizing, g(z, n) determines σ(H) except for the discrete eigenvalues which are also (discrete) simultaneous eigenvalues of H_{−} and H_{+}.

To say more about the spectral multiplicity of H, we need to investigate the determinant of R = dρ/dρ^{tr}. By (B.61) we infer

(3.16)  R_{i,j}(λ) = lim_{ε↓0} \frac{Im(M_{i,j}(λ + iε))}{Im(g(λ + iε, 0) + g(λ + iε, 1))}.

Hence, using (2.143) we see

det R(λ) =
lim_{ε↓0} \frac{Im(m̃_{+}(λ + iε))Im(m̃_{−}(λ + iε))}{a(0)^{2}|m̃_{+}(λ + iε) + m̃_{−}(λ + iε)|^{2}} \frac{1}{Im(g(λ + iε, 0) + g(λ + iε, 1))^{2}} (3.17)

Observe that the first factor is bounded by (2a(0))^{−2}. Now Lemma B.14 immediately gives

Lemma 3.6. The singular spectrum of H has spectral multiplicity one. The absolutely continuous spectrum of H has multiplicity two on σ_{ac}(H_{+}) ∩ σ_{ac}(H_{−}) and multiplicity one on σ_{ac}(H)\(σ_{ac}(H_{+}) ∩ σ_{ac}(H_{−})).

Proof. Recall from Appendix B that the set

(3.18)  M_{s} = {λ ∈ ℝ| lim_{ε↓0} Im(g(λ + iε, 0) + g(λ + iε, 1)) = ∞}

is a minimal support for the singular spectrum (singular continuous plus pure point spectrum) of H. Hence the singular part of H is given by H_{s} = H_{pp} ⊕ H_{sc} = P_{M_{s}}(H)H and the absolutely continuous part is given by H_{ac} = (1 − P_{M_{s}}(H))H. So we see that the singular part has multiplicity one by Lemma B.14.

For the absolutely continuous part use that

(3.19)  M_{ac,±} = {λ ∈ ℝ|0 < lim_{ε↓0} Im(m̃_{±}(λ + iε)) < ∞}

are minimal supports for the absolutely continuous spectrum of H_{±}. Again the remaining result follows from Lemma B.14. □

3.2. Locating the essential spectrum

The objective of this section is to locate the essential spectrum (which by the way is always nonempty since H is bounded) for certain classes of Jacobi operators. Since our operators have spectral multiplicity at most two, the essential spectrum is the set of all accumulation points of the spectrum.

We first show that it suffices to consider H_{±}. In fact, even H_{+} is enough since the corresponding results for H_{−} follow from reflection.

If we embed H_{n_{0}}^{∞} into ℓ^{2}(ℤ) by defining (e.g.) H_{n_{0}}^{∞}u(n_{0}) = 0, we see from (1.106) that the kernel of the resolvent corresponding to this new operator H̃_{n_{0}}^{∞} is given by

(3.20) G̃_{n_{0}}^{∞}(z, n, m) = G(z, n, m) − \frac{G(z, n, n_{0})G(z, n_{0}, m)}{G(z, n_{0}, n_{0})} − \frac{1}{z}δ_{n_{0}}(n)δ_{n_{0}}(m).

Hence H̃_{n_{0}}^{∞} is a rank two resolvent perturbation of H implying σ_{ess}(H) = σ_{ess}(H̃_{n_{0}}^{∞}). However, the last term in (3.20) is clearly artificial and it is more natural to consider H_{n_{0}}^{∞} as rank one perturbation.

Lemma 3.7. For any Jacobi operator H we have

(3.21)  σ_{ess}(H) = σ_{ess}(H_{0}^{∞}) = σ_{ess}(H_{−}) ∪ σ_{ess}(H_{+})

and, in addition, σ_{ess}(H_{±}) = σ_{ess}(H_{±,n_{0}}^{β}) for any n_{0} ∈ ℤ, β ∈ ℝ ∪ {∞}.

Moreover, for λ_{0} < λ_{1} we have (3.22)

dim Ran P_{(λ_{0},λ_{1})}(H_{n_{0}}^{∞}) − 1 ≤ dim Ran P_{(λ_{0},λ_{1})}(H) ≤ dim Ran P_{(λ_{0},λ_{1})}(H_{n_{0}}^{∞}) + 1.

Proof. The first part follows since finite rank perturbations have equivalent essential spectra.

For the second claim, there is nothing to prove if (λ_{0}, λ_{1}) ∩ σ_{ess}(H) ≠ ∅. If this is not the case, let us view g(., n_{0}) as a function from (λ_{0}, λ_{1}) → ℝ ∪ {∞} ≅ S^{1}. Then g(λ, n_{0}) = ∞ implies that λ is a simple eigenvalue of H but not of H_{n_{0}}^{∞}. On the other hand, if g(λ, n_{0}) = 0, then either λ is a single eigenvalue of H_{n_{0}}^{∞} (but not of H) or a double eigenvalue of H_{n_{0}}^{∞} and a single eigenvalue of H. Combining this observation with the fact that g(., n_{0}) is monotone (cf. equation (2.37)) finishes the proof. □

Note that σ(H_{+}) ∩ σ(H_{+}^{β}) = σ_{ess}(H_{+}) for β ≠ ∞.

The following lemma shows that λ is in the essential spectrum of H if there exists a non square summable solution which does not grow exponentially. This is a strengthening of (2.48).

Lemma 3.8. Suppose there is a solution of τu = λu such that

(3.23)  lim_{k→∞} \frac{1}{n_{k}} ln(u(2n_{k})^{2} + u(2n_{k} − 1)^{2}) = 0

for some increasing sequence 0 < n_{k} < n_{k+1}. Then λ ∈ σ_{ess}(H_{+}). Similarly for H_{−}.

Proof. Normalizing u(0)^{2} + u(1)^{2} = 1 we have u(n) = s_{β}(λ, n) (up to sign). Altering b(1) we can assume β = ∞ and shifting b(n) it is no restriction to set λ = 0 (i.e., u(n) = s(0, n)). We will try to construct a Weyl sequence (cf. [192], Thm. VII.12) as follows. Set

(3.24)  u_{N}(n) = \{ \begin{matrix}_{\frac{s(0,n)}{\sqrt{∑_{j=1}^{N} s(0,j)^{2}}}} , & n ≤ N \\ 0  , & n > N \end{matrix} .

Then ∥u_{N}∥ = 1 and

H_{+}u_{N}(n) = \frac{1}{\sqrt{∑_{j=1}^{N} s(0, j)^{2}}} (a(N)s(0, N)δ_{N}(n)

+ (a(N − 1)s(0, N − 1) + b(N)s(0, N))δ_{N+1}(n)). (3.25)

Hence

(3.26)  ∥H_{+}u_{2N}∥^{2} ≤ const\frac{f(N)}{∑_{j=1}^{N} f(j)},

where f(n) = s(0, 2n)^{2} + s(0, 2n − 1)^{2}. If the right hand side does not tend to zero, we must have

(3.27)  f(n) ≥ ε ∑_{j=1}^{n} f(j),  ε > 0,

and hence f(n) ≥ (1 − ε)^{1−n}. But this contradicts our assumption (3.23)

(3.28)  0 = lim_{k→∞} \frac{1}{n_{k}} ln f(n_{k}) ≥ lim_{k→∞} \frac{1 − n_{k}}{n_{k}} ln(1 − ε) = − ln(1 − ε) > 0.

Hence ∥H_{+}u_{2N}∥ → 0 implying 0 ∈ σ(H_{+}) by Weyl’s theorem. If 0 were a discrete eigenvalue, s(0, n) would decrease exponentially by Lemma 2.5 and the limit in (3.23) would be negative. So 0 ∈ σ_{ess}(H_{+}) as claimed. □

Next, we note that the essential spectrum depends only on the asymptotic behavior of the sequences a(n) and b(n).

Lemma 3.9. Let H_{1,2} be two Jacobi operators and suppose

(3.29)  lim_{n→∞}(|a_{2}(n)| − |a_{1}(n)|) = lim_{n→∞}(b_{2}(n) − b_{1}(n)) = 0.

Then

(3.30)  σ_{ess}(H_{1,+}) = σ_{ess}(H_{2,+}).

Proof. Clearly we can assume a_{1,2}(n) > 0. The claim now follows easily from the fact that, by assumption, H_{2,+} − H_{1,+} is compact (approximate H_{2,+} − H_{1,+} by finite matrices and use that norm limits of compact operators are compact). □

Finally, let us have a closer look at the location of the essential spectrum. We abbreviate

(3.31)  c_{±}(n) = b(n) ± (|a(n)| + |a(n − 1)|)

and

(3.32)  \underline{c}_{−} = lim inf_{n→∞} c_{−}(n), \overline{c}_{+} = lim sup_{n→∞} c_{+}(n).

Lemma 3.10. We have

(3.33)  σ_{ess}(H_{+}) ⊆ [\underline{c}_{−}, \overline{c}_{+}].

If, in addition, a(n), b(n) are slowly oscillating, that is,

(3.34)  lim_{n→∞}(|a(n + 1)| − |a(n)|) = 0,  lim_{n→∞}(b(n + 1) − b(n)) = 0,

then we have

(3.35)  σ_{ess}(H_{+}) = [\underline{c}_{−}, \overline{c}_{+}].

Proof. Clearly we can assume a(n) > 0. But we may even assume c_{+}(n) ≤ \overline{c}_{+} and c_{−}(n) ≥ \underline{c}_{−}. In fact, consider Ĥ_{+} associated with the sequences â(n) = min\{a(n), \overline{c}_{+}−a(n−1)−b(n), −\underline{c}_{−}−a(n−1)+b(n)\}, b̂(n) = b(n). Since a(n)−â(n) = max\{0, c_{+}(n) − \overline{c}_{+}, \underline{c}_{−} − c_{−}(n)\} we infer σ_{ess}(H_{+}) = σ_{ess}(Ĥ_{+}) from Lemma 3.9.

For the first claim we use

⟨f, H_{+}f⟩ = −a(0)|f(1)|^{2} + ∑_{n=1}^{∞} ( − a(n)|f(n + 1) − f(n)|^{2}

+ (a(n − 1) + a(n) + b(n))|f(n)|^{2}) (3.36)

and proceed as in Lemma 1.8, proving (3.33).

Suppose (3.34) holds and fix λ ∈ (\underline{c}_{−}, \overline{c}_{+}). To prove (3.35) we need to show λ ∈ σ_{ess}(H_{+}). By Weyl’s criterion (cf. [192], Thm. VII.12) it suffices to exhibit an orthonormal sequence u_{N} ∈ ℓ(ℕ) with ∥(H_{+} − λ)u_{N}∥ → 0 as N → ∞. For each N ∈ ℕ we can find n_{N} > N(N − 1)/2 such that (i). there is a λ_{N} ∈ [c_{−}(n_{N}), c_{+}(n_{N})] such that |λ − λ_{N}| ≤ 1/\sqrt{N} and (ii). |a(n) − a_{N}| + |a(n − 1) − a_{N}| + |b(n) − b_{N}| ≤ 1/\sqrt{N} for n_{N} ≤ n ≤ n_{N} + N, where a_{N} = (a(n_{N}) + a(n_{N} − 1))/2), b_{N} = b(n_{N}).

If we set

(3.37)  k_{N} = \frac{λ_{N} − b_{N} + \sqrt{(λ_{N} − b_{N})^{2} − 4a_{N}^{2}}}{2a_{N}} ,

we have |k_{N}| = 1 since |λ_{N} − b_{N}| ≤ 2a_{N}. Now define u_{N}(n) = (k_{N})^{n}/\sqrt{N} for n_{N} ≤ n < n_{N} +N and u_{N}(n) = 0 otherwise. Then ∥u_{N}∥ = 1 and u_{N}(n)u_{M}(n) = 0, N ≠ M. Moreover, we have

∥(H_{+} − λ)u_{N}∥ ≤ ∥(H_{N,+} − λ_{N})u_{N}∥ + ∥(λ − λ_{N})u_{N}∥ + ∥(H_{+} − H_{N,+})u_{N}∥
(3.38)  ≤ \frac{4∥a∥_{∞}}{\sqrt{N}} + \frac{1}{\sqrt{N}} + \frac{1}{\sqrt{N}} → 0

as N → ∞, where in the last estimate we have used τ_{N}(k_{N})^{n} = λ_{N}(k_{N})^{n}, (i), and (ii), respectively. This proves λ ∈ σ_{ess}(H_{+}) and hence (3.35). □

3.3. Locating the absolutely continuous spectrum

In this section we will try to locate the absolutely continuous spectrum of H in some situations. We will first investigate the half line operator H_{+} whose spectral properties are related to the boundary behavior of m_{+}(λ + iε) as ε ↓ 0 (see Appendix B). Our strategy in this section will be to relate this boundary behavior to the behavior of solutions of τu = λu.

But first we start with a preliminary lemma which shows that the absolutely continuous spectrum of H is completely determined by those of H_{−} and H_{+}.

Lemma 3.11. For any n_{0} ∈ ℤ, β ∈ ℝ ∪ {∞}, the absolutely continuous part of the spectral measures corresponding to H_{±} and H_{±,n_{0}}^{β}, respectively, are equivalent. For fixed n_{0} and different β, the singular parts are mutually disjoint. In particular,

(3.39)  σ_{ac}(H_{±}) = σ_{ac}(H_{±,n_{0}}^{β}).

The same is true for the trace measure dρ^{tr} corresponding to H and dρ̃_{+} + dρ̃_{−}. In particular,

(3.40)  σ_{ac}(H) = σ_{ac}(H_{−}) ∪ σ_{ac}(H_{+}).

In addition, let us abbreviate g(λ, n) = lim sup_{ε↓0} Im(g(λ + iε, n)), m_{±}(λ, n) = lim sup_{ε↓0} Im(m_{±}(λ+iε, n)), and ξ(λ, n) = π^{−1} lim sup_{ε↓0} arg g(λ+iε, n). Then for any fixed n_{0} ∈ ℤ the following sets

M_{ac,1} = {λ ∈ ℝ | 0 < Im(g(λ, n_{0}))},
M_{ac,2} = {λ ∈ ℝ | 0 < ξ(λ, n_{0}) < 1},
(3.41)  M_{ac,3} = ⋃_{σ∈{±}}{λ ∈ ℝ | 0 < Im(m_{σ}(λ, n_{0}))}

are minimal supports for the absolutely continuous spectrum of H.

Proof. Using (2.16) and (i) of Theorem B.8 we infer

(3.42)  dρ_{+,n_{0},ac}^{β}(λ) = \frac{β^{2}}{|β − a(n_{0})m_{+}(λ, n_{0})|^{2}} dρ_{+,n_{0},ac}(λ)

and since H_{+,n_{0}} = H_{+,n_{0}+1}^{0} the equivalence of the absolutely continuous parts of the spectral measures follows for the + sign. That the singular parts are disjoint follows from (2.92) since m_{+}^{β_{1}}(λ) = ∞ implies m_{+}^{β_{2}}(λ) ≠ ∞ for all β_{2} ≠ β_{1}. The − part is similar.

It remains to establish the claims for H. It is no restriction to set n_{0} = 0. By Theorem B.8 (ii) the set L of all λ ∈ ℝ for which both limits lim_{ε↓0} m_{±}(λ + iε) exist and are finite is of full Lebesgue measure. For λ ∈ L we can compute lim_{ε↓0} M(λ + iε) from (2.143). Hence we obtain M_{ac,1} ∩ L = M_{ac,2} ∩ L = M_{ac,3} ∩ L. Moreover, M_{ac,1} ∩ L = {λ|0 < Im(g(λ, 0) + g(λ, 1))} and the claim is immediate from Lemma B.7 and Lemma B.12. □

In particular, this says that if lim sup_{ε↓0} Im(m_{+}(λ + iε)) < ∞, λ ∈ (λ_{0}, λ_{1}), then the spectrum of H_{+} is purely absolutely continuous in (λ_{0}, λ_{1}). Note that we could replace m_{±}(z, n) by m̃_{±}(z, n) in the above theorem.

Now recall the split-up (2.108), dρ_{+} = dρ_{+,αc} + dρ_{+,αs}, with respect to the Hausdorff measure h^{α}, α ∈ [0, 1]. In view of Theorem B.9 a support C^{α}(ρ_{+}) of dρ_{+,αc} can be found by investigating the boundary values of m_{+}(λ + iε) as ε ↓ 0. We set

(3.43)  ∥f∥_{(1,n)} = \sqrt{∑_{j=1}^{n} |f(j)|^{2}}

and extend this definition for real n ≥ 1 by linear interpolation of ∥f∥_{(1,n)}^{2}. That is, we have ∥f∥_{(1,n)}^{2} = ∫_{0}^{n} |f([[x]] + 1)|^{2}dx, where [[.]] denotes the integer part (see (4.14)).

Lemma 3.12. Let ε = a(0)(2∥s(λ)∥_{(1,n)}∥c(λ)∥_{(1,n)})^{−1}, then

(3.44)  5 − \sqrt{24} ≤ |a(0)m_{+}(λ + iε)|\frac{∥s(λ)∥_{(1,n)}}{∥c(λ)∥_{(1,n)}} ≤ 5 + \sqrt{24}.

Proof. Let n ∈ ℕ, then by (2.158)

u_{+}(λ + iε, n) = \frac{c(λ, n)}{a(0)} − m_{+}(λ + iε)s(λ, n)

−\frac{iε}{a(0)} ∑_{j=1}^{n}(c(λ, n)s(λ, j) − c(λ, j)s(λ, n))u_{+}(λ + iε, j). (3.45)

Hence one obtains after a little calculation (as in the proof of Lemma 2.15)

∥u_{+}(λ + iε)∥_{(1,n)} ≥ ∥\frac{c(λ)}{a(0)} − m_{+}(λ + iε)s(λ)∥_{(1,n)}

− 2ε∥s(λ)∥_{(1,n)}∥\frac{c(λ)}{a(0)}∥_{(1,n)}∥u_{+}(λ + iε)∥_{(1,n)}. (3.46)

Using the definition of ε and (2.89) we obtain

∥\frac{c(λ)}{a(0)} − m_{+}(λ + iε)s(λ)∥_{(1,n)}^{2} ≤ 4∥u_{+}(λ + iε)∥_{(1,n)}^{2}

≤ 4∥u_{+}(λ + iε)∥_{(1,∞)}^{2}

= \frac{4}{ε}Im(m_{+}(λ + iε))

≤ 8∥s(λ)∥_{(1,n)}∥\frac{c(λ)}{a(0)}∥_{(1,n)}Im(m_{+}(λ + iε)). (3.47)

Combining this estimate with

(3.48) ∥\frac{c(λ)}{a(0)} − m_{+}(λ + iε)s(λ)∥_{(1,n)}^{2} ≥ (∥\frac{c(λ)}{a(0)}∥_{(1,n)} − |m_{+}(λ + iε)|∥s(λ)∥_{(1,n)})^{2}

shows (1 − x)^{2} ≤ 8x, where x = |a(0)m_{+}(λ + iε)|∥s(λ)∥_{(1,n)}∥c(λ)∥_{(1,n)}^{−1}. □

We now introduce the concept of α-subordinacy, α ∈ [0, 1]. A nonzero solution u of τu = zu is called α-subordinate at +∞ with respect to another solution v if

(3.49)  lim inf_{n→+∞} \frac{∥u∥_{(1,n)}^{2−α}}{∥v∥_{(1,n)}^{α}} = 0.

Moreover, u is called α-subordinate at +∞ if (3.49) holds for for any linearly independent solution v.

An α_{1}-subordinate solution is also an α_{2}-subordinate solution for any α_{1} ≤ α_{2} and if there is no α_{2}-subordinate solution, then there is also no α_{1}-subordinate solution.

For example, if u ∈ ℓ_{±}^{2}(ℕ), then u is α-subordinate for α ∈ (0, 1] (there can be no 0-subordinate solution). In fact, let v(λ) be a second linearly independent solution. Since W_{n}(u(λ), v(λ)) is a nonzero constant, we infer that v(n) cannot be bounded since otherwise W_{n}(u(λ), v(λ)) → 0 as n → ±∞. Moreover, by (2.47) we see that the set of all λ ∈ ℝ for which no α-subordinate (α ∈ (0, 1]) solution exists must be a subset of the essential spectrum.

A similar definition applies to −∞.

The following theorem is an immediate consequence of our previous lemma and Theorem B.9.

Theorem 3.13. Let α ∈ [0, 1], λ ∈ ℝ. Then λ ∈ C^{α}(ρ_{+}) if s(λ) is not αsubordinate with respect to c(λ). If α ∈ [0, 1), the converse is also true.

In particular, if no α-subordinate solution at +∞ exists for any λ ∈ (λ_{1}, λ_{2}), then ρ_{+} is purely α-continuous on (λ_{1}, λ_{2}), that is, [λ_{1}, λ_{2}] ⊆ σ_{αc}(H_{+}) and (λ_{1}, λ_{2})∩ σ_{αs}(H_{+}) = ∅, α ∈ (0, 1].

The case α = 1 deserves special attention, since 1-continuous is the same as absolutely continuous. We will call u subordinate with respect to v if

(3.50)  lim_{n→+∞} \frac{∥u∥_{(1,n)}}{∥v∥_{(1,n)}} = 0.

We use the ordinary limit (instead of lim inf) in the case α = 1 for convenience later on. We will infer from Lemma 3.14 below that, for our purpose, this makes no difference.

It is easy to see that if u is subordinate with respect to v, then it is subordinate with respect to any linearly independent solution. In particular, a subordinate solution is unique up to a constant. Moreover, if a solution u of τu = λu, λ ∈ ℝ, is subordinate, then it is real up to a constant, since both the real and the imaginary part are subordinate. For z ∈ ℂ\ℝ we know that there is always a subordinate solution near ±∞, namely u_{±}(z, n). The following result considers the case z ∈ ℝ.

Lemma 3.14. Let λ ∈ ℝ. There is a near +∞ subordinate solution u(λ) if and only if m_{+}(λ + iε) converges to a limit in ℝ ∪ {∞} as ε ↓ 0. Moreover,

(3.51)  lim_{ε↓0} m_{+}(λ + iε) = −\frac{u(λ, 1)}{a(0)u(λ, 0)}

in this case. A similar result holds near −∞.

Proof. If u(λ) = s(λ) is subordinate or m_{+}(λ + i0) = ∞, the claim follows from Lemma 3.12. Similarly, if u(λ) = c(λ) is subordinate or m_{+}(λ + i0) = 0. If u(λ) = s_{β}(λ) is subordinate or m_{+}(λ + i0) = β/a(0), we can reduce it to the first case using the transform b(1) → b(1) − a(0)/β (see (2.16)). □

We are interested in N_{±}(τ), the set of all λ ∈ ℝ for which no subordinate solution exists, that is,

(3.52)  N_{±}(τ) = {λ ∈ ℝ|No solution of τu = λu is subordinate near ±∞}.

Since the set, for which the limit lim_{ε↓0} m_{+}(λ+iε) does not exist, is of zero spectral and Lebesgue measure (Theorem B.8 (ii)), changing the lim in (3.50) to a lim inf will affect N_{±}(τ) only on such a set (which is irrelevant for our purpose).

Then, as consequence of the previous lemma, we have

Theorem 3.15. The set N_{±}(τ) is a minimal support for the absolutely continuous spectrum of H_{±}. In particular,

(3.53)  σ_{ac}(H_{±}) = \overline{N_{±}(τ)}^{ess}.

Proof. By Theorem B.8 (ii) we may assume m_{+}(λ + i0) exists and is finite almost everywhere. But for those values λ we have 0 < Im(m_{±}(λ + i0)) < ∞ if and only if λ ∈ N_{±}(τ) and the result follows from Lemma B.7. □

Using Lemma 3.1 we can now show Theorem 3.16. Let f ∈ ℓ^{2}(ℤ) and

(3.54) B_{f,±}(τ) = {λ ∈ ℝ|f(n)u(n) ∈ ℓ^{2}(±ℕ) for all solutions of τu = λu}.

Then we have

(3.55)  σ_{ac}(H_{±}) ⊆ \overline{B_{f,±}(τ)}^{ess}.

Choosing f(n) = (1 + |n|)^{−1/2−δ}, δ > 0, we even see

(3.56)  σ_{ac}(H_{±}) ⊆ \overline{\{λ ∈ ℝ|\overline{γ}^{±}(λ) = 0\}}^{ess}.

In addition, let (3.57)

B_{±}(τ) = \{λ ∈ ℝ| lim inf_{n→∞} \frac{1}{n} ∑_{m=1}^{n} |u(±m)|^{2} < ∞ for all solutions of τu = λu\},

then B_{±}(τ) ⊆ N_{±}(τ) and

(3.58)  σ_{ac}(H_{±}) = \overline{B_{±}(τ)}^{ess}.

Proof. We only consider the + sign. By Lemma 3.1 the sets

(3.59)  S_{f,±}^{β}(τ) = {λ ∈ ℝ|f(n)s_{β}(λ, n) ∈ ℓ^{2}(±ℕ)}

are of full H_{±}^{β} spectral measure. Next, note that B_{f,+}(τ) = S_{f,+}^{∞}(τ)∩S_{f,+}^{β}(τ) if β ≠ ∞. Moreover, since dρ_{+,ac} and dρ_{+,ac}^{β} are equivalent, we infer ρ_{+,ac}(ℝ\S_{f,+}^{β}(τ)) = ρ_{+,ac}^{β}(ℝ\S_{f,+}^{β}(τ)) = 0. Thus S_{f,+}^{∞}(τ) and S_{f,+}^{β}(τ) are both supports for dρ_{+,ac} and so is their intersection.

The second assertion follows since λ ∈ S_{f,+}(τ) implies γ^{+}(λ) = 0 for our particular choice of f.

Finally, as before B_{±}(τ) is a support for σ_{ac}(H_{±}) and hence, by the previous theorem, it suffices to verify B_{±}(τ) ⊆ N_{±}(τ). Suppose λ ∈ B_{+}(τ) is such that a subordinate solution u exists, that is, lim_{n→+∞} ∥u∥_{(1,n)}/∥v∥_{(1,n)} = 0. By assumption, there is a subsequence n_{j} such that n_{j}^{−1}∥v∥_{(1,n_{j})} is bounded. Hence n_{j}^{−1}∥u∥_{(1,n_{j})} must tend to zero. But this contradicts constancy of the Wronskian

(3.60) 0 < |W (u, v)| = \frac{1}{n_{j}} ∑_{m=1}^{n_{j}} |W_{n}(u, v)| ≤ const\sqrt{\frac{∥u∥_{(1,n_{j}+1)}}{n_{j}} \frac{∥v∥_{(1,n_{j}+1)}}{n_{j}}} → 0

(use Cauchy-Schwarz). The case λ ∈ B_{−}(τ) is similar. □

Now, we want to extend our results to operators H on ℓ^{2}(ℤ). The spectrum of H is related to the boundary behavior of g(λ + iε, n), n = 0, 1, which again is related to the boundary behavior of m̃_{±}(λ + iε) by equation (2.201).

Lemma 3.17. Suppose lim inf_{ε↓0} Im(m_{σ}(λ+iε)) > 0 and lim sup_{ε↓0} |m_{σ}(λ+iε)| < ∞ for λ ∈ (λ_{1}, λ_{2}), σ = − or σ = +. Then [λ_{1}, λ_{2}] ⊆ σ_{ac}(H) and (λ_{1}, λ_{2})∩σ_{s}(H) = ∅. Here σ_{s}(H) = σ_{sc}(H) ∪ σ_{pp}(H).

Proof. Consider (e.g.) σ = +. Our assumption implies that [λ_{1}, λ_{2}] ⊆ σ_{ess}(H_{+}) ⊆ σ_{ess}(H). Moreover, using (2.201) one easily shows

(3.61)  lim sup_{ε↓0} Im(g(λ + iε, n)) < ∞, n = 0, 1.

Together with Lemma B.7 this implies that the singular spectrum of H is not supported on (λ_{1}, λ_{2}). □

This last result is the key to our next theorem.

Theorem 3.18. On every subinterval (λ_{1}, λ_{2}) ⊆ N_{±}(τ), the spectrum of H_{±}^{β} is purely absolutely continuous in the sense that

(3.62)  [λ_{1}, λ_{2}] ⊆ σ_{ac}(H_{±}^{β}),  (λ_{1}, λ_{2}) ∩ σ_{s}(H_{±}^{β}) = ∅.

In addition, the spectrum of H is purely absolutely continuous on every subinterval (λ_{1}, λ_{2}) ⊆ N_{−}(τ) ∪ N_{+}(τ), that is,

(3.63)  [λ_{1}, λ_{2}] ⊆ σ_{ac}(H),  (λ_{1}, λ_{2}) ∩ σ_{s}(H) = ∅.

Proof. Clearly Lemma 3.12 (and all other considerations) holds for any other boundary condition β. Hence only the claim concerning H needs to be proven. If λ ∈ N_{+}(τ), we know lim sup_{ε↓0} |m̃_{+}^{β}(λ + iε)| = C(β) < ∞ and we need to show lim inf_{ε↓0} Im(m̃_{+}(λ + iε)) > 0. Suppose that we can find a sequence ε_{n} → 0 such that Im(m̃_{+}(λ + iε_{n})) → 0 as n → ∞. Now set m̃_{+}(λ + iε_{n}) = x_{n} + iy_{n} and β_{n} = a(0)x_{n}. Since lim sup |x_{n}| < ∞ we can pass to a subsequence such that x_{n} converges. This implies (x_{n}, y_{n}, β_{n}) → (x_{0}, 0, β_{0} = a(0)x_{0}) and by (2.92)

|m̃_{+}^{β_{0}}(λ + iε_{n})| = \frac{1}{|a(0)|} |\frac{1 + a(0)β_{0}(x_{n} + iy_{n})}{β_{0} − a(0)(x_{n} + iy_{n})}|
(3.64)  ≥ \frac{1}{|a(0)|} \frac{|1 + β_{0}β_{n}|}{\sqrt{(β_{0} − β_{n})^{2} + y_{n}^{2}}}

contradicting our assumption. The case λ ∈ N_{−}(τ) is similar. □

The relation between the absolutely continuous spectrum and (non-)existence of subordinate solutions is often referred to as principle of subordinacy.

Finally, let us show how these results can be applied. Note that λ ∈ B_{±}(τ) if all solutions of τu = λu are bounded near ±∞.

Theorem 3.19. Suppose the sequences a(n) > 0, b(n) satisfy

(3.65)  ∑_{n∈ℤ} |a(n + 1) − a(n)| + |b(n + 1) − b(n)| < ∞

and set

(3.66)  lim_{n→±∞} a(n) = a_{±}, lim_{n→±∞} b(n) = b_{±}.

Then for any λ ∈ (−2a_{±} + b_{±}, 2a_{±} + b_{±}) all solutions of τu = λu are bounded near ±∞. Hence we have

σ_{ess}(H_{±}) = σ_{ac}(H_{±}) = [−2a_{±} + b_{±}, 2a_{±} + b_{±}], σ_{sc}(H_{±}) = ∅,
(3.67) σ_{pp}(H_{±}) ⊂ (−∞, −2a_{±} + b_{±}] ∪ [2a_{±} + b_{±}, ∞).

Moreover,

σ_{ess}(H) = σ_{ac}(H) = σ_{ac}(H_{+}) ∪ σ_{ac}(H_{−}), σ_{sc}(H) = ∅,
(3.68)  σ_{pp}(H) ⊂ \overline{ℝ\σ_{ess}(H)}.

Proof. After the transformation z → 2a_{+}z + b_{+} we can assume a_{+} = 1/2, b_{+} = 0. Consider the quantity

(3.69)  K(n) = a(n)(u(n + 1)^{2} + u(n)^{2}) + (b(n) − λ)u(n)u(n + 1),

where u is a solution of τu = λu. For λ ∈ (−1, 1) we can find N ∈ ℕ, ε > 0 such that

(3.70)  \frac{λ − b(n)}{2a(n)} ≤ 1 − ε, n ≥ N.

Whence

\frac{K(n)}{a(n)} ≥ u(n + 1)^{2} + u(n)^{2} − (1 − ε)2u(n)u(n + 1)
= ε(u(n + 1)^{2} + u(n)^{2}) + (1 − ε)(u(n + 1) − u(n))^{2}
(3.71)  ≥ ε(u(n + 1)^{2} + u(n)^{2}).

Thus to show u bounded near +∞ it suffices to show this result for K. One computes

K(n + 1) − K(n) = \frac{a(n + 1) − a(n)}{a(n + 1)}(K(n) + (a(n + 1) + a(n))u(n + 1)^{2})

+ \frac{a(n)(b(n + 1) − b(n))}{a(n + 1)}u(n)u(n + 1) (3.72)

which implies using (3.71)

(3.73)  K(n + 1) ≤ (1 + C(n))K(n), C(n) ∈ ℓ^{1}(ℤ),

where

(3.74) C(n) = \frac{|a(n + 1) − a(n)|}{a(n + 1)}(1 + \frac{a(n + 1) + a(n)}{εa(n)}) + \frac{|b(n + 1) − b(n)|}{2εa(n + 1)}.

Hence

(3.75)  K(n) ≤ K(n_{0}) ∏_{m=n_{0}}^{∞}(1 + C(m)), n ≥ n_{0},

is bounded. A similar argument shows u bounded near −∞. The rest follows from Lemma 3.10 and Theorem 3.18. □

Note that (3.65) does not exclude eigenvalues at the boundary of the essential spectrum. For example, taking

(3.76)  a(n) = \frac{1}{2},  b(n) = \frac{2 − 3n^{2}}{4 + n^{4}} ,

there is an eigenvalue at 1 with corresponding eigenfunction u_{±}(1, n) = (1 + n^{2})^{−1}. The reason is that the first moment ∑_{n∈ℤ} |nb(n)| is not finite. We will learn more about this case in Chapter 10.

3.4. A bound on the number of eigenvalues

In this section we will derive a Birman-Schwinger type bound on the number of eigenvalues below the essential spectrum of certain Jacobi operators.

Theorem 3.20. Let H_{0}, H be operators in a Hilbert space ℌ. Suppose H_{0} ≥ λ_{0} and λ_{0} ∈ σ_{ess}(H_{0}). Let H be such that H −H_{0} is Hilbert-Schmidt, then, for λ < λ_{0} we have

(3.77)  dim Ran P_{(−∞,λ)}(H) ≤ tr((H − H_{0})(H_{0} − λ)^{−1})^{2}.

Similarly, if H_{0} ≤ λ_{0} and λ_{0} ∈ σ_{ess}(H_{0}). Then

(3.78)  dim Ran P_{(λ,∞)}(H) ≤ tr((H − H_{0})(λ − H_{0})^{−1})^{2}

for λ > λ_{0}.

Proof. Since H − H_{0} is compact we have σ_{ess}(H) = σ_{ess}(H_{0}). Using the same notation as in [195], Theorem XIII.1 we infer as in the proof of the proposition after Theorem XIII.2 that μ_{n}(H_{0} + β(H − H_{0})) is monotone nonincreasing with respect to β ∈ [0, ∞). Now we use that

(3.79)  (H_{0} + β(H − H_{0}))u = λu,  u ∈ ℌ, λ < λ_{0},

is equivalent to

(3.80)  v = βK_{λ}v,  v ∈ ℌ,

where K_{λ} = \sqrt{(H_{0} − λ)^{−1}}(H_{0} − H)\sqrt{(H_{0} − λ)^{−1}} and u = \sqrt{(H_{0} − λ)^{−1}}v. Notice that K_{λ} is a self-adjoint operator. By [192], Theorem VI.25 we have tr K_{λ}^{2} = tr((H_{0}−H)(H−λ)^{−1})^{2} < ∞. Denote the nonzero eigenvalues of K_{λ} by 1/β_{m}. Then each eigenvalue μ_{n}(H_{0}+β(H−H_{0})) with μ_{n}(H) < λ passes λ when β = β_{m} ∈ (0, 1). Hence

(3.81)  dim Ran P_{(−∞,λ)}(H) ≤ ∑_{n∈ℕ} \frac{1}{β_{n}^{2}} = tr K_{λ}^{2}

concluding the proof of the first assertion. Using P_{(−∞,λ)}(−H) = P_{(−λ,∞)}(H) one obtains the second. □

Finally, let us consider a case where this estimate can be made somewhat more explicit. Namely, we will assume that we can find a function G̃_{0}(λ, n, m) such that

(3.82)  |G_{0}(λ, n, m)| ≤ G̃_{0}(λ, n, m) and G̃_{0}(λ, m + n, m) ≤ G̃_{0}(λ, m, m).

This assumption seems reasonable since G_{0}(λ, n, .) ∈ ℓ^{2}(ℤ). Then we have

(3.83)  |G_{0}(λ, n, m)| ≤ min(g̃_{0}(λ, n), g̃_{0}(λ, m)) ≤ \sqrt{g̃_{0}(λ, n)}\sqrt{g̃_{0}(λ, m)},

where g̃_{0}(λ, n) = G̃_{0}(λ, n, n).

Corollary 3.21. Let H_{0} and H be two given Jacobi operators for which the assumptions of Theorem 3.20 hold. Moreover, suppose there is a g̃_{0}(λ, n) such that

(3.84)  |G_{0}(λ, n, m)| ≤ \sqrt{g̃_{0}(λ, n)}\sqrt{g̃_{0}(λ, m)}.

Then we have

dim Ran P_{(λ,±∞)}(H) ≤ ( ∑_{n∈ℤ} 2|a(n) − a_{0}(n)|\sqrt{g̃_{0}(λ, n + 1)}\sqrt{g̃_{0}(λ, n)}

+ |b(n) − b_{0}(n)|g̃_{0}(λ, n))^{2} (3.85) for λ < λ_{0} ≤ H_{0}, λ > λ_{0} ≥ H_{0}, respectively. Proof. The kernel of (H − H_{0})(H_{0} − λ)^{−1} is given by

K(λ, n, m) = (a(n) − a_{0}(n))G(λ, n + 1, m) + (b(n) − b_{0}(n))G(λ, n, m)

+ (a(n − 1) − a_{0}(n − 1))G(λ, n − 1, m) (3.86) which can be estimated by

K̃(λ, n, m) = |a(n) − a_{0}(n)|\sqrt{g̃_{0}(λ, n + 1)}\sqrt{g̃_{0}(λ, m)}
+ |b(n) − b_{0}(n)|\sqrt{g̃_{0}(λ, n)}\sqrt{g̃_{0}(λ, m)}

+ |a(n − 1) − a_{0}(n − 1)|\sqrt{g̃_{0}(λ, n − 1)}\sqrt{g̃_{0}(λ, m)}. (3.87) Now

tr((H − H_{0})(λ − H_{0})^{−1})^{2} = ∑_{n,m∈ℤ} K(λ, n, m)K(λ, m, n)

≤ ∑_{n,m∈ℤ} K̃(λ, n, m)K̃(λ, m, n)

= ∑_{n,m∈ℤ} K̃(λ, n, n)K̃(λ, m, m) = ( ∑_{n∈ℤ} K̃(λ, n, n))^{2} (3.88)

concludes the proof. □ Applications of these results will be given in Section 10.2.

Oscillation theory

There is a close connection between the number of sign flips of solutions of Jacobi equations and the spectra of the corresponding Jacobi operators. The investigation of this interconnection was started by Sturm (in the case of differential equations) and is now known as oscillation theory.

4.1. Prüfer variables and Sturm’s separation theorem

In this section we will study oscillation properties of solutions of (1.19). To be more precise, we are interested in the number of sign flips of solutions. It will be convenient to assume

(4.1)  a(n) < 0,  b(n) ∈ ℝ.

In the sequel, a solution of (1.19) will always mean a real-valued, nonzero solution of (1.19). Given a solution u(λ, .) of τu = λu, λ ∈ ℝ, we introduce Prüfer variables ρ_{u}(λ, .), θ_{u}(λ, .) via

u(λ, n) = ρ_{u}(λ, n) sin θ_{u}(λ, n),
(4.2)  u(λ, n + 1) = ρ_{u}(λ, n) cos θ_{u}(λ, n).

Notice that the Prüfer angle θ_{u}(λ, n) is only defined up to an additive integer multiple of 2π (which depends on n).

Inserting (4.2) into (τ − λ)u = 0 yields

(4.3)  a(n) cot θ_{u}(λ, n) + a(n − 1) tan θ_{u}(λ, n − 1) = λ − b(n)

and

(4.4)  ρ_{u}(λ, n) sin θ_{u}(λ, n) = ρ_{u}(λ, n − 1) cos θ_{u}(λ, n − 1).

Equation (4.3) is a discrete Riccati equation (cf. (1.52)) for cot θ_{u}(n) and (4.4) can be solved if θ_{u}(n) is known provided it is replaced by

(4.5)  a(n)ρ_{u}(λ, n) = a(n − 1)ρ_{u}(λ, n − 1) = 0

if sin θ_{u}(λ, n) = cos θ_{u}(λ, n − 1) = 0 (use τu = λu and (4.8) below). Explicitly,

(4.6)  ρ_{u}(λ, n) = ρ_{u}(λ, 0) \begin{matrix}n−1 \\ ∏ & ∗ \\ m=0 \end{matrix} \frac{cos θ_{u}(λ, m)}{sin θ_{u}(λ, m + 1)},

where cos θ_{u}(λ, m)/ sin θ_{u}(λ, m + 1) has to be replaced by a(m + 1)/a(m) whenever sin θ_{u}(λ, m + 1) = 0.

The Wronskian of two solutions u_{i}(λ_{i}, n), i = 1, 2, reads

(4.7) W_{n}(u_{1}(λ_{1}), u_{2}(λ_{2})) = a(n)ρ_{u_{1}}(λ_{1}, n)ρ_{u_{2}}(λ_{2}, n) sin(θ_{u_{1}}(λ_{1}, n) − θ_{u_{2}}(λ_{2}, n)).

The next lemma considers zeros of solutions and their Wronskians more closely. In particular, we will show that a solutions (resp. their Wronskians) must switch sign at a zero.

Lemma 4.1. Let u_{1,2} be solutions of τu_{1,2} = λ_{1,2}u_{1,2} corresponding to λ_{1} ≠ λ_{2}, respectively. Then

(4.8)  u_{1}(n) = 0 ⇒ u_{1}(n − 1)u_{1}(n + 1) < 0.

Moreover, suppose W_{n}(u_{1}, u_{2}) = 0 but W_{n−1}(u_{1}, u_{2})W_{n+1}(u_{1}, u_{2}) ≠ 0, then

(4.9)  W_{n−1}(u_{1}, u_{2})W_{n+1}(u_{1}, u_{2}) < 0.

Otherwise, if W_{n}(u_{1}, u_{2}) = W_{n+1}(u_{1}, u_{2}) = 0, then necessarily

(4.10)  u_{1}(n + 1) = u_{2}(n + 1) = 0 and W_{n−1}(u_{1}, u_{2})W_{n+2}(u_{1}, u_{2}) < 0.

Proof. The fact u_{1}(n) = 0 implies u_{1}(n − 1)u_{1}(n + 1) ≠ 0 (otherwise u_{1} vanishes identically) and a(n)u_{1}(n + 1) = −a(n − 1)u_{1}(n − 1) (from τu_{1} = λu_{1}) shows u_{1}(n − 1)u_{1}(n + 1) < 0.

Next, from (1.20) we infer

(4.11)  W_{n+1}(u_{1}, u_{2}) − W_{n}(u_{1}, u_{2}) = (λ_{2} − λ_{1})u_{1}(n + 1)u_{2}(n + 1).

And W_{n}(u_{1}, u_{2}) = 0 is equivalent to u_{1}(n) = c u_{2}(n), u_{1}(n + 1) = c u_{2}(n + 1) for some c ≠ 0. Hence applying the above formula gives

(4.12)  W_{n−1}(u_{1}, u_{2})W_{n+1}(u_{1}, u_{2}) = −c^{2}(λ_{2} − λ_{1})^{2}u_{1}(n)^{2}u_{1}(n + 1)^{2},

proving the first claim. If W_{n}(u_{1}, u_{2}), W_{n+1}(u_{1}, u_{2}) are both zero, we must have u_{1}(n + 1) = u_{2}(n + 1) = 0 and as before W_{n−1}(u_{1}, u_{2})W_{n+2}(u_{1}, u_{2}) = −(λ_{2} − λ_{1})^{2}u_{1}(n)u_{1}(n + 2)u_{2}(n)u_{2}(n + 2). Hence the claim follows from the first part (4.8). □

We can make the Prüfer angle θ_{u}(λ, .) unique by fixing, for instance, θ_{u}(λ, 0) and requiring

(4.13)  [[θ_{u}(λ, n)/π]] ≤ [[θ_{u}(λ, n + 1)/π]] ≤ [[θ_{u}(λ, n)/π]] + 1,

where

(4.14)  [[x]] = sup{n ∈ ℤ | n < x}.

Since our solutions u(λ, n) will usually depend continuously on λ, the same should be true for θ_{u}(λ, n). In particular, continuity in λ should be compatible with the requirement (4.13).

Lemma 4.2. Let Λ ⊆ ℝ be an interval. Suppose u(λ, n) is continuous with respect to λ ∈ Λ and (4.13) holds for one λ_{0} ∈ Λ. Then it holds for all λ ∈ Λ if we require θ_{u}(., n) ∈ C(Λ, ℝ).

Proof. Fix n and set

(4.15)  θ_{u}(λ, n) = kπ + δ(λ), θ_{u}(λ, n + 1) = kπ + Δ(λ), k ∈ ℤ,

where δ(λ) ∈ (0, π], Δ(λ) ∈ (0, 2π]. If (4.13) should break down, then by continuity we must have one of the following cases for some λ_{1} ∈ Λ. (i) δ(λ_{1}) = 0 and Δ(λ_{1}) ∈ (π, 2π), (ii) δ(λ_{1}) = π and Δ(λ_{1}) ∈ (0, π), (iii) Δ(λ_{1}) = 0 and δ(λ_{1}) ∈ (0, π), (iv) Δ(λ_{1}) = 2π and δ(λ_{1}) ∈ (0, π). Abbreviate R = ρ(λ_{1}, n)ρ(λ_{1}, n + 1). Case (i) implies 0 > sin(Δ(λ_{1})) = cos(kπ) sin(kπ + Δ(λ_{1})) = R^{−1}u(λ_{1}, n + 1)^{2} > 0, contradicting (i). Case (ii) is similar. Case (iii) implies δ(λ_{1}) = π/2 and hence 1 = sin(kπ + π/2) cos(kπ) = R^{−1}u(λ_{1}, n)u(λ_{1}, n + 2) contradicting (4.8). Again, case (iv) is similar. □

Let us call a point n ∈ ℤ a node of a solution u if either u(n) = 0 or a(n)u(n)u(n + 1) > 0. Then, [[θ_{u}(n)/π]] = [[θ_{u}(n + 1)/π]] implies no node at n. Conversely, if [[θ_{u}(n + 1)/π]] = [[θ_{u}(n)/π]] + 1, then n is a node by (4.8). Denote by #(u) the total number of nodes of u and by #_{(m,n)}(u) the number of nodes of u between m and n. More precisely, we will say that a node n_{0} of u lies between m and n if either m < n_{0} < n or if n_{0} = m but u(m) ≠ 0. Hence we conclude that the Prüfer angle of a solution counts the number of nodes.

Lemma 4.3. Let m < n. Then we have for any solution u

(4.16)  #_{(m,n)}(u) = [[θ_{u}(n)/π]] − lim_{ε↓0}[[θ_{u}(m)/π + ε]]

and

(4.17)  #(u) = lim_{n→∞} ([[θ_{u}(n)/π]] − [[θ_{u}(−n)/π]]).

Next, we prove the analog of Sturm’s separation theorem for differential equations.

Lemma 4.4. Let u_{1,2} be solutions of τu = λu corresponding to λ_{1} ≤ λ_{2}. Suppose m < n are two consecutive points which are either nodes of u_{1} or zeros of W_{.}(u_{1}, u_{2}) (the cases m = −∞ or n = +∞ are allowed if u_{1} and u_{2} are both in ℓ_{±}^{2}(ℤ) and W_{±∞}(u_{1}, u_{2}) = 0, respectively) such that u_{1} has no further node between m and n. Then u_{2} has at least one node between m and n + 1. Moreover, suppose m_{1} < ⋯ < m_{k} are consecutive nodes of u_{1}. Then u_{2} has at least k − 1 nodes between m_{1} and m_{k}. Hence we even have

(4.18)  #_{(m,n)}(u_{2}) ≥ #_{(m,n)}(u_{1}) − 1.

Proof. Suppose u_{2} has no node between m and n + 1. Hence we may assume (perhaps after flipping signs) that u_{1}(j) > 0 for m < j < n, u_{1}(n) ≥ 0, and u_{2}(j) > 0 for m ≤ j ≤ n. Moreover, u_{1}(m) ≤ 0, u_{1}(n + 1) < 0, and u_{2}(n + 1) ≥ 0 provided m, n are finite. By Green’s formula (1.20)

(4.19)  0 ≤ (λ_{2} − λ_{1}) ∑_{j=m+1}^{n} u_{1}(j)u_{2}(j) = W_{n}(u_{1}, u_{2}) − W_{m}(u_{1}, u_{2}).

Evaluating the Wronskians shows W_{n}(u_{1}, u_{2}) < 0, W_{m}(u_{1}, u_{2}) > 0, which is a contradiction.

It remains to prove the last part. We will use induction on k. The case k = 1 is trivial and k = 2 has already been proven. Denote the nodes of u_{2} which are lower or equal than m_{k+1} by n_{k} > n_{k−1} > · · ·. If n_{k} > m_{k} we are done since there are k − 1 nodes n such that m_{1} ≤ n ≤ m_{k} by induction hypothesis. Otherwise we can find k_{0}, 0 ≤ k_{0} ≤ k, such that m_{j} = n_{j} for 1 + k_{0} ≤ j ≤ k. If k_{0} = 0, we are done and hence we can suppose k_{0} ≥ 1. By induction hypothesis it suffices to show that there are k − k_{0} nodes n of u_{2} with m_{k_{0}} ≤ n ≤ m_{k+1}. By assumption m_{j} = n_{j}, 1 + k_{0} ≤ j ≤ k, are the only nodes n of u_{2} such that m_{k_{0}} ≤ n ≤ m_{k+1}. Abbreviate m = m_{k_{0}}, n = m_{k+1} and assume without restriction u_{1}(m + 1) > 0, u_{2}(m) > 0. Since the nodes of u_{1} and u_{2} coincide we infer 0 < ∑_{j=m+1}^{n} u_{1}(j)u_{2}(j) and we can proceed as in the first part to obtain a contradiction. □

We call τ oscillatory if one solution of τu = 0 has an infinite number of nodes. In addition, we call τ oscillatory at ±∞ if one solution of τu = 0 has an infinite number of nodes near ±∞. We remark that if one solution of (τ − λ)u = 0 has infinitely many nodes, so has any other (corresponding to the same λ) by (4.18). Furthermore, τ − λ_{1} oscillatory implies τ − λ_{2} oscillatory for all λ_{2} ≥ λ_{1} (again by (4.18)).

Now we turn to the special solution s(λ, n) characterized by the initial conditions s(λ, 0) = 0, s(λ, 1) = 1. As in Lemma 2.4 we infer

(4.20)  W_{n}(s(λ), s′(λ)) = \begin{matrix}n \\ ∑ & ∗ \\ j=1 \end{matrix} s(λ, j)^{2}.

Here the prime denotes the derivative with respect to λ. Evaluating the above equation using Prüfer variables shows

(4.21)  θ′_{s}(λ, n) = \frac{1}{−a(n)ρ_{s}(λ, n)^{2}} \begin{matrix}n \\ ∑ & ∗ \\ j=1 \end{matrix} s(λ, j)^{2}.

In particular, θ′_{s}(λ, n) < 0 for n < −1, θ′_{s}(λ, −1) = θ′_{s}(λ, 0) = 0, and θ′_{s}(λ, n) > 0 for n > 0. Equation (4.21) implies that nodes of s(λ, n) for n ∈ ℕ move monotonically to the left without colliding. In addition, since s(λ, n) cannot pick up nodes locally by (4.8), all nodes must enter at ∞ and since θ′_{s}(λ, 0) = 0 they are trapped inside (0, ∞).

We will normalize θ_{s}(λ, 0) = 0 implying θ_{s}(λ, −1) = −π/2. Since s(λ, n) is a polynomial in λ, we easily infer s(λ, n_{0}) > < 0 for fixed n_{0} > < 0 and λ sufficiently small (see (1.68) and (1.69)). This implies

(4.22)  − π < θ_{s}(λ, n_{0}) < −π/2, n_{0} < −1, 0 < θ_{s}(λ, n_{0}) < π, n_{0} ≥ 1,

for fixed n_{0} and λ sufficiently small. Moreover, dividing (4.3) by λ and letting λ → −∞ using (4.22) shows

(4.23)  lim_{λ→−∞} \frac{cot(θ_{s}(λ, n))^{±1}}{λ} = \frac{1}{a(n)}, n \begin{matrix}≥ +1 \\ < −1 \end{matrix} ,

and hence

(4.24)  θ_{s}(λ, n) = \{ \begin{matrix}−\frac{π}{2} − \frac{a(n)}{λ} + o(\frac{1}{λ}), & n < −1 \\ \frac{a(n)}{λ} + o(\frac{1}{λ}), & n ≥ 1 \end{matrix} ,

as λ → −∞.

Now what happens with θ_{s}(λ, n) as λ increases? Suppose n ≥ 1 for simplicity. We already know that for λ small, θ_{s}(λ, n) starts near 0. Then it increases as λ increases by (4.21). It crosses π/2 at the first zero of s(λ, n+1). Next it crosses π at the first zero of s(λ, n). This process continues until it finally crosses nπ at the last zero of s(λ, n + 1). Here we have assumed that s(λ, n), n ∈ ℕ, has precisely n − 1 distinct real zeros. However, this is easily seen to hold, since s(λ_{0}, n) = 0 implies that λ_{0} is an eigenvalue for H_{0,n} (see (1.65)). Hence λ_{0} must be real and simple since H_{0,n} is self-adjoint and its spectrum is simple (cf. Remark 1.10), respectively. Let us summarize these findings. The interlacing of zeros is a classical result from the theory orthogonal polynomials.

Theorem 4.5. The polynomial s(λ, n), n ∈ ℕ, has n − 1 real and distinct roots denoted by

(4.25)  λ_{1}^{n} < λ_{2}^{n} < ⋯ λ_{n−1}^{n}.

The zeros of s(λ, n) and s(λ, n + 1) are interlacing, that is,

(4.26)  λ_{1}^{n+1} < λ_{1}^{n} < λ_{2}^{n+1} < ⋯ < λ_{n−1}^{n} < λ_{n}^{n+1}.

Moreover,

(4.27)  σ(H_{0,n}) = {λ_{j}^{n}}_{j=1}^{n−1}.

Proceeding as for (4.24) we also note

(4.28)  θ_{s}(λ, n) = \{ \begin{matrix}\frac{(2n+1)π}{2} + \frac{a(n)}{λ} + o(\frac{1}{λ}), & n < −1 \\ nπ − \frac{a(n)}{λ} + o(\frac{1}{λ}), & n ≥ 1 \end{matrix} ,

as λ → +∞.

Analogously, let u_{±}(λ, n) be solutions of τu = λu as in Lemma 2.2. Then Lemma 2.4 implies

θ′_{+}(λ, n) = \frac{∑_{j=n+1}^{∞} u_{+}(λ, j)^{2}}{a(n)ρ_{+}(λ, n)^{2}} < 0,
(4.29)  θ′_{−}(λ, n) = \frac{∑_{j=−∞}^{n} u_{−}(λ, j)^{2}}{−a(n)ρ_{−}(λ, n)^{2}} > 0,

where we have abbreviated ρ_{u_{±}} = ρ_{±}, θ_{u_{±}} = θ_{±}.

Since H is bounded from below we can normalize

(4.30)  0 < θ_{∓}(λ, n) < π/2, n ∈ ℤ, λ < inf σ(H),

and we get as before

(4.31)  θ_{−}(λ, n) = \frac{a(n)}{λ} + o(\frac{1}{λ}), θ_{+}(λ, n) = \frac{π}{2} − \frac{a(n)}{λ} + o(\frac{1}{λ}), n ∈ ℤ,

as λ → −∞.

4.2. Classical oscillation theory

Before we come to the first applications we recall a lemma from functional analysis. It will be one of our main ingredients in the following theorems.

Lemma 4.6. Let H be a bounded self-adjoint operator and η_{j}, 1 ≤ j ≤ k, be linearly independent elements of a (separable) Hilbert space ℌ. (i). Let λ ∈ ℝ. If

(4.32)  ⟨η, Hη⟩ < λ∥η∥^{2}

for any nonzero linear combination η = ∑_{j=1}^{k} c_{j}η_{j}, then

(4.33)  dim Ran P_{(−∞,λ)}(H) ≥ k.

Similarly, ⟨η, Hη⟩ > λ∥η∥^{2} implies dim Ran P_{(λ,∞)}(H) ≥ k. (ii). Let λ_{1} < λ_{2}. If

(4.34)  ∥(H − \frac{λ_{2} + λ_{1}}{2} )η∥ < \frac{λ_{2} − λ_{1}}{2} ∥η∥

for any nonzero linear combination η = ∑_{j=1}^{k} c_{j}η_{j}, then

(4.35)  dim Ran P_{(λ_{1},λ_{2})}(H) ≥ k.

Proof. (i). Let M = span{η_{j}} ⊆ ℌ. We claim dim P_{(−∞,λ)}(H)M = dim M = k. For this it suffices to show KerP_{(−∞,λ)}(H)|_{M} = {0}. Suppose P_{(−∞,λ)}(H)η = 0, η ≠ 0. Then, abbreviating dρ_{η}(x) = d⟨η, P_{(−∞,x)}(H)η⟩, we see that for any nonzero linear combination η

⟨η, Hη⟩ = ∫_{ℝ} x dρ_{η}(x) = ∫_{[λ,∞)} x dρ_{η}(x)
(4.36)  ≥ λ ∫_{[λ,∞)} dρ_{η}(x) = λ∥η∥^{2}.

This contradicts our assumption (4.32). (ii). Using the same notation as before we need to show KerP_{(λ_{1},λ_{2})}(H)|_{M} = {0}. If P_{(λ_{1},λ_{2})}(H)η = 0, η ≠ 0, then,

∥(H − \frac{λ_{2} + λ_{1}}{2} )η∥^{2} = ∫_{ℝ}(x − \frac{λ_{2} + λ_{1}}{2} )^{2}dρ_{η}(x) = ∫_{Λ} x^{2}dρ_{η}(x + \frac{λ_{2} + λ_{1}}{2} )

≥ \frac{(λ_{2} − λ_{1})^{2}}{4} ∫_{Λ} dρ_{η}(x + \frac{λ_{2} + λ_{1}}{2} ) = \frac{(λ_{2} − λ_{1})^{2}}{4} ∥η∥^{2}, (4.37)

where Λ = (−∞, −(λ_{2} − λ_{1})/2] ∪ [(λ_{2} − λ_{1})/2, ∞). But this is a contradiction as before. □

Using this result we can now show that the number of nodes of s(λ, n) equals the number of eigenvalues below λ. We begin with the case of finite and half-line operators.

Theorem 4.7. Let λ ∈ ℝ. Then we have

(4.38)  dim Ran P_{(−∞,λ)}(H_{0,n}) = #_{(0,n)}(s(λ)), n > 1,

and

(4.39)  dim Ran P_{(−∞,λ)}(H_{+}) = #_{(0,+∞)}(s(λ)).

The same theorem holds if + is replaced by −.

Proof. We only carry out the proof for the plus sign (the other part following from reflection). By virtue of (4.21), (4.24), and Lemma 4.3 we infer

(4.40)  dim Ran P_{(−∞,λ)}(H_{0,n}) = [[θ_{s}(λ, n)/π]] = #_{(0,n)}(s(λ)), n > 1,

since λ ∈ σ(H_{0,n}) if and only if θ_{s}(λ, n) = 0 mod π. Let k = #(s(λ)) if #(s(λ)) < ∞, otherwise the following argument works for arbitrary k ∈ ℕ. If we pick n so large that k nodes of s(λ) are to the left of n, we have k eigenvalues λ̂_{1} < ⋯ < λ̂_{k} < λ of H_{0,n}. Taking an arbitrary linear combination η(m) = ∑_{j=1}^{k} c_{j}s(λ̂_{j}, n), c_{j} ∈ ℂ, for m < n and η(m) = 0 for m ≥ n a straightforward calculation (using orthogonality of s(λ̂_{j})) verifies

(4.41)  ⟨η, H_{+}η⟩ < λ∥η∥^{2}.

Invoking Lemma 4.6 shows

(4.42)  dim Ran P_{(−∞,λ)}(H_{+}) ≥ k.

For the reversed inequality we can assume k = #(s(λ)) < ∞. Consider H̃_{0,n} = H_{0,n} ⊕ λ1l on ℓ^{2}(0, n) ⊕ ℓ^{2}(n − 1, ∞). Since H̃_{0,n} → H strongly as n → ∞ this implies ([116], Lemma 5.2)

(4.43)  dim Ran P_{(−∞,λ)}(H_{+}) ≤ lim_{n→∞} dim Ran P_{(−∞,λ)}(H_{0,n}) = k

completing the proof. □

Remark 4.8. (i). Consider the following example

a(n) = −\frac{1}{2}, n ∈ ℕ,
(4.44)  b(1) = −1, b(2) = −b_{2}, b(3) = −\frac{1}{2}, b(n) = 0, n ≥ 4.

The essential spectrum of H_{+} is given by σ_{ess}(H_{+}) = [−1, 1] and one might expect that H_{+} has no eigenvalues below the essential spectrum if b_{2} → −∞. However, since we have

(4.45)  s(−1, 0) = 0, s(−1, 1) = 1, s(−1, 2) = 0, s(−1, n) = −1, n ≥ 3,

Theorem 4.7 shows that, independent of b_{2} ∈ ℝ, there is always precisely one eigenvalue below the essential spectrum. (ii). By a simple transformation we obtain the corresponding result for H_{+,n_{0}}^{β}, β ≠ 0,

(4.46)  dim Ran P_{(−∞,λ)}(H_{+,n_{0}}^{β}) = #_{(0,+∞)}(s_{β}(λ, ., n_{0})),

where s_{β}(λ, ., n_{0}) is the solution satisfying the boundary condition in (1.94) (see (2.71)). Similar modifications apply to Theorems 4.13, 4.16, and 4.17 below.

As a consequence of Theorem 4.7 we infer the following connection between being oscillatory or not and the infimum of the essential spectrum.

Corollary 4.9. We have

(4.47)  dim Ran P_{(−∞,λ)}(H_{±}) < ∞

if and only if τ − λ is non-oscillatory near ±∞ and hence

(4.48)  inf σ_{ess}(H_{±}) = inf{λ ∈ ℝ | (τ − λ) is oscillatory at ± ∞}.

Moreover, let λ_{0} < ⋯ < λ_{k} < … be the eigenvalues of H_{±} below the essential spectrum of H_{±}. Then the eigenfunction corresponding to λ_{k} has precisely k nodes inside (0, ±∞).

In a similar way we obtain

Theorem 4.10. Let λ < inf σ_{ess}(H). Then

(4.49)  dim Ran P_{(−∞,λ)}(H) = #(u_{+}(λ)) = #(u_{−}(λ)).

Proof. Again it suffices to prove the first equality. By virtue of (4.29) and (4.31) we infer

(4.50)  dim Ran P_{(−∞,λ)}(H_{−,n}) = [[θ_{−}(λ, n)/π]], n ∈ ℤ.

We first want to show that [[θ_{−}(λ, n)/π]] = #_{(−∞,n)}(u_{−}(λ)) or equivalently that lim_{n→∞}[[θ_{−}(λ, n)/π]] = 0. Suppose lim_{n→∞}[[θ_{−}(λ_{1}, n)/π]] = k ≥ 1 for some λ_{1} ∈ ℝ (saying that u_{−}(., n) looses at least one node at −∞). In this case we can find n such that θ_{−}(λ_{1}, n) > kπ for m ≥ n. Now pick λ_{0} such that θ_{−}(λ_{0}, n) = kπ. Then u_{−}(λ_{0}, .) has a node at n but no node between −∞ and n (by Lemma 4.3). Now apply Lemma 4.4 to u_{−}(λ_{0}, .), u_{−}(λ_{1}, .) to obtain a contradiction. The rest follows as in the proof of Theorem 4.7. □

As before we obtain

Corollary 4.11. We have

(4.51)  dim Ran P_{(−∞,λ)}(H) < ∞

if and only if τ − λ is non-oscillatory and hence

(4.52)  inf σ_{ess}(H) = inf{λ ∈ ℝ | (τ − λ) is oscillatory}.

Furthermore, let λ_{0} < ⋯ < λ_{k} < … be the eigenvalues of H below the essential spectrum of H. Then the eigenfunction corresponding to λ_{k} has precisely k − 1 nodes.

Remark 4.12. (i). Corresponding results for the projection P_{(λ,∞)}(H) can be obtained from P_{(λ,∞)}(H) = P_{(−∞,−λ)}(−H). In fact, it suffices to change the definition of a node according to u(n) = 0 or a(n)u(n)u(n + 1) < 0 and P_{(−∞,λ)}(H) to P_{(λ,∞)}(H) in all results of this section. (ii). Defining u_{±}(λ, n) as in Remark 2.3 for λ = inf σ_{ess}(H), one sees that Theorem 4.10 holds for λ ≤ inf σ_{ess}(H).

Theorem 4.13. Let λ_{1} < λ_{2}. Suppose τ − λ_{2} is oscillatory near +∞. Then

(4.53)  dim Ran P_{(λ_{1},λ_{2})}(H_{+}) = lim inf_{n→+∞} (#_{(0,n)}(s(λ_{2})) − #_{(0,n)}(s(λ_{1}))).

The same theorem holds if + is replaced by −.

Proof. As before we only carry out the proof for the plus sign. Abbreviate Δ(n) = [[θ_{s}(λ_{2}, n)/π]] − [[θ_{s}(λ_{1}, n)/π]] = #_{(0,n)}(s(λ_{2})) − #_{(0,n)}(s(λ_{1})). By (4.40) we infer

(4.54)  dim Ran P_{[λ_{1},λ_{2})}(H_{0,n}) = Δ(n), n > 2.

Let k = lim inf Δ(n) if lim sup Δ(n) < ∞ and k ∈ ℕ otherwise. We contend that there exists n ∈ ℕ such that

(4.55)  dim Ran P_{(λ_{1},λ_{2})}(H_{0,n}) ≥ k.

In fact, if k = lim sup Δ(n) < ∞, it follows that Δ(n) is eventually equal to k and since λ_{1} ∉ σ(H_{0,m}) ∩ σ(H_{0,m+1}), m ∈ ℕ, we are done in this case. Otherwise we can pick n such that dim Ran P_{[λ_{1},λ_{2})}(H_{0,n}) ≥ k + 1. Hence H_{0,n} has at least k eigenvalues λ̂_{j} with λ_{1} < λ̂_{1} < ⋯ < λ̂_{k} < λ_{2}. Again let η(m) = ∑_{j=1}^{k} c_{j}s(λ̂_{j}, n), c_{j} ∈ ℂ for m < n and η(m) = 0 for n ≥ m be an arbitrary linear combination. Then

(4.56)  ∥(H_{+} − \frac{λ_{2} + λ_{1}}{2} )η∥ < \frac{λ_{2} − λ_{1}}{2} ∥η∥

together with the Lemma 4.6 implies

(4.57)  dim Ran P_{(λ_{1},λ_{2})}(H_{+}) ≥ k.

To prove the second inequality we use H̃_{0,n} = H_{0,n} ⊕λ_{2}1l → H_{+} strongly as n → ∞ and proceed as before

(4.58)  dim Ran P_{(λ_{1},λ_{2})}(H_{+}) ≤ lim inf_{n→∞} P_{[λ_{1},λ_{2})}(H̃_{0,n}) = k

since P_{[λ_{1},λ_{2})}(H̃_{0,n}) = P_{[λ_{1},λ_{2})}(H_{0,n}). □

4.3. Renormalized oscillation theory

The objective of this section is to look at the nodes of the Wronskian of two solutions u_{1,2} corresponding to λ_{1,2}, respectively. We call n ∈ ℤ a node of the Wronskian if W_{n}(u_{1}, u_{2}) = 0 and W_{n+1}(u_{1}, u_{2}) ≠ 0 or if W_{n}(u_{1}, u_{2})W_{n+1}(u_{1}, u_{2}) < 0. Again we will say that a node n_{0} of W (u_{1}, u_{2}) lies between m and n if either m < n_{0} < n or if n_{0} = m but W_{n_{0}}(u_{1}, u_{2}) ≠ 0. We abbreviate

(4.59)  Δ_{u_{1},u_{2}}(n) = (θ_{u_{2}}(n) − θ_{u_{1}}(n)) mod 2π.

and require

(4.60)  [[Δ_{u_{1},u_{2}}(n)/π]] ≤ [[Δ_{u_{1},u_{2}}(n + 1)/π]] ≤ [[Δ_{u_{1},u_{2}}(n)/π]] + 1.

We will fix λ_{1} ∈ ℝ and a corresponding solution u_{1} and choose a second solution u(λ, n) with λ ∈ [λ_{1}, λ_{2}]. Now let us consider

(4.61)  W_{n}(u_{1}, u(λ)) = −a(n)ρ_{u_{1}}(n)ρ_{u}(λ, n) sin(Δ_{u_{1},u}(λ, n))

as a function of λ ∈ [λ_{1}, λ_{2}]. As in the case of solutions, we first show that the normalization (4.60) is compatible with continuity in λ.

Lemma 4.14. Suppose Δ_{u_{1},u}(λ_{1}, .) satisfies (4.60), then we have

(4.62)  Δ_{u_{1},u}(λ, n) = θ_{u}(λ, n) − θ_{u_{1}}(n),

where θ_{u}(λ, .), θ_{u_{1}}(.) both satisfy (4.13). That is, Δ_{u_{1},u}(., n) ∈ C([λ_{1}, λ_{2}], ℝ) and (4.60) holds for all Δ_{u_{1},u}(λ, .) with λ ∈ [λ_{1}, λ_{2}]. In particular, the second inequality in (4.13) is attained if and only if n is a node of W_{.}(u_{1}, u(λ)). Moreover, denote by #_{(m,n)}W (u_{1}, u_{2}) the total number of nodes of W_{.}(u_{1}, u_{2}) between m and n. Then

(4.63)  #_{(m,n)}W (u_{1}, u_{2}) = [[Δ_{u_{1},u_{2}}(n)/π]] − lim_{ε↓0}[[Δ_{u_{1},u_{2}}(m)/π + ε]]

and

#W (u_{1}, u_{2}) = #_{(−∞,∞)}W (u_{1}, u_{2})
(4.64)  = lim_{n→∞} ([[Δ_{u_{1},u_{2}}(n)/π]] − [[Δ_{u_{1},u_{2}}(−n)/π]]).

Proof. We fix n and set

(4.65)  Δ_{u_{1},u}(λ, n) = kπ + δ(λ), Δ_{u_{1},u}(λ, n + 1) = kπ + Δ(λ),

where k ∈ ℤ, δ(λ_{1}) ∈ (0, π] and Δ(λ_{1}) ∈ (0, 2π]. Clearly (4.62) holds for λ = λ_{1} since W_{.}(u_{1}, u(λ_{1})) is constant. If (4.60) should break down, we must have one of the following cases for some λ_{0} ≥ λ_{1}. (i) δ(λ_{0}) = 0, Δ(λ_{0}) ∈ (π, 2π], or (ii) δ(λ_{0}) = π, Δ(λ_{0}) ∈ (0, π], or (iii) Δ(λ_{0}) = 2π, δ(λ_{0}) ∈ (π, π], or (iv) Δ(λ_{0}) = 0, δ(λ_{0}) ∈ (π, π]. For notational convenience let us set δ = δ(λ_{0}), Δ = Δ(λ_{0}) and θ_{u_{1}}(n) = θ_{1}(n), θ_{u}(λ_{0}, n) = θ_{2}(n). Furthermore, we can assume θ_{1,2}(n) = k_{1,2}π + δ_{1,2}, θ_{1,2}(n + 1) = k_{1,2}π + Δ_{1,2} with k_{1,2} ∈ ℤ, δ_{1,2} ∈ (0, π] and Δ_{1,2} ∈ (0, 2π].

Suppose (i). Then

(4.66)  W_{n+1}(u_{1}, u(λ_{0})) = (λ_{0} − λ_{1})u_{1}(n + 1)u(λ_{0}, n + 1).

Inserting Prüfer variables shows

(4.67)  sin(Δ_{2} − Δ_{1}) = ρ cos^{2}(δ_{1}) ≥ 0

for some ρ > 0 since δ = 0 implies δ_{1} = δ_{2}. Moreover, k = (k_{2} − k_{1}) mod 2 and kπ + Δ = (k_{2} − k_{1})π + Δ_{2} − Δ_{1} implies Δ = (Δ_{2} − Δ_{1}) mod 2π. Hence we have sin Δ ≥ 0 and Δ ∈ (π, 2π] implies Δ = 2π. But this says δ_{1} = δ_{2} = π/2 and Δ_{1} = Δ_{2} = π. Since we have at least δ(λ_{2} − ε) > 0 and hence δ_{2}(λ_{2} − ε) > π/2, Δ_{2}(λ_{2} − ε) > π for ε > 0 sufficiently small. Thus from Δ(λ_{2} − ε) ∈ (π, 2π) we get

(4.68)  0 > sin Δ(λ_{2} − ε) = sin(Δ_{2}(λ_{2} − ε) − π) > 0,

contradicting (i).

Suppose (ii). Again by (4.66) we have sin(Δ_{2} − Δ_{1}) ≥ 0 since δ_{1} = δ_{2}. But now (k + 1) = (k_{1} − k_{2}) mod 2. Furthermore, sin(Δ_{2} − Δ_{1}) = − sin(Δ) ≥ 0 says Δ = π since Δ ∈ (0, π]. Again this implies δ_{1} = δ_{2} = π/2 and Δ_{1} = Δ_{2} = π. But since δ(λ) increases/decreases precisely if Δ(λ) increases/decreases for λ near λ_{0}, (4.60) stays valid.

Suppose (iii) or (iv). Then

(4.69)  W_{n}(u_{1}, u(λ_{0})) = −(λ_{0} − λ_{1})u_{1}(n + 1)u(λ_{0}, n + 1).

Inserting Prüfer variables gives

(4.70)  sin(δ_{2} − δ_{1}) = −ρ sin(Δ_{1}) sin(Δ_{2})

for some ρ > 0. We first assume δ_{2} > δ_{1}. In this case we infer k = (k_{2} − k_{1}) mod 2 implying Δ_{2} − Δ_{1} = 0 mod 2π contradicting (4.70). Next assume δ_{2} ≤ δ_{1}. Then we obtain (k + 1) = (k_{2} − k_{1}) mod 2 implying Δ_{2} − Δ_{1} = π mod 2π and hence sin(δ_{2} − δ_{1}) ≥ 0 from (4.70). Thus we get δ_{1} = δ_{2} = π/2 Δ_{1} = Δ_{2} = π, and hence Δ_{2} − Δ_{1} = 0 mod 2π contradicting (iii), (iv). This settles (4.62).

Furthermore, if Δ(λ) ∈ (0, π], we have no node at n since δ(λ) = π implies Δ(λ) = π by (ii). Conversely, if Δ(λ) ∈ (π, 2π] we have a node at n since Δ(λ) = 2π is impossible by (iii). The rest being straightforward. □

Equations (4.16), (4.62), and (4.63) imply

Corollary 4.15. Let λ_{1} ≤ λ_{2} and suppose u_{1,2} satisfy τu_{1,2} = λ_{1,2}u_{1,2}, respectively. Then we have

(4.71)  |#_{(n,m)}W (u_{1}, u_{2}) − (#_{(n,m)}(u_{2}) − #_{(n,m)}(u_{1}))| ≤ 2

Now we come to a renormalized version of Theorem 4.13. We first need the result for a finite interval.

Theorem 4.16. Fix n_{1} < n_{2} and λ_{1} < λ_{2}. Then

(4.72)  dim Ran P_{(λ_{1},λ_{2})}(H_{n_{1},n_{2}}) = #_{(n_{1},n_{2})}W (s(λ_{1}, ., n_{1}), s(λ_{2}, ., n_{2})).

Proof. We abbreviate

(4.73)  Δ(λ, n) = Δ_{s(λ_{1},.,n_{1}),s(λ,.,n_{2})}(n)

and normalize (perhaps after flipping the sign of s(λ_{1}, ., n_{1})) Δ(λ_{1}, n) ∈ (0, π]. From (4.21) we infer

(4.74)  dim Ran P_{(λ_{1},λ_{2})}(H_{n_{1},n_{2}}) = − lim_{ε↓0}[[Δ(λ_{2}, n_{1})/π + ε]]

since λ ∈ σ(H_{n_{1},n_{2}}) is equivalent to Δ(λ, n_{1}) = 0 mod π. Using (4.63) completes the proof. □

Theorem 4.17. Fix λ_{1} < λ_{2}. Then

(4.75)  dim Ran P_{(λ_{1},λ_{2})}(H_{+}) = #_{(0,+∞)}W (s(λ_{1}), s(λ_{2})).

The same theorem holds if + is replaced by −.

Proof. Again, we only prove the result for H_{+}. Set k = #_{(0,∞)}W (s(λ_{1}), s(λ_{2})) provided this number is finite and k ∈ ℕ otherwise. We abbreviate

(4.76)  Δ(λ, n) = Δ_{s(λ_{1}),s(λ)}(n)

and normalize Δ(λ_{1}, n) = 0 implying Δ(λ, n) > 0 for λ > λ_{1}. Hence, if we choose n so large that all k nodes are to the left of n, we have

(4.77)  Δ(λ, n) > kπ.

Thus we can find λ_{1} < λ̂_{1} < ⋯ < λ̂_{k} < λ_{2} with Δ(λ̂_{j}, n) = jπ. Now define

(4.78)  η_{j}(m) = \{ \begin{matrix}s(λ̂_{j}, m) − ρ_{j}s(λ_{1}, m), & m ≤ n \\ 0, & m ≥ n \end{matrix} ,

where ρ_{j} ≠ 0 is chosen such that s(λ̂_{j}, m) = ρ_{j}s(λ_{1}, m) for m = n, n + 1. Furthermore observe that

(4.79)  τη_{j}(m) = \{ \begin{matrix}λ̂_{j}s(λ̂_{j}, m) − λ_{1}ρ_{1}s(λ_{1}, m), & m ≤ n \\ 0, & m ≥ n \end{matrix}

and that s(λ_{1}, m), s(λ̂_{j}, .), 1 ≤ j ≤ k, are orthogonal on 1, …, n. Next, let η = ∑_{j=1}^{k} c_{j}η_{j}, c_{j} ∈ ℂ, be an arbitrary linear combination, then a short calculation verifies

(4.80)  ∥(H_{+} − \frac{λ_{2} + λ_{1}}{2} )η∥ < \frac{λ_{2} − λ_{1}}{2} ∥η∥.

Invoking Lemma 4.6 gives

(4.81)  dim Ran P_{(λ_{1},λ_{2})}(H_{+}) ≥ k.

To prove the reversed inequality is only necessary if #_{(0,∞)}W (s(λ_{1}),s(λ_{2})) < ∞. In this case we look at H_{0,n}^{∞,β} with β = s(λ_{2}, n + 1)/s(λ_{2}, n). By Theorem 4.16 and Remark 4.8 (ii) we have

(4.82)  dim Ran P_{(λ_{1},λ_{2})}(H̃_{0,n}^{∞,β}) = #_{(0,n)}W (s(λ_{1}), s(λ_{2})).

Now use strong convergence of H̃_{0,n}^{∞,β} = H_{0,n}^{∞,β} ⊕ λ_{1}1l to H_{+} as n → ∞ to obtain

(4.83)  dim Ran P_{(λ_{1},λ_{2})}(H_{+}) ≤ lim inf_{n→∞} dim Ran P_{(λ_{1},λ_{2})}(H̃_{0,n}^{∞,β}) = k

completing the proof. □

As a consequence we obtain the analog of Corollary 4.9.

Corollary 4.18. Let u_{1,2} satisfy τu_{1,2} = λ_{1,2}u_{1,2}. Then

(4.84)  #_{(0,±∞)}W (u_{1}, u_{2}) < ∞ ⇔ dim Ran P_{(λ_{1},λ_{2})}(H_{±}) < ∞.

Proof. By (4.18) and Corollary 4.15 we learn that #_{(0,±∞)}W (u_{1}, u_{2}) is finite if and only if #_{(0,±∞)}W (s(λ_{1}), s(λ_{2})) is finite. □

Finally, we turn to our main result for Jacobi operators H on ℤ.

Theorem 4.19. Fix λ_{1} < λ_{2} and suppose [λ_{1}, λ_{2}] ∩ σ_{ess}(H) = ∅. Then

dim Ran P_{(λ_{1},λ_{2})}(H) = #W (u_{∓}(λ_{1}), u_{±}(λ_{2}))
(4.85)  = #W (u_{±}(λ_{1}), u_{±}(λ_{2})).

Proof. Since the proof is similar to the proof of Theorem 4.17 we will only outline the first part. Let k = #W (u_{+}(λ_{1}), u_{−}(λ_{2})) if this number is finite and k ∈ ℕ otherwise. Pick n > 0 so large that all zeros of the Wronskian are between −n and n. We abbreviate

(4.86)  Δ(λ, n) = Δ_{u_{+}(λ_{1}),u_{−}(λ)}(n)

and normalize Δ(λ_{1}, n) ∈ [0, π) implying Δ(λ, n) > 0 for λ > λ_{1}. Hence, if we choose n ∈ ℕ so large that all k nodes are between −n and n, we can assume

(4.87)  Δ(λ, n) > kπ.

Thus we can find λ_{1} < λ̂_{1} < ⋯ < λ̂_{k} < λ_{2} with Δ(λ̂_{j}, n) = 0 mod π. Now define

(4.88)  η_{j}(m) = \{ \begin{matrix}u_{−}(λ̂_{j}, m) & m ≤ n \\ ρ_{j}u_{+}(λ_{1}, m) & m ≥ n \end{matrix} ,

where ρ_{j} ≠ 0 is chosen such that u_{−}(λ̂_{j}, m) = ρ_{j}u_{+}(λ_{1}, m) for m = n, n + 1. Now proceed as in the previous theorems. □

Again, we infer as a consequence.

Corollary 4.20. Let u_{1,2} satisfy τu_{1,2} = λ_{1,2}u_{1,2}. Then

(4.89)  #W (u_{1}, u_{2}) < ∞ ⇔ dim Ran P_{(λ_{1},λ_{2})}(H) < ∞.

Proof. Follows from Corollary 4.18 and dim Ran P_{(λ_{1},λ_{2})}(H) finite if and only if both dim Ran P_{(λ_{1},λ_{2})}(H_{−}) and dim Ran P_{(λ_{1},λ_{2})}(H_{+}) finite (see (3.22)). □

Remark 4.21. (i). Lemma 1.6 shows that all theorems remain valid if our hypothesis a(n) < 0 is replaced by a(n) ≠ 0. (ii). Defining u_{±}(λ, n) as in Remark 2.3 for λ at the boundary of σ_{ess}(H), one sees that Theorem 4.19 holds for [λ_{1}, λ_{2}] ∩ σ_{ess}(H) ⊆ {λ_{1}, λ_{2}}.

Random Jacobi operators

Up to this point we have only considered the case of Jacobi operators where the sequences a and b are assumed to be known. However, this assumption is often not fulfilled in physical applications. For example, look at the one-dimensional crystal model in Section 1.5. In a more realistic setting such a crystal will contain impurities, the precise locations of which are in general unknown. All one might know is that these impurities occur with a certain probability. This leads us to the study of Jacobi operators, where the coefficients are random variables. These random Jacobi operators serve also as a model in solid state physics of disordered systems, such as alloys, glasses, and amorphous materials in the so called tight binding approximation.

5.1. Random Jacobi operators

To begin with, let us introduce the proper setting for this new model. We consider the probability space (Ω, F, μ), where F is a σ-algebra on Ω and μ a probability measure (i.e., μ(Ω) = 1) on (Ω, F). A random variable f is a measurable functions on Ω and its expectation is 𝔼(f) = ∫_{Ω} f(ω)dμ(ω).

If you are not familiar with the basics of probability (respectively measure) theory, you might want to refer to any textbook (e.g., [39]) first.

We will choose

(5.1)  Ω = Ω_{0}^{ℤ} = {ω = (ω(j))_{j∈ℤ}|ω(j) ∈ Ω_{0}},

where Ω_{0} is a bounded Borel subset of (ℝ\{0})×ℝ, and F is the σ-algebra generated by the cylinder sets (i.e., by sets of the form {ω ∈ Ω|ω(j) ∈ B_{i}, 1 ≤ j ≤ n} with B_{i} Borel subsets of Ω_{0}).

On Ω we have a discrete invertible dynamical system, namely the shift operators

(5.2)  T^{i} : \begin{matrix}Ω & → & Ω \\ ω(j) & ↦ & ω(j − i) \end{matrix} .

Since the shift of a cylinder set is again one, T (= T^{1}) is measurable.

The measure μ is called invariant (or stationary) if μ(T F) = μ(F) for any F ∈ F. Alternatively, T is called measure preserving. In addition, our dynamical system T is called ergodic (with respect to μ) if any shift invariant set F = T F has probability μ(F) zero or one.

Given (Ω, F, μ) as above we can define random variables

a : \begin{matrix}Ω & → & ℓ^{∞}(ℤ, ℝ\\{0\}) \\ ω(j) & ↦ & a_{ω}(n) = ω_{1}(n) \end{matrix}
(5.3)  b : \begin{matrix}Ω & → & ℓ^{∞}(ℤ, ℝ\\{0\}) \\ ω(j) & ↦ & b_{ω}(n) = ω_{2}(n) \end{matrix}

and a corresponding random Jacobi operator

(5.4)  H_{ω} = a_{ω}S^{+} + a_{ω}^{−}S^{−} + b_{ω}.

Since we assumed Ω_{0} bounded, there is a constant C (independent of ω) such that ∥H_{ω}∥ ≤ C. The simplest example is to take a probability measure μ_{0} on Ω_{0} and consider the product measure μ = ×_{i∈ℤ}μ_{0}. It is easy to see that μ is ergodic. This is known as Anderson model.

Now what are the questions we should ask about H_{ω}? Clearly, if we look for features shared by all operators H_{ω}, ω ∈ Ω, we could as well think of ω as being fixed. So we will look for properties which only hold with a certain probability. For example, properties which hold almost surely (a.s.), that is with probability one.

From now on we will assume the following hypothesis throughout this entire chapter.

Hypothesis H.5.1. Let (Ω, F, μ) be as described in (5.1) and let T, as introduced in (5.2), be ergodic with respect to μ.

As a warm up we recall a simple lemma concerning invariant random variables. A random variable f is called invariant (with respect to T^{i}) if f(T^{i}ω) = f(ω).

Lemma 5.2. An invariant random variable f : Ω → ℝ ∪ {∞} is constant almost surely (i.e., there is a set with probability one on which f is constant).

Proof. Let I_{i}^{n} = [\frac{i}{2^{n}} , \frac{i+1}{2^{n}}), I_{∞}^{n} = \{∞\} and Ω_{i}^{n} = f^{−1}(I_{i}^{n}) = \{ω|\frac{i}{2^{n}} ≤ f < \frac{i+1}{2^{n}}\}, Ω_{∞}^{n} = f^{−1}(∞). The sets Ω_{i}^{n}, i ∈ ℤ ∪ {∞}, are disjoint and μ(Ω_{i}^{n}) ∈ {0, 1} due to ergodicity. Moreover, since

(5.5)  ∑_{i∈ℤ∪{∞}} μ(Ω_{i}^{n}) = μ( ⋃_{i∈ℤ∪{∞}} Ω_{i}^{n}) = μ(Ω) = 1

there exists i_{n} ∈ ℤ ∪ {∞} such that μ(Ω_{i_{n}}^{n}) = 1 and μ(Ω_{i}^{n}) = 0 , i ≠ i_{n}. And since the nesting intervals I_{i_{n+1}}^{n+1} ⊂ I_{i_{n}}^{n} must converge to a point {x_{0}} = ⋂_{n∈ℕ} I_{i_{n}}^{n} we have

(5.6)  μ(f^{−1}(x_{0})) = μ( ⋂_{n∈ℕ} Ω_{i_{n}}^{n}) = lim_{n→∞} μ(Ω_{i_{n}}^{n}) = 1.

□

This result clearly suggests to look at properties of H_{ω} which are invariant under the shift T^{i}. Since we have

(5.7)  H_{T^{i}ω} = S^{i}H_{ω}S^{−i},

we see that H_{T^{i}ω} and H_{ω} are unitarily equivalent. Hence, one might expect that the spectrum is constant a.s.. Since the spectrum itself is no random variable we need to look for random variables which contain information on the location of the spectrum. Promising candidates are traces of the spectral projections

(5.8)  tr(P_{Λ}(H_{ω})) = dim RanP_{Λ}(H_{ω}).

Based on this observation we obtain the following theorem.

Theorem 5.3. There exists a set Σ ⊂ ℝ such that

(5.9)  σ(H_{ω}) = Σ a.s. .

In addition

(5.10)  σ_{d}(H_{ω}) = ∅ a.s. .

Proof. We start by claiming that the function ω ↦ P_{Λ}(H_{ω}) is weakly measurable (i.e., ⟨f, P_{Λ}(H_{ω})g⟩ is measurable for any f, g ∈ ℓ^{2}(ℤ); by the polarization identity it suffices to consider the case f = g). In fact, ω ↦ H_{ω} is weakly measurable and so is any polynomial of H_{ω}, since products of weakly measurable functions are again weakly measurable. Approximating P_{Λ}(H_{ω}) by (ω-independent) polynomials in the strong topology, the claim follows. As a consequence, the trace

(5.11)  tr(P_{Λ}(H_{ω})) = ∑_{n∈ℤ}⟨δ_{n}, P_{Λ}(H_{ω})δ_{n}⟩

is an invariant (by (5.7)) random variable and hence a.s. constant by the lemma. Moreover,

tr(P_{Λ}(H_{ω})) = 𝔼(tr(P_{Λ}(H_{ω})))
= 𝔼( ∑_{n∈ℤ}⟨δ_{n}, P_{Λ}(H_{ω})δ_{n}⟩) = ∑_{n∈ℤ} 𝔼(⟨δ_{n}, P_{Λ}(H_{ω})δ_{n}⟩)
(5.12)  = ∑_{n∈ℤ} 𝔼(⟨δ_{0}, P_{Λ}(H_{T^{n}ω})δ_{0}⟩) = ∑_{n∈ℤ} 𝔼(⟨δ_{0}, P_{Λ}(H_{ω})δ_{0}⟩)

is zero or infinity depending on 𝔼(⟨δ_{0}, P_{Λ}(H_{ω})δ_{0}⟩) = 0 or not. Here we have used invariance of μ with respect to T^{n} in the last equality.

For any pair of rational numbers (p, q) ∈ ℚ^{2} we set d_{(p,q)} = 0, ∞ if the almost sure value of trP_{(p,q)}(H_{ω}) is 0, ∞, respectively. Set

(5.13)  Ω̃ = ⋂_{(p,q)∈ℚ^{2}} Ω_{(p,q)}, Ω_{(p,q)} = {ω|trP_{(p,q)}(H_{ω}) = d_{(p,q)}}.

Then μ(Ω̃) = 1 for μ(Ω_{(p,q)}) = 1 and the intersection is countable.

Now for ω_{1}, ω_{2} ∈ Ω̃ we claim σ(H_{ω_{1}}) = σ(H_{ω_{2}}). Indeed, if λ ∉ σ(H_{ω_{1}}), there exist two rational numbers λ_{1} < λ < λ_{2} such that

(5.14)  0 = dim RanP_{(λ_{1},λ_{2})}(H_{ω_{1}}) = d_{(λ_{1},λ_{2})} = dim RanP_{(λ_{1},λ_{2})}(H_{ω_{1}})

implying λ ∉ σ(H_{ω_{2}}). Interchanging ω_{1} and ω_{2} finishes the first part.

For the remaining one suppose λ ∈ σ_{d}(H_{ω}). Then

(5.15)  0 < dim RanP_{(λ_{1},λ_{2})}(H_{ω}) < ∞

with λ_{1} < λ < λ_{2} sufficiently close to λ. This is impossible for ω ∈ Ω̃. So σ_{d}(H_{ω}) = ∅, ω ∈ Ω̃. □

Considering the corresponding projections onto the absolutely continuous, singular continuous, and pure point subspaces we expect a similar result for the corresponding spectra. The tricky part is to establish measurability of these projections.

Theorem 5.4. There exist sets Σ_{ac}, Σ_{sc}, and Σ_{pp} such that

σ_{ac}(H_{ω}) = Σ_{ac} a.s. ,
σ_{sc}(H_{ω}) = Σ_{sc} a.s. ,
(5.16)  σ_{pp}(H_{ω}) = Σ_{pp} a.s. .

Proof. Given the proof of the previous theorem, we only need to show that two of the projections P^{ac}(H), P^{sc}(H), and P^{pp}(H) are weakly measurable (for fixed ω, which is omitted here for notational simplicity). Indeed, we have P_{Λ}^{ac}(H) = P^{ac}(H)P_{Λ}(H), etc.. We will show that P^{c}(H) = P^{ac}(H) + P^{sc}(H) and P^{s}(H) = P^{sc}(H) + P^{pp}(H) are weakly measurable.

By the RAGE Theorem ([50], Theorem 5.8) we have

(5.17)  lim_{T →∞} \frac{1}{T} ∥∫_{0}^{T} e^{itH}χ_{N}P^{c}(H)e^{−itH} ∥ = 0,

where χ_{N}(n) = 1 if |n| ≤ N and χ_{N}(n) = 0 if |n| > N. Hence

⟨f, P^{c}(H)g⟩ = \frac{1}{T} ∫_{0}^{T} ⟨f, e^{itH}(1l − χ_{N} + χ_{N})P^{c}(H)e^{−itH}g⟩dt
= lim_{T →∞} \frac{1}{T} ∫_{0}^{T} ⟨f, e^{itH}(1l − χ_{N})P^{c}(H)e^{−itH}g⟩dt
= lim_{T →∞} \frac{1}{T} ∫_{0}^{T} ⟨f, e^{itH}(1l − χ_{N})e^{−itH}g⟩dt

(5.18) − lim_{T →∞} \frac{1}{T} ∫_{0}^{T} ⟨f, e^{itH}(1l − χ_{N})P^{pp}(H)e^{−itH}g⟩dt

Now we estimate the last term using P^{pp}(H)g = ∑_{j} g_{j}, where g_{j} are eigenfunctions of H, that is, Hg_{j} = λ_{j}g_{j}.

⟨f, e^{itH}(1l − χ_{N})P^{pp}(H)e^{−itH}g⟩ ≤ ∥e^{−itH}f∥∥(1l − χ_{N})e^{−itH}P^{pp}(H)g∥

≤ ∥f∥∥ ∑_{j}(1l − χ_{N})e^{−itλ_{j}}g_{j}∥ ≤ ∥f∥ ∑_{j} ∥(1l − χ_{N})g_{j}∥. (5.19)

Hence we have

(5.20)  ⟨f, P^{c}(H)g⟩ = lim_{N→∞} lim_{T →∞} \frac{1}{T} ∫_{0}^{T} ⟨f, e^{itH}(1l − χ_{N})e^{−itH}g⟩dt

which is measurable since e^{itH} = ∑_{j=0}^{∞}(itH)^{j}/j!.

Finally, P^{s}(H) is measurable because of

(5.21)  ⟨f, P^{s}(H)g⟩ = lim_{n→∞} sup_{I∈I,|I|<1/n}⟨f, P_{I}(H)g⟩

(see Lemma B.6). □

Since eigenvalues of H_{ω} are simple we get another interesting result.

Theorem 5.5. For any λ ∈ ℝ we have μ({ω|λ ∈ σ_{p}(H_{ω})}) = 0.

Proof. Since trP_{{λ}}(H_{ω}) equals zero or infinity a.s. its probability of being one is zero. □

This does not say that Σ_{pp} is empty. But if it is non-empty, it must be locally uncountable.

5.2. The Lyapunov exponent and the density of states

In this section we will need that a_{ω} is bounded away from zero.

Hypothesis H. 5.6. Suppose Ω_{0} ⊆ [M^{−1}, M] × [−M, M] for some M ≥ 1, that is,

(5.22)  \frac{1}{M} ≤ a_{ω} ≤ M,  −M ≤ b_{ω} ≤ M.

In particular, we have ∥H_{ω}∥ ≤ 3M.

We recall the definition of the Lyapunov exponent of H_{ω} from Section 1.1

(5.23)  γ_{ω}^{±}(z) = lim_{n→±∞} \frac{1}{|n|} ln ∥Φ_{ω}(z, n)∥

provided this limit exists. For its investigation we need some standard results from ergodic theory.

A sequence of random variables f_{n} : Ω → ℝ is called a subadditive process if

(5.24)  f_{m+n}(ω) ≤ f_{m}(ω) + f_{n}(T^{m}ω),

where T is measure preserving. If equality holds, the process is called additive. Then we have the subadditive ergodic theorem by Kingman [148] (see [214] for a simple proof).

Theorem 5.7. If {f_{n}}_{n∈ℕ} is a subadditive process satisfying 𝔼(|f_{n}|) < ∞, n ∈ ℕ, and inf_{n} 𝔼(f_{n})/n > −∞, then f_{n}(ω)/n converges almost surely and, if T is ergodic,

(5.25)  lim_{n→∞} \frac{1}{n}f_{n}(ω) = lim_{n→∞} \frac{1}{n}𝔼(f_{n}) = inf_{n∈ℕ} \frac{1}{n}𝔼(f_{n}) a.s. .

In the additive case we have f_{n}(ω) = ∑_{m=0}^{n−1} f_{1}(T^{m}ω) and the theorem says that lim_{n→∞} f_{n}(ω)/n = 𝔼(f_{1}) almost surely if 𝔼(|f_{1}|) < ∞. This is known as Birkhoff’s ergodic theorem.

Now we can prove the following result.

Theorem 5.8. For each z ∈ ℂ there exists a number

(5.26)  γ(z) = inf_{n∈ℕ} \frac{1}{n}𝔼(ln ∥Φ_{ω}(z, n)∥)

(independent of ω) such that

(5.27)  lim_{n→±∞} \frac{1}{|n|} ln ∥Φ_{ω}(z, n)∥ = γ(z) a.s. .

Proof. Abbreviate f_{n}(ω) = ln ∥Φ_{ω}(z, n)∥, then f_{n} is a subadditive process since by (1.32)

f_{m+n}(ω) = ln ∥Φ_{ω}(z, n + m, m)Φ_{ω}(z, m)∥
(5.28)  ≤ f_{n}(T^{−m}ω) + f_{m}(ω).

Moreover, for n > 0 (recall (1.33))

(5.29)  \frac{1}{M} ≤ ∥Φ_{ω}(z, n)∥ ≤ \begin{matrix}n \\ ∏ & ∗ \\ j=1 \end{matrix} ∥U_{ω}(z, j)∥ ≤ (M(M + |z|))^{n}

implies 𝔼(|f_{n}|) ≤ nM(M + |z|) and inf_{n∈ℕ} 𝔼(f_{n})/n ≥ 0. Since the same considerations apply to n < 0 we infer from Theorem 5.7

(5.30)  lim_{n→±∞} \frac{f_{n}(ω)}{|n|} = inf_{n∈±ℕ} \frac{1}{|n|}𝔼(f_{n})

almost surely. It remains to show that both limits are equal, provided both exist. This follows using (1.32), (1.45), and invariance of our measure

𝔼(f_{−n}) = 𝔼(ln ∥Φ_{ω}(z, −n, 0)∥) = 𝔼(ln ∥Φ_{ω}(z, 0, −n)^{−1}∥)
= 𝔼(ln ∥Φ_{T^{n}ω}(z, n, 0)^{−1}∥) = 𝔼(ln ∥Φ_{ω}(z, n, 0)^{−1}∥)
(5.31)  = 𝔼(ln a_{ω}(n) − ln a_{ω}(0) + ln ∥Φ_{ω}(z, n, 0)∥) = 𝔼(f_{n}).
□

In particular this result says that for fixed z ∈ ℂ we have γ_{ω}^{+}(z) = γ_{ω}^{−}(z) = γ(z) almost surely.

We call γ(z) the Lyapunov exponent. It has several interesting properties. First of all, by the considerations after (1.36) we have γ(z) ≥ 0. Before we can proceed, we need to review the concept of a subharmonic function.

A measurable function f : ℂ → ℝ ∪ {−∞} is called submean if max(f, 0) is locally integrable and

(5.32)  f(z_{0}) ≤ \frac{1}{2πir^{2}} ∫_{|z−z_{0}|≤r} f(z) d\overline{z} ∧ dz, r > 0.

It is called uppersemicontinuous (u.s.c.) if for any sequence z_{n} → z_{0} we have

(5.33)  lim sup_{z_{n}→z_{0}} f(z) ≤ f(z_{0}).

If f is both, submean and u.s.c., it is called subharmonic. As an immediate consequence we obtain that

(5.34)  f(z_{0}) = lim_{r↓0} \frac{1}{2πir^{2}} ∫_{|z−z_{0}|≤r} f(z) d\overline{z} ∧ dz

if f is subharmonic. The properties of being submean or subharmonic are preserved by certain limiting operations.

Lemma 5.9. (i). Suppose f_{n} are submean and f_{n+1} ≤ f_{n}. Then f(z) = inf_{n} f_{n}(z) is submean. If, in addition, f_{n} are subharmonic, then so is f. (ii). Suppose f_{n} are submean and sup_{n} max(f_{n}, 0) is locally integrable. Then sup_{n} f_{n} and lim sup_{n} f_{n} are submean.

Proof. (i). Switching to f_{n} − f_{1} it is no restriction to assume f_{n} ≤ 0 and hence the first assertion follows from the monotone convergence theorem. The second one follows since the infimum of u.s.c. functions is again u.s.c.. (ii). Since

sup_{n} f_{n}(z_{0}) ≤ \frac{1}{2πir^{2}} sup_{n} ∫_{|z−z_{0}|≤r} f_{n}(z) d\overline{z} ∧ dz

(5.35)  ≤ \frac{1}{2πir^{2}} ∫_{|z−z_{0}|≤r} sup_{n} f_{n}(z) d\overline{z} ∧ dz,

we see that sup_{n} f_{n} is submean. Combining this result with the previous one and lim sup_{n} f_{n} = inf_{N} sup_{n≥N} f_{n} finishes the proof. □

As a first observation note that the norm ∥Φ_{ω}(z, n)∥ is subharmonic. In fact, ∥Φ_{ω}(z, n)∥ ≥ 1/M is continuous and submean by (5.36)

∥Φ_{ω}(z_{0}, n)∥ = ∥\frac{1}{2πir^{2}} ∫_{|z−z_{0}|≤r}Φ_{ω}(z, n)d\overline{z} ∧ dz∥ ≤ \frac{1}{2πir^{2}} ∫_{|z−z_{0}|≤r}∥Φ_{ω}(z, n)∥d\overline{z} ∧ dz.

Moreover, since the logarithm is convex, ln ∥Φ_{ω}(z, n)∥ is submean by Jensen’s inequality.

This is already half of the proof that γ(z) is subharmonic.

Lemma 5.10. The Lyapunov exponent γ(z) is subharmonic. Furthermore, the upper Lyapunov exponents \overline{γ}_{ω}^{±}(z) of H_{ω} are submean.

Proof. By Fubini’s theorem the function

(5.37)  𝔼(ln ∥Φ_{ω}(z, n)∥)

is submean. By the estimate (5.29) and the dominated convergence theorem it is also continuous and hence subharmonic. Thus, γ(z) is the limit of a monotone not increasing sequence of subharmonic functions

(5.38)  \frac{1}{2^{n}} 𝔼(ln ∥Φ_{ω}(z, 2^{n})∥) ↘ γ(z).

The rest follows from our previous lemma. □

This result looks pretty technical at the first sight, however, in connection with (5.34) it says that in order to show γ(z) = γ̃(z) for some subharmonic function γ̃(z) it suffices to show equality for z ∈ ℂ\ℝ. This will be used in what follows since the case z ∈ ℂ\ℝ is usually easier to handle as we will see below.

For z ∈ ℂ\ℝ we know the asymptotic behavior of the solutions of τ_{ω}u = zu. There are solutions u_{ω,±}(z, n) which decay exponentially as n → ±∞ and grow exponentially in the other direction. Hence it is not hard to see that

(5.39)  γ(z) = lim_{n→±∞} \frac{1}{2n} ln (|u_{ω}(z, n)|^{2} + |u_{ω}(z, n + 1)|^{2})

for any solution u_{ω}(z, n) which is not a multiple of u_{ω,±}(z, n) and the limits exist whenever the limits in (5.27) exist. Based on this observation we obtain.

Theorem 5.11. Let E_{0} = min Σ and E_{∞} = max Σ with Σ from (5.9). For z ∈ ℂ\ℝ we have

γ(z) = lim_{n→∓∞} \frac{1}{|n|} ln |u_{ω,±}(z, n)| = − lim_{n→±∞} \frac{1}{|n|} ln |u_{ω,±}(z, n)|
(5.40)  = −𝔼(ln a_{ω}) − 𝔼(ln |m_{ω,±}(z)|)

almost surely. In addition,

(5.41)  γ(z) = −𝔼(ln a_{ω}) + ln |E_{∞} − z| − Re ∫_{E_{0}}^{E_{∞}} \frac{ξ(λ)}{λ − z} dλ,

where ξ(λ) = 𝔼(ξ_{ω,±}(λ, n)) and ξ_{ω,±}(λ, n) are the ξ-functions associated with the Weyl m-functions m_{ω,±}(z, n) (see Theorem B.11).

Proof. We first compute

lim_{n→∞} \frac{1}{2n} ln (|u_{ω,+}(z, n)|^{2} + |u_{ω,+}(z, n + 1)|^{2}) =

= lim_{n→∞} \frac{1}{2n} (ln(1 + |m_{ω,+}(z, n)|^{2}) + 2 ln |u_{ω,+}(z, n + 1)|)

= lim_{n→∞} \frac{1}{n} ∑_{j=0}^{n−1} ln |a(j)m_{ω,+}(z, j)|, (5.42)

where we have used (2.3). Since m_{ω,+}(z, n) is measurable and satisfies (2.3) we can invoke Birkhoff’s ergodic theorem to see that the limit is 𝔼(ln a_{ω})+𝔼(ln |m_{ω,+}(z)|). Moreover, using (compare (6.56) below)

(5.43)  ln m_{ω,+}(z, n) = − ln(E_{ω,∞} − z) + ∫_{E_{ω,0}}^{E_{ω,∞}} \frac{ξ_{ω,+}(λ, n)}{λ − z}dλ,

and Fubini’s theorem we see

(5.44)  𝔼(ln |m_{ω,+}(z)|) = − ln |E_{∞} − z| + Re ∫_{E_{0}}^{E_{∞}} \frac{𝔼(ξ_{ω,+}(λ, n))}{λ − z}dλ.

Computing the remaining three limits in a similar fashion the result follows from (5.39). □

Note that (5.41) implies

(5.45)  γ(λ + iε) = γ(λ − iε),  ε ∈ ℝ.

Next we want to shed some additional light on the function ξ(λ) which appeared in the last theorem. To do this we first introduce the so-called density of states. We consider the finite operators H_{ω,0,n} with Dirichlet boundary conditions at 0 and n. The integrated density of states for this operator is defined by

(5.46)  N_{n}^{ω}(λ) = \frac{1}{n} dim RanP_{(−∞,λ)}(H_{ω,0,n+1}) = \frac{1}{n}trP_{(−∞,λ)}(H_{ω,0,n+1}).

That is we take the number of eigenstates below λ and divide it by the total number of eigenstates. In particular, N_{n}^{ω}(λ) is zero below the spectrum of H_{ω,0,n+1}, one above the spectrum of H_{ω,0,n+1}, and increasing in between. By Theorem 4.7 we also have

(5.47)  N_{n}^{ω}(λ) = \frac{1}{n}#_{(0,n+1)}(s_{ω}(λ))

which is why N_{n}^{ω}(λ) is also called rotation number.

Our hope is that N_{n}^{ω}(λ) will tend to some limit as n → ∞.

If the process

(5.48)  f_{n}(ω) = trP_{(−∞,λ)}(H_{ω,0,n+1})

were subadditive, the desired result would follow directly from Kingman’s Theorem. Unfortunately this is not true! To see this observe

(5.49)  H_{ω,0,n+m+1} = H_{ω,0,m+1} ⊕ H_{T^{−m}ω,0,n+1} + M_{ω,m},

where M_{ω,m} = a_{ω}(m)(⟨δ_{m}, .⟩δ_{m+1} + ⟨δ_{m+1}, .⟩δ_{m}). In the simplest case m = n = 1 the above equation reads

(5.50)  ( \begin{matrix}b_{1} & a_{1} \\ a_{1} & b_{2} \end{matrix} ) = ( \begin{matrix}b_{1} & 0 \\ 0 & 0 \end{matrix} ) ⊕ ( \begin{matrix}0 & 0 \\ 0 & b_{2} \end{matrix} ) + a_{1} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

Let b_{+} = max(b_{1}, b_{2}) and b_{−} = min(b_{1}, b_{2}). Then the eigenvalues of this matrix are b_{−} − c and b_{+} + c for suitable c > 0. Hence

(5.51)  f_{2}(ω) = f_{1}(ω) + f_{1}(T^{−1}ω) + \{ \begin{matrix}0 , & λ ≤ b_{−} − c \\ 1 , & b_{−} − c < λ ≤ b_{−} \\ 0 , & b_{−} < λ ≤ b_{+} \\ −1, & b_{+} < λ ≤ b_{+} + c \\ 0 , & b_{+} + c < λ \end{matrix}

shows that neither f_{n}(ω) nor −f_{n}(ω) is subadditive in general.

However, since M_{ω,m} is a rank two perturbation, we have at least

(5.52)  f_{n+m}(ω) ≤ f_{m}(ω) + f_{n}(T^{−m}ω) + 2

and hence f̃_{n}(ω) = f_{n}(ω) + 2 is subadditive. Clearly f_{n}(ω)/n converges if and only if f̃_{n}(ω)/n converges in which case the limits are equal. Summarizing,

(5.53)  lim_{n→∞} N_{n}^{ω}(λ) = N(λ)

almost surely. The quantity N(λ) is called integrated density of states. Because of (5.47) it is also known as rotation number. Some of its properties are collected in the following lemma.

Lemma 5.12. There is a set Ω̃ ⊆ Ω of probability one such that for continuous f,

(5.54)  lim_{n→∞} ∫ f(λ)dN_{n}^{ω}(λ) = ∫ f(λ)dN(λ) = 𝔼(⟨δ_{0}, f(H_{ω})δ_{0}⟩), ω ∈ Ω̃.

In other words, dN_{n}^{ω} converges weakly to dN almost surely. Moreover, N(λ) is continuous and (5.54) also holds for f = χ_{(−∞,λ_{0})}.

Proof. Note that f ↦ 𝔼(⟨δ_{0}, f(H_{ω})δ_{0}⟩) is a positive linear functional on the space C([−3M, 3M]) and hence by the Riez-Markov Theorem there is a (positive probability) measure dÑ such that

(5.55)  ∫ f(λ)dÑ(λ) = 𝔼(⟨δ_{0}, f(H_{ω})δ_{0}⟩).

We begin by showing that (5.54) holds with dN replaced by dÑ. Let us choose f(λ) = λ^{k}, k ∈ ℕ_{0}, first and consider the process

(5.56)  f_{n}(ω) = tr (H_{ω,0,n+1})^{k} = ∑_{m=1}^{n}⟨δ_{m}, (H_{ω,0,n+1})^{k}δ_{m}⟩.

Since each summand depends on n (unless k = 0, 1) it is not clear that this process is (sub)additive. Hence we will try to replace it by

(5.57)  f̃_{n}(ω) = tr (H_{ω})^{k} = ∑_{m=1}^{n}⟨δ_{m}, (H_{ω})^{k}δ_{m}⟩,

which is additive. By Birkhoff’s theorem we have lim_{n→∞} f̃_{n}(ω)/n = ∫ λ^{k}dÑ(λ) for ω in a set Ω̃_{k} of probability one. Now what about the error we make when replacing f_{n} by f̃_{n}? The matrix elements ⟨δ_{m}, (H_{ω})^{k}δ_{m}⟩ and ⟨δ_{m}, (H_{ω})^{k}δ_{m}⟩ only differ for m = n, n − 1, …, n − n(k) and the difference can be estimated using a_{ω} ≤ M and |b_{ω}| ≤ M. Hence we obtain

(5.58)  |f̃_{n}(ω) − f_{n}(ω)| ≤ C_{k}(M)

and so f_{n}(ω)/n converges if and only if f̃_{n}(ω)/n converges in which case the limits are equal. Choosing Ω̃ = ∩_{k=0}^{∞}Ω̃_{k} we see that (5.54) holds if f is a polynomial.

Now let f be continuous and choose a polynomial f_{ε}(λ) such that the difference |f(λ) − f_{ε}(λ)| is at most ε/2 for all λ ∈ [−3M, 3M]. Then we have

|∫ f(λ)dÑ(λ) − ∫ f(λ)dN_{n}^{ω}(λ)| ≤ |∫ f_{ε}(λ)dN_{n}^{ω}(λ) − ∫ f_{ε}(λ)dÑ(λ)| +

+ ∫ |f(λ) − f_{ε}(λ)|dN_{n}^{ω}(λ) + ∫ |f(λ) − f_{ε}(λ)|dÑ(λ). (5.59)

Performing the limit n → ∞ we see

(5.60)  lim_{n→∞} |∫ f(λ)dÑ(λ) − ∫ f(λ)dN_{n}^{ω}(λ)| ≤ ε

and letting ε ↓ 0 establishes (5.54) for continuous f.

Let f_{ε}(λ) = ε^{2}/(λ^{2} + ε^{2}). Using dominated convergence we obtain

Ñ({λ_{0}}) = lim_{ε↓0} ∫ f_{ε}(λ)dÑ(λ) = lim_{ε↓0} 𝔼(⟨δ_{0}, f_{ε}(H_{ω})δ_{0}⟩)
(5.61)  = 𝔼(lim_{ε↓0}⟨δ_{0}, f_{ε}(H_{ω})δ_{0}⟩) = 𝔼(⟨δ_{0}, P_{{λ_{0}}}(H_{ω})δ_{0}⟩) = 0,

where the last equality follows from Theorem 5.5. This shows that Ñ(λ) is continuous.

To show (5.54) for f(λ) = χ_{(−∞,λ_{0})}(λ) we approximate f by continuous functions f_{ε} given by f_{ε}(λ) = 1 − (λ − λ_{0})/δ for λ_{0} < λ < λ_{0} + δ and f_{ε}(λ) = f(λ) else. The parameter δ is chosen such that Ñ((λ_{0} − δ, λ_{0} − δ)) ≤ ε/2 (this is possible since Ñ is continuous). Now observe (5.62)

lim_{n→∞} ∫ |f(λ) − f_{ε}(λ)|dN_{n}^{ω}(λ) ≤ lim_{n→∞} ∫ g_{ε}(λ)dN_{n}^{ω}(λ) = ∫ g_{ε}(λ)dÑ(λ) ≤ \frac{ε}{2},

where g_{ε}(λ) = (1 − |λ − λ_{0}|/δ)χ_{(λ_{0}−δ,λ_{0}+δ)}(λ), and proceed as before.

Equation (5.54) for f = χ_{(−∞,λ)} finally tells us N(λ) = Ñ(λ). □

Remark 5.13. (i). The proof of the previous lemma shows that (5.63)

N(λ) = lim_{n→∞} \frac{1}{n}tr(χ_{(0,n+1)}P_{(−∞,λ)}(H_{ω})) = lim_{n→∞} \frac{1}{n} ∑_{m=1}^{n}⟨δ_{m}, P_{(−∞,λ)}(H_{ω})δ_{m}⟩,

where the right hand side is an additive process. (ii). We can replace H_{ω,0,n+1} by H_{ω,−n−1,n+1}, that is,

(5.64)  N(λ) = lim_{n→∞} \frac{1}{2n + 1}trP_{(−∞,λ)}(H_{ω,−n−1,n+1})),

or we could choose different boundary conditions.

In particular,

(5.65)  N(λ) = lim_{n→∞} \frac{1}{n}#_{(0,n+1)}(u_{ω}(λ))

for any real (nontrivial) solution τ_{ω}u_{ω}(λ) = λu_{ω}(λ) (compare (4.18)).

Note that N(λ) can also be viewed as the average of the spectral measures dρ_{ω,n,n}(λ) since by (5.54) for f = χ_{(−∞,λ)},

(5.66)  N(λ) = 𝔼(⟨δ_{n}, P_{(−∞,λ)}(H_{ω})δ_{n}⟩) = 𝔼(ρ_{ω,n,n}((−∞, λ))).

In particular, we obtain the following corollary.

Corollary 5.14. We have σ(dN) = Σ. Moreover, N(λ) = 0 for λ ≤ E_{0} and N(λ) = 1 for λ ≥ E_{∞}.

Another consequence is the Thouless formula.

Theorem 5.15. For any z ∈ ℂ we have

(5.67)  γ(z) = −𝔼(ln a_{ω}) + ∫_{E_{0}}^{E_{∞}} ln |λ − z|dN(λ)

and N(λ) = ξ(λ).

Proof. Invoking (1.65) we get

(5.68)  \frac{1}{n} ln |s_{ω}(λ, n)| = \frac{1}{n} ∑_{m=1}^{n} ln a_{ω}(m) + ∫ ln |λ − z|dN_{n}^{ω}(λ)

for z ∈ ℂ\ℝ. Using (5.39) with u_{ω} = s_{ω} to compute γ(z) proves (5.67) for z ∈ ℂ\ℝ. Since both quantities are subharmonic (replace ln |λ − z| by ln(ε + |λ − z|) and let ε ↓ 0), equality holds for all z ∈ ℂ. Comparing (5.41) and (5.67) we obtain N(λ) = ξ(λ) after an integration by parts. □

There are a few more useful formulas. Set

(5.69)  \overline{g}(z) = 𝔼(g_{ω}(z, n)) = 𝔼(g_{ω}(z, 0)) = ∫ \frac{dN(λ)}{λ − z},

then

(5.70)  \frac{∂}{∂ε}γ(λ + iε) = ∫ \frac{ε dN(x)}{(x − λ)^{2} − ε^{2}} = Im(\overline{g}(λ + iε)), |ε| > 0.

In particular, the moments of dN are the expectations of the coefficients of the Laurent expansion of g_{ω}(z, 0) around infinity (compare Section 6.1 below)

(5.71)  𝔼(⟨δ_{0}, (H_{ω})^{j}δ_{0}⟩) = ∫ λ^{j}dN(λ).

Explicitly we have (∫ dN(λ) = 1)

∫ λ dN(λ) = 𝔼(b_{ω}(0)), ∫ λ^{2}dN(λ) = 2𝔼(a_{ω}(0)^{2}) + 𝔼(b_{ω}(0)^{2}), etc. . (5.72)

In addition, solving (2.11) for the \frac{1}{m_{ω,±}(z,n)} term, taking imaginary parts, logs, and expectations we find

(5.73)  2γ(λ + iε) = 𝔼( ln(1 + \frac{ε}{a_{ω}({0 \atop −1})Im(m_{ω,±}(λ + iε))})), |ε| > 0.

Finally, let us try to apply the results found in this section. To begin with, note that we know \underline{γ}_{ω}^{±}(z) = \overline{γ}_{ω}^{±}(z) = γ(z) a.s. only for fixed z ∈ ℂ. On the other hand, to say something about H_{ω} we need information on the (upper, lower) Lyapunov exponents of H_{ω} for all (or at least most z).

Lemma 5.16. The Lebesgue measure of the complement of the set \{λ ∈ ℝ|\underline{γ}_{ω}^{±}(λ) = \overline{γ}_{ω}^{±}(λ) = γ(λ)\} is zero almost surely. The same is true for λ ∈ ℂ. Moreover,

(5.74)  \overline{γ}_{ω}^{±}(z) ≤ γ(z) for all z a.s..

Proof. The set L = \{(λ, ω)|\underline{γ}_{ω}^{±}(z) = \overline{γ}_{ω}^{±}(z) = γ(z)\} is measurable in the product space ℝ × Ω. The bad set is B = (ℝ × Ω)\L and the corresponding sections are B_{ω} = {λ|(λ, ω) ∈ B}, B_{λ} = {ω|(λ, ω) ∈ B}. Since we know μ(B_{λ}) = 0 we obtain from Fubini’s theorem

(5.75)  (λ × μ)(B) = ∫_{Ω} |B_{ω}|dμ(ω) = ∫_{ℝ} μ(B_{λ})dλ = 0

that |B_{ω}| = 0 a.s. as claimed. Similar for λ ∈ ℂ. To prove the remaining assertion, we use that \overline{γ}_{ω}^{±}(z) is submean

\overline{γ}_{ω}^{±}(z_{0}) ≤ \frac{1}{2πir^{2}} ∫_{|z−z_{0}|≤r}\overline{γ}_{ω}^{±}(z) d\overline{z} ∧ dz
(5.76)  = \frac{1}{2πir^{2}} ∫_{|z−z_{0}|≤r}γ(z) d\overline{z} ∧ dz = γ(z_{0}).
□

Now we are ready to prove the following theorem which should be compared with Theorem 3.16.

Theorem 5.17. The absolutely continuous spectrum of H_{ω} is given by

(5.77)  Σ_{ac} = \overline{\{λ ∈ ℝ|γ(λ) = 0\}}^{ess}

almost surely.

Proof. Combining Theorem 3.16 and the previous lemma we have immediately Σ_{ac} ⊆ \overline{\{λ ∈ ℝ|γ(λ) = 0\}}^{ess}. The converse is a little harder.

Abbreviate N = {λ ∈ ℝ|γ(λ) = 0}. Using x(1 + x/2)^{−1} ≤ ln(1 + x), x ≥ 0, we obtain from (5.73)

(5.78)  𝔼(\frac{ε}{n_{ω,±}(λ + iε)}) ≤ 𝔼( ln(1 + \frac{ε}{n_{ω,±}(λ + iε) − \frac{ε}{2}} )) = 2γ(λ + iε),

where n_{ω,±}(z) = a({0 \atop −1})Im(m_{ω,±}(z)) + \frac{Im(z)}{2}. Moreover, we have by (5.70) and Fatou’s lemma

𝔼( lim inf_{ε↓0} \frac{1}{n_{ω,±}(λ + iε)}) ≤ lim inf_{ε↓0} 𝔼(\frac{1}{n_{ω,±}(λ + iε)})
(5.79)  = 2 lim inf_{ε↓0} \frac{γ(λ + iε)}{ε} = Im(\overline{g}(λ + i0)) < ∞

for λ ∈ N for which the limit lim_{ε↓0} Im(\overline{g}(λ + iε)) exists and is finite. Denote this set by N′ and observe |N\N′| = 0 (see Theorem B.8 (ii)). Thus we see μ({ω| lim sup_{ε↓0} Im(m_{ω,±}(λ + iε)) = 0}) = 0 for λ ∈ N′. And as in the proof of the previous lemma this implies |{λ| lim sup_{ε↓0} Im(m_{ω,±}(λ + iε)) = 0} ∩ N| = 0 almost surely. A look at Lemma B.7 completes the proof. □

Next, let us extend this result.

Lemma 5.18. If γ(λ) = 0 for λ in a set N of positive Lebesgue measure, then we have

a(0)Im(m_{ω,+}(λ + i0)) = a(−1)Im(m_{ω,−}(λ + i0)),

Re(\frac{−1}{g_{ω}(λ+i0,0)}) = 0. (5.80)

almost surely for a.e. λ ∈ N.

Proof. Let us abbreviate n_{±} = n_{ω,±}(λ + iε) (see the notation from the previous proof) and n = n_{ω}(λ + iε) = −g_{ω}(λ + iε, 0)^{−1}. An elementary calculation using Im(n) = n_{+} + n_{−} verifies

(\frac{1}{n_{+}} + \frac{1}{n_{−}} ) \frac{(n_{+} − n_{−})^{2} + (Re(n))^{2}}{|n|^{2}} = (\frac{1}{n_{+}} + \frac{1}{n_{−}} ) (1 − 4\frac{n_{+}n_{−}}{|n|^{2}} )

= \frac{1}{n_{+}} + \frac{1}{n_{−}} + 4Im(\frac{1}{n}). (5.81)

Invoking (5.70) and (5.78) we see

(5.82) 𝔼 ((\frac{1}{n_{+}} + \frac{1}{n_{−}} ) \frac{(n_{+} − n_{−})^{2} + (Re(n))^{2}}{|n|^{2}} ) ≤ 4 (\frac{γ(λ + iε)}{ε} − \frac{∂γ(λ + iε)}{∂ε}) .

Moreover, since for a.e. λ the limits n_{ω,±}(λ+i0) and n_{ω}(λ+i0) exist by Theorem B.8 (ii), we are done. □

The condition (5.80) says precisely that m̃_{ω,+}(λ + i0) = −\overline{m̃_{ω,−}(λ + i0)}. This should be compared with the reflectionless condition in Lemma 8.1 (iii) below.

Theorem 5.19. If γ(λ) vanishes on an open interval I = (λ_{1}, λ_{2}), the spectrum of H_{ω} is purely absolutely continuous on I almost surely, that is, (Σ_{sc} ∪ Σ_{pp}) ∩ I = ∅.

Proof. By (5.80) and Lemma B.4 we see that −g_{ω}(λ + i0, 0)^{−1} is holomorphic on I and nonzero. So the same is true for g_{ω}(λ + i0, 0) and the result follows from (3.41). □

As a consequence we have that the class of random Jacobi operators where γ(λ) vanishes on a set of positive Lebesgue measure is rather special.

Theorem 5.20. If γ vanishes on a set of positive Lebesgue measure, then for almost surely ω ∈ Ω, the sequences (a_{ω}(n), b_{ω}(n))_{n∈ℤ} are determined by either (a_{ω}(n), b_{ω}(n))_{n≥n_{0}} or (a_{ω}(n), b_{ω}(n))_{n≤n_{0}} for any n_{0} ∈ ℤ.

Proof. It is no restriction to assume n_{0} = 0. By (5.80) and Lemma B.4 we learn that m_{ω,+}(z) uniquely determines m_{ω,−}(z) and vice versa. Hence the claim follows from Theorem 2.29. □

Sequences (a_{ω}(n), b_{ω}(n))_{n∈ℤ} of the above type are referred to as deterministic sequences. Hence the theorem says that for nondeterministic sequences we have Σ_{ac} = ∅. In particular this holds for the Anderson model. An example for deterministic sequences are almost periodic ones which will be discussed in the next section.

5.3. Almost periodic Jacobi operators

Now we will consider the case where the coefficients a, b are almost periodic sequences. We review some basic facts first.

Let c^{0} = (a^{0}, b^{0}) ∈ ℓ^{∞}(ℤ, ℝ_{+}) × ℓ^{∞}(ℤ) and let us abbreviate c^{m} = S^{m}c^{0} = (S^{m}a^{0}, S^{m}b^{0}) (S^{m} being the shift operator from (1.8)). We set Ω(c^{0}) = {c^{m}}_{m∈ℤ} and call the closure the hull of c^{0}, hull(c^{0}) = \overline{Ω(c^{0})}. The sequence c^{0} is called almost periodic if its hull is compact. Note that compactness implies that (H.5.1) holds for almost periodic sequences.

For an almost periodic sequence c^{0} we can define a product

(5.83)  ∘ : \begin{matrix}Ω(c^{0}) × Ω(c^{0}) & → & Ω(c^{0}) \\ (c^{m}, c^{n}) & ↦ & c^{m} ∘ c^{n} = c^{m+n} \end{matrix} ,

which makes Ω(c^{0}) an abelian group. Moreover, this product is continuous since

∥c^{n_{1}} ∘ c^{m_{1}} − c^{n_{0}} ∘ c^{m_{0}}∥ ≤ ∥c^{n_{1}+m_{1}} − c^{n_{1}+m_{0}}∥ + ∥c^{n_{1}+m_{0}} − c^{n_{0}+m_{0}}∥
(5.84)  = ∥c^{m_{1}} − c^{m_{0}}∥ + ∥c^{n_{1}} − c^{n_{0}}∥,

where we have used the triangle inequality and translation invariance of the norm in ℓ^{∞}(ℤ) × ℓ^{∞}(ℤ). Hence our product can be uniquely extended to the hull of c^{0} such that (hull(c^{0}), ∘) becomes a compact topological group. On such a compact topological group we have a unique invariant

(5.85)  ∫ f(cc′)dμ(c′) = ∫ f(c′)dμ(c′),  c ∈ hull(c^{0}),

Baire measure μ, the Haar measure, if we normalize μ(hull(c^{0})) = 1. The invariance property (5.85) implies

(5.86)  μ(S^{n}F) = μ(F),  F ⊆ hull(c^{0}),

since S^{n}c = c^{n} ∘ c. If we knew that μ is ergodic, we could apply the theory of the first section with Ω = hull(c^{0}). Now let S^{n}F = F, n ∈ ℤ, and consider

(5.87)  μ̃(G) = ∫_{G} χ_{F} dμ.

Then μ̃ also satisfies (5.85) and hence must be equal to μ up to normalization by uniqueness of the Haar measure. Hence μ̃(G) = μ̃(hull(c^{0}))μ(G) and using μ̃(F) = μ̃(hull(c^{0})) = μ(F) we obtain for G = F that μ(F) = μ(F)^{2} implying μ(F) ∈ {0, 1}. Thus μ is ergodic.

As a consequence we note

Theorem 5.21. Let (a, b) be almost periodic. For all Jacobi operators corresponding to sequences in the hull of (a, b), the spectrum is the same and the discrete spectrum is empty.

Proof. By Theorem 5.3 the claim holds for a set of full Haar measure. Since such a set is dense we can approximate any point (a, b) in the hull by a sequence (a_{n}, b_{n}) within this set in the sup norm. Hence the corresponding operators H_{n} converge to H in norm and the result follows. □

Since the absolutely continuous part is not stable under small perturbations, Theorem 5.4 cannot be directly extended to the the entire hull of an almost periodic sequence.

Theorem 5.22. Let (a, b) be almost periodic. For all Jacobi operators corresponding to sequences in a subset of full Haar measure of hull((a, b)), the pure point, singular continuous, absolutely continuous spectrum is the same, respectively.

One way of constructing examples of almost periodic operators is to choose periodic functions p_{1}(x), p_{2}(x) (with period one) and consider

(5.88) (a_{θ}(n), b_{θ}(n)) = (p_{1}(2πα_{1}n + θ_{1}), p_{1}(2πα_{2}n + θ_{2})), α ∈ ℝ^{2}, θ ∈ [0, 2π)^{2}.

Note that such sequences are even quasi periodic by construction.

We will restrict our attention to the most prominent model where p_{1}(x) = 1 and p_{2}(x) = β cos(x). The corresponding operator H_{θ} is known as almost Mathieu operator and given by

(5.89)  (a_{θ}(n), b_{θ}(n)) = (1, β cos(2παn + θ)), α ∈ ℝ, θ ∈ [0, 2π).

Since b_{θ+π} = −b_{θ} we can assume β > 0 without loss of generality. We need to distinguish two cases.

If α ∈ ℚ, say α = \frac{M}{N} (with N minimal), then b_{θ}(n + N) = b_{θ}(n) and

(5.90)  hull(1, b_{θ}) = {(1, b_{θ+2παn})}_{n=1}^{N}.

The underlying group is ℤ_{N} and the Haar measure is the normalized counting measure. This is a special case of periodic sequences and will be investigated in Chapter 7.

The case we are interested in here is α ∉ ℚ. Then

(5.91)  hull(1, b_{θ}) = {(1, b_{θ})|θ ∈ [0, 2π)},

where the right hand side is compact as the continuous image of the circle. The underlying group is S^{1} = \{e^{iθ}|θ ∈ [0, 2π)\} and the Haar measure is \frac{dθ}{2π}. Note that the shift satisfies S^{m}b_{θ} = b_{θ+2παm}.

We start our investigation with a bound for the Lyapunov exponent from below.

Lemma 5.23. Suppose α ∉ ℚ, then the Lyapunov exponent of H_{θ} satisfies

(5.92)  γ(z) ≥ ln \frac{β}{2} .

Proof. Observe

(5.93)  b_{θ}(n) = \frac{β}{2} (e^{2πiαn}w + e^{2πiαn}w^{−1}), w = e^{2πiθ},

and (cf. (1.29))

(5.94)  U_{θ}(z, n) = \frac{1}{w} ( \begin{matrix}0 & w \\ −w & zw − \frac{β}{2} (e^{2πiαn}w^{2} + e^{2πiαn}) \end{matrix} ) .

Hence f_{n}(w) = w^{n}Φ_{θ}(z, n) is holomorphic with respect to w and we have by Jensen’s inequality that

(5.95)  \frac{1}{2π} ∫_{0}^{2π} ln ∥f_{n}(e^{2πiθ})∥dθ ≥ ln ∥f_{n}(0)∥ = n ln \frac{β}{2} .

Using (5.26) we compute

(5.96)  γ(z) = inf_{n∈ℕ} \frac{1}{n}𝔼(ln ∥f_{n}∥) = inf_{n∈ℕ} \frac{1}{n} ∫_{0}^{2π} ln ∥f_{n}(e^{2πiθ})∥\frac{dθ}{2π} ≥ ln \frac{β}{2}

since ∥Φ_{θ}(z, n)∥ = ∥f_{n}(w)∥. □

In particular, applying Theorem 5.17 we obtain

Theorem 5.24. If β > 2 and α ∉ ℚ, then σ_{ac}(H_{θ}) = ∅ for a.e. θ.

Moreover, we can even exclude point spectrum if α is well approximated by rational numbers and hence b_{θ} is well approximated by periodic sequences. The key observation is the following lemma.

Lemma 5.25. Let a(n) ≥ δ^{−1} > 0. Suppose there are periodic sequences a_{k} and b_{k}, k ∈ ℕ, with (not necessarily minimal) period N_{k} tending to infinity as k → ∞ such that

(5.97) sup_{|n|≤2N_{k}+1} |a(n) − a_{k}(n)| ≤ (C_{k})^{N_{k}}, sup_{|n|≤2N_{k}} |b(n) − b_{k}(n)| ≤ (C_{k})^{N_{k}}

for some positive sequence C_{k} tending to zero as k → ∞. Then any (nontrivial) solution of τu = zu satisfies (see the notation in (1.28))

(5.98)  lim sup_{n→∞} \frac{∥\underline{u}(n)∥ + ∥\underline{u}(−n)∥}{∥\underline{u}(0)∥} ≥ 1.

Proof. It is no restriction to assume δ^{−1} ≤ |a_{k}(n)| ≤ sup |a(n)| and inf b(n) ≤ b_{k}(n) ≤ sup b(n). Setting M(z) = δ max(∥a∥_{∞}, ∥b−z∥_{∞}) we obtain for the transfer matrices

(5.99)  ∥Φ(z, n, m)∥ ≤ M(z)^{|n−m|},  ∥Φ_{k}(z, n, m)∥ ≤ M(z)^{|n−m|}.

Now using

(5.100) Φ(z, n) − Φ_{k}(z, n) = \begin{matrix}n \\ ∑ & ∗ \\ m=1 \end{matrix} Φ_{k}(z, n, m)(U(z, m) − U_{k}(z, m))Φ(z, m − 1)

we see

(5.101)  sup_{|n|≤2N_{k}} ∥Φ(z, n) − Φ_{k}(z, n)∥ ≤ 2δN_{k}M(z)(M(z)^{2}C_{k})^{N_{k}}

which tends to zero as k → ∞.

Abbreviate x_{j} = ∥\underline{u}_{k}(jN_{k})∥ = ∥Φ_{k}(z, jN_{k})\underline{u}(0)∥. Periodicity of a_{k}, b_{k} implies det Φ_{k}(z, N_{k}) = 1 and Φ_{k}(z, jN_{k}) = Φ_{k}(z, N_{k})^{j} (compare also Section 7.1). Hence using the characteristic equation (A^{2} + tr(A)A + det(A) = 0) and the triangle inequality it is not hard to show (set A = Φ_{k}(z, N_{k})^{±1}) that x_{1} + x_{−1} ≥ τ and x_{±2} + τx_{±1} ≥ 1, where τ = trΦ_{k}(z, N_{k}). Hence max(x_{2} + x_{−2}, x_{1} + x_{−1}) ≥ 1 proving the claim. □

In order for b_{θ} to satisfy the assumptions of the previous lemma we need an appropriate condition for α. A suitable one is that α is a Liouville number, that is, α ∈ ℝ\ℚ and for any k ∈ ℕ there exists M_{k}, N_{k} ∈ ℕ such that

(5.102)  |α − \frac{M_{k}}{N_{k}} | ≤ \frac{1}{k^{N_{k}}} .

Theorem 5.26. Suppose α is a Liouville number, then σ_{pp}(H_{θ}) = ∅.

Proof. For b_{θ,k}(n) = β cos(2π \frac{M_{k}}{N_{k}} n + θ) we estimate

sup_{|n|≤2N_{k}} |b_{θ}(n) − b_{θ,k}(n)| = β sup_{|n|≤2N_{k}} | cos(2παn + θ) − cos(2π \frac{M_{k}}{N_{k}} n + θ)|

(5.103)  ≤ 2πβ sup_{|n|≤2N_{k}} |n| |α − \frac{M_{k}}{N_{k}} | ≤ \frac{4πβN_{k}}{k^{N_{k}}}

and the result follows from the previous lemma. □

Combining Theorem 5.24 and Theorem 5.26 we obtain the

Corollary 5.27. The spectrum of H_{θ} is purely singular continuous for a.e. θ if α is a Liouville number and β > 2.

Trace formulas

Trace formulas are an important tool for both spectral and inverse spectral theory. As a first ingredient we consider asymptotic expansions which have already turned out useful in Section 2.7 and are of independent interest. In the remaining section we will develop the theory of xi functions and trace formulas based on Krein’s spectral shift theory.

6.1. Asymptotic expansions

Our aim is to derive asymptotic expansions for g(z, n) = G(z, n, n) and γ^{β}(z, n). Since both quantities are defined as expectations of resolvents

g(z, n) = ⟨δ_{n}, (H − z)^{−1}δ_{n}⟩,
(6.1)  γ^{β}(z, n) = (1 + β^{2})⟨δ_{n}^{β}, (H − z)^{−1}δ_{n}^{β}⟩ − \frac{β}{a(n)},

all we have to do is invoking Neumann’s expansion for the resolvent

(H − z)^{−1} = − ∑_{j=0}^{N−1} \frac{H^{j}}{z^{j+1}} + \frac{1}{z^{N}} H^{N}(H − z)^{−1}

(6.2)  = − ∑_{j=0}^{∞} \frac{H^{j}}{z^{j+1}} , |z| > ∥H∥.

In summary, we obtain the following result as simple consequence (see also Section 2.1) of the spectral theorem.

Lemma 6.1. Suppose δ ∈ ℓ^{2}(ℤ) with ∥δ∥ = 1. Then

(6.3)  g(z) = ⟨δ, (H − z)^{−1}δ⟩

is Herglotz, that is,

(6.4)  g(z) = ∫_{ℝ} \frac{1}{λ − z} dρ_{δ}(λ),

where dρ_{δ}(λ) = d⟨δ, P_{(−∞,λ]}(H)δ⟩ is the spectral measure of H associated to the sequence δ. Moreover,

Im(g(z)) = Im(z)∥(H − z)^{−1}δ∥^{2},

g(\overline{z}) = \overline{g(z)}, |g(z)| ≤ ∥(H − z)^{−1}∥ ≤ \frac{1}{|Im(z)|}, (6.5)

and

(6.6)  g(z) = − ∑_{j=0}^{∞} \frac{⟨δ, H^{j}δ⟩}{z^{j+1}}.

We note that solving (1.107) for h(z, n) shows that h(z, n) can be written as the difference of two Herglotz functions.

Lemma 6.1 implies the following asymptotic expansions for g(z, n), h(z, n), and γ^{β}(z, n).

Theorem 6.2. The quantities g(z, n), h(z, n), and γ^{β}(z, n) have the following Laurent expansions

g(z, n) = − ∑_{j=0}^{∞} \frac{g_{j}(n)}{z^{j+1}}, g_{0} = 1,

h(z, n) = −1 − ∑_{j=0}^{∞} \frac{h_{j}(n)}{z^{j+1}}, h_{0} = 0,

(6.7)  γ^{β}(z, n) = −\frac{β}{a(n)} − ∑_{j=0}^{∞} \frac{γ_{j}^{β}(n)}{z^{j+1}}, γ_{0}^{β} = 1 + β^{2}.

Moreover, the coefficients are given by

g_{j}(n) = ⟨δ_{n}, H^{j}δ_{n}⟩,
h_{j}(n) = 2a(n)⟨δ_{n+1}, H^{j}δ_{n}⟩,
γ_{j}^{β}(n) = ⟨(δ_{n+1} + βδ_{n}), H^{j}(δ_{n+1} + βδ_{n})⟩
(6.8)  = g_{j}(n + 1) + \frac{β}{a(n)}h_{j}(n) + β^{2}g_{j}(n), j ∈ ℕ_{0}.

Remark 6.3. Using the unitary transform U_{ε} of Lemma 1.6 shows that g_{j}(n), h_{j}(n) do not depend on the sign of a(n), that is, they only depend on a(n)^{2}.

The next lemma shows how to compute g_{j}, h_{j} recursively.

Lemma 6.4. The coefficients g_{j}(n) and h_{j}(n) for j ∈ ℕ_{0} satisfy the following recursion relations

g_{j+1} = \frac{h_{j} + h_{j}^{−}}{2} + bg_{j},
(6.9)  h_{j+1} − h_{j+1}^{−} = 2(a^{2}g_{j}^{+} − (a^{−})^{2}g_{j}^{−}) + b(h_{j} − h_{j}^{−}).

Proof. The first equation follows from

(6.10)  g_{j+1}(n) = ⟨Hδ_{n}, H^{j}δ_{n}⟩ = \frac{h_{j}(n) + h_{j}(n − 1)}{2} + b(n)g_{j}(n)

using Hδ_{n} = a(n)δ_{n+1} + a(n − 1)δ_{n−1} + b(n)δ_{n}. Similarly,

h_{j+1}(n) = b(n)h_{j}(n) + 2a(n)^{2}g_{j}(n + 1) + 2a(n − 1)a(n)⟨δ_{n+1}, H^{j}δ_{n−1}⟩
(6.11)  = b(n + 1)h_{j}(n) + 2a^{2}g_{j}(n) + 2a(n)a(n + 1)⟨δ_{n+2}, H^{j}δ_{n}⟩

and eliminating ⟨δ_{n+1}, H^{j}δ_{n−1}⟩ completes the proof. □

The system (6.9) does not determine g_{j}(n), h_{j}(n) uniquely since it requires solving a first order recurrence relation at each step, producing an unknown summation constant each time

h_{j+1}(n) − h_{j+1}(n_{0}) =\begin{matrix}n \\ ∑ & ∗ \\ m=n_{0}+1 \end{matrix} (2a(m)^{2}g_{j}(m + 1) − 2a(m − 1)^{2}g_{j}(m − 1)

+ b(m)h_{j}(m) − b(m)h_{j}(m − 1)). (6.12)

To determine the constant observe that the right hand side must consists of two summands, the first one involving a(n + k), b(n + k) with |k| < j and the second one involving a(n_{0} + k), b(n_{0} + k) with |k| < j. This follows since the left hand side is of this form (use Hδ_{n} = a(n)δ_{n+1} + a(n − 1)δ_{n−1} + b(n)δ_{n} and induction). In particular, the first summand must be equal to h_{j+1}(n) and the second to h_{j+1}(n_{0}). This determines the unknown constant.

Since this procedure is not very straightforward, it is desirable to avoid these constants right from the outset. This can be done at the expense of giving up linearity. If we compare powers of z in (2.187) we obtain

(6.13)  h_{j+1} = 2a^{2} ∑_{ℓ=0}^{j} g_{j−ℓ}g_{ℓ}^{+} − \frac{1}{2} ∑_{ℓ=0}^{j} h_{j−ℓ}h_{ℓ}, j ∈ ℕ,

which determines g_{j}, h_{j} recursively together with the first equation of (6.9). Explicitly we obtain

g_{0} = 1, g_{1} = b, g_{2} = a^{2} + (a^{−})^{2} + b^{2},
(6.14)  h_{0} = 0, h_{1} = 2a^{2}, h_{2} = 2a^{2}(b^{+} + b)

and hence from (6.8)

γ_{0}^{β} = 1 + β^{2}, γ_{1}^{β} = b^{+} + 2aβ + bβ^{2},
(6.15) γ_{2}^{β} = (a^{+})^{2} + a^{2} + (b^{+})^{2} + 2a(b^{+} + b)β + (a^{2} + (a^{−})^{2} + b^{2})β^{2}.

Remark 6.5. (i). Let us advocate another approach which produces a recursion for g_{j} only. Inserting the expansion (6.7) for g(z, n) into (1.110) and comparing coefficients of z^{j} one infers

g_{0} = 1, g_{1} = b, g_{2} = a^{2} + (a^{−})^{2} + b^{2},
g_{3} = a^{2}(b^{+} + 2b) + (a^{−})^{2}(2b + b^{−}) + b^{3},
g_{j+1} = 2bg_{j} − a^{2}g_{j−1}^{+} + (a^{−})^{2}g_{j−1}^{−} − b^{2}g_{j−1} − \frac{1}{2} ∑_{ℓ=0}^{j−1} k_{j−ℓ−1}k_{ℓ}

+ 2a^{2}( ∑_{ℓ=0}^{j−1} g_{j−ℓ−1}g_{ℓ}^{+} − 2b ∑_{ℓ=0}^{j−2} g_{j−ℓ−2}g_{ℓ}^{+} + b^{2} ∑_{ℓ=0}^{j−3} g_{j−ℓ−3}g_{ℓ}^{+}), (6.16)

for j ≥ 3, where k_{0}(n) = −b(n) and

(6.17)  k_{j} = a^{2}g_{j−1}^{+} − (a^{−})^{2}g_{j−1}^{−} + b^{2}g_{j−1} − 2bg_{j} + g_{j+1}, j ∈ ℕ.

If g_{j}(n) is known, h_{j}(n) can be determined using

(6.18)  h_{j+1} = bh_{j} + g_{j+2} − 2bg_{j+1} + a^{2}g_{j}^{+} − (a^{−})^{2}g_{j}^{−} + b^{2}g_{j}, j ∈ ℕ_{0}.

Equation (6.18) follows after inserting the first equation of (6.9) into the second. (ii). Analogously, one can get a recurrence relation for γ_{j}^{β} using (1.112). Since this approach gets too cumbersome we omit further details at this point.

Next, we turn to Weyl m-functions. As before we obtain

Lemma 6.6. The quantities m_{±}(z, n) have the Laurent expansions

(6.19)  m_{±}(z, n) = − ∑_{j=0}^{∞} \frac{m_{±,j}(n)}{z^{j+1}}, m_{±,0}(n) = 1.

The coefficients m_{±,j}(n) are given by

(6.20)  m_{±,j}(n) = ⟨δ_{n±1}, (H_{±,n})^{j}δ_{n±1}⟩, j ∈ ℕ,

and satisfy

m_{±,0} = 1, m_{±,1} = b^{±},

m_{±,j+1} = b^{±}m_{±,j} + (a^{{+ \atop −−}})^{2} ∑_{ℓ=0}^{j−1} m_{±,j−ℓ−1}m_{±,ℓ}^{+}, j ∈ ℕ. (6.21)

Finally, we consider the solutions ϕ_{±}(z, n) = ∓a(n)m̃_{±}(z, n) of the Riccati equation (1.52). We will show uniqueness of solutions with particular asymptotic behavior.

Lemma 6.7. The Riccati equation (1.52) has unique solutions ϕ_{±}(z, n) having the Laurent expansions

(6.22)  ϕ_{±}(z, n) = (\frac{a(n)}{z})^{±1} ∑_{j=0}^{∞} \frac{ϕ_{±,j}(n)}{z^{j}}.

The coefficients ϕ_{±,j}(n) satisfy

ϕ_{+,0} = 1, ϕ_{+,1} = b^{+},

ϕ_{+,j+1} = b^{+}ϕ_{+,j} + (a^{+})^{2} ∑_{ℓ=0}^{j−1} ϕ_{+,j−ℓ−1}ϕ_{+,ℓ}^{+}, j ≥ 1, (6.23)

and

ϕ_{−,0} = 1, ϕ_{−,1} = −b,

ϕ_{−,j+1} = −bϕ_{−,j}^{−} − ∑_{ℓ=1}^{j} ϕ_{−,j−ℓ+1}ϕ_{−,ℓ}^{−} − (a^{−})^{2}δ_{1,j}, j ≥ 1. (6.24)

Proof. Existence is clear from

(6.25) ϕ_{+}(z, n) = −a(n)m_{+}(z, n),  ϕ_{−}(z, n) = \frac{z + b(n) + a(n − 1)^{2}m_{−}(z, n)}{a(n)},

and the previous lemma. Moreover, inserting (6.22) into the Riccati equation (1.52) produces the relations (6.23), (6.24) which determine the coefficients ϕ_{±,j} uniquely. Since ϕ_{±}(z) is uniquely determined by its Laurent expansion we are done. □

Explicitly one computes

ϕ_{−}(z, .) = \frac{z}{a}(1 − \frac{b}{z} − \frac{(a^{−})^{2}}{z^{2}} − \frac{(a^{−})^{2}b^{−}}{z^{3}} − \frac{(a^{−})^{2}((a^{−−})^{2} + (b^{−})^{2})}{z^{4}}

+ O(\frac{1}{z^{5}} )),

ϕ_{+}(z, .) = \frac{a}{z}(1 + \frac{b^{+}}{z} + \frac{(a^{+})^{2} + (b^{+})^{2}}{z^{2}} + \frac{(a^{+})^{2}(2b^{+} + b^{++}) + (b^{+})^{3}}{z^{3}}

+ \frac{((a^{+})^{2} + (b^{+})^{2})^{2} + (a^{+})^{2}((b^{+} + b^{++})^{2} + (a^{++})^{2})}{z^{4}}

+ O(\frac{1}{z^{5}} )). (6.26)

This lemma has some interesting consequences for u_{±}(z, n). If we have a solution ϕ_{±}(z, n) of the Riccati equation with asymptotic behavior (6.22), we know that u_{±}(z, n, n_{0}) = c(z, n, n_{0})+ϕ_{±}(z, n_{0})s(z, n, n_{0}) are square summable near ±∞. Moreover, we obtain

u_{−}(z, n, n_{0}) = (\begin{matrix}n−1 \\ ∏ & ∗ \\ j=n_{0} \end{matrix} \frac{z}{a(j)}) (1 − \frac{1}{z} \begin{matrix}n−1 \\ ∑ & ∗ \\ j=n_{0} \end{matrix} b(j) + O(\frac{1}{z^{2}} )) ,

(6.27)  u_{+}(z, n, n_{0}) = (\begin{matrix}n−1 \\ ∏ & ∗ \\ j=n_{0} \end{matrix} \frac{a(j)}{z}) (1 + \frac{1}{z}\begin{matrix}n \\ ∑ & ∗ \\ j=n_{0}+1 \end{matrix} b(j) + O(\frac{1}{z^{2}} )) .

Remark 6.8. If we reflect at a point n_{0} (cf. (1.82)) we obtain from (6.8)

(6.28)  g_{R,j}(n_{0} + k) = g_{j}(n_{0} − k),  h_{R,j}(n_{0} + k) = h_{j}(n_{0} − k − 1)

and thus

(6.29)  γ_{R,j}^{β}(n_{0} + k) = β^{2}γ_{j}^{1/β}(n_{0} − k).

In addition, since we have m_{R,+}(z, n_{0} + k) = m_{−}(z, n_{0} − k) we infer

(6.30)  ϕ_{+,ℓ+2}(n_{0} + k) = −a_{R}(n_{0} − k)^{2}ϕ_{R,−,ℓ}(n_{0} − k).

6.2. General trace formulas and xi functions

The objective of the present section is to develop trace formulas for arbitrary Jacobi operators. Applications will be given at the end of Section 7.5 and in Section 8.1. Our main ingredient will be Krein’s spectral shift theory (for the special case of rank one perturbations). Hence we will recall some basic results first.

Consider the following rank one perturbation

(6.31)  H_{n,θ} = H + θ⟨δ_{n}, .⟩δ_{n}, θ ≥ 0,

and abbreviate

(6.32)  g_{θ}(z, n) = ⟨δ_{n}, (H_{n,θ} − z)^{−1}δ_{n}⟩.

By virtue of the second resolvent formula one infers

(6.33)  (H_{n,θ} − z)^{−1} = (H − z)^{−1} − θ⟨(H_{n,θ} − \overline{z})^{−1}δ_{n}, .⟩(H − z)^{−1}δ_{n}

and hence

(6.34)  g_{θ}(z, n) = \frac{g(z, n)}{1 + θg(z, n)}.

Furthermore, we compute (H_{n,θ} − z)^{−1}δ_{n} = (1 + θg_{θ}(z, n))(H − z)^{−1}δ_{n} and thus

(6.35) (H_{n,θ} − z)^{−1} = (H − z)^{−1} − \frac{θ}{1 + θg(z, n)}⟨(H − \overline{z})^{−1}δ_{n}, .⟩(H − z)^{−1}δ_{n}

(a special case of Krein’s resolvent formula). We have σ_{ess}(H_{n,θ}) = σ_{ess}(H) and an investigation of the discrete poles of (6.35) shows

(6.36)  σ_{d}(H_{n,θ}) = \{λ ∈ ℝ\σ_{ess}(H)|g(λ, n) = −\frac{1}{θ}\}.

Moreover, since g(λ, n) is increasing (w.r.t. λ) in each spectral gap, there can be at most one discrete eigenvalue of H_{n,θ} in each spectral gap.

Computing traces we obtain

tr((H_{n,θ} − z)^{−1} − (H − z)^{−1}) = −\frac{θ}{1 + θg(z, n)}⟨δ_{n}, (H − z)^{−2}δ_{n}⟩
(6.37)  = −\frac{d}{dz} ln(1 + θg(z, n)).

Next, note that g(z, n) and hence 1 + θg(z, n) is Herglotz. Hence we have the following exponential representation (cf. Lemma B.12 (iii))

(6.38)  1 + θg(z, n) = exp ( ∫_{ℝ} \frac{ξ_{θ}(λ, n)}{λ − z}dλ),

where

(6.39)  ξ_{θ}(λ, n) = \frac{1}{π} lim_{ε↓0} arg (1 + θg(λ + iε, n)).

Now we can rewrite (6.37) as

(6.40)  tr((H_{n,θ} − z)^{−1} − (H − z)^{−1}) = − ∫_{ℝ} \frac{ξ_{θ}(λ, n)}{(λ − z)^{2}} dλ.

Moreover, ξ_{θ}(λ, n) (which is only defined a.e.) is compactly supported (see also Lemma B.12 (iii)) and ∫_{ℝ} ξ_{θ}(λ, n)dλ = θ. Comparing the Laurent expansions around ∞ of both sides in (6.40) gives (formally)

(6.41)  tr((H_{n,θ})^{ℓ} − H^{ℓ}) = ℓ ∫_{ℝ} λ^{ℓ−1}ξ_{θ}(λ, n)dλ.

However, (6.41) can be easily obtained rigorously by comparing coefficients in (6.35) and then taking traces of the resulting equations.

More general, Krein’s result implies

(6.42)  tr(f(H_{n,θ}) − f(H)) = ∫_{ℝ} f′(λ)ξ_{θ}(λ, n)dλ

for continuously differentiable f (f can be assumed compactly supported since only values in the support of ξ_{θ}(., n) contribute). The functions 1 + θg(z, n) and ξ_{θ}(λ, n) are called perturbation determinant and spectral shift function of the pair (H, H_{n,θ}), respectively.

Let us collect some parts of these results in our first theorem.

Theorem 6.9. Let ξ_{θ}(λ, n) be defined as above. Then we have

(6.43)  b_{θ}^{(ℓ)}(n) = tr((H_{n,θ})^{ℓ+1} − H^{ℓ+1}) = (ℓ + 1) ∫_{ℝ} λ^{ℓ}ξ_{θ}(λ, n)dλ,

with (cf. (6.8))

(6.44) b_{θ}^{(0)}(n) = θ, b_{θ}^{(ℓ)}(n) = θ(ℓ + 1)g_{ℓ}(n) + θ ∑_{j=1}^{ℓ} g_{ℓ−j}(n)b_{θ}^{(j−1)}(n), ℓ ∈ ℕ.

Proof. The claim follows after expanding both sides of

(6.45)  ln(1 + θg(z, n)) = ∫_{ℝ} \frac{ξ_{θ}(λ, n)}{λ − z}dλ

and comparing coefficients using the following connections between the series of g(z) and ln(1 + g(z)) (cf., e.g., [185]). Let g(z) have the Laurent expansion

(6.46)  g(z) = ∑_{ℓ=1}^{∞} \frac{g_{ℓ}}{z^{ℓ}}

as z → ∞. Then we have

(6.47)  ln(1 + g(z)) = ∑_{ℓ=1}^{∞} \frac{c_{ℓ}}{z^{ℓ}} ,

where

(6.48)  c_{1} = g_{1}, c_{ℓ} = g_{ℓ} − ∑_{j=1}^{ℓ−1} \frac{j}{ℓ} g_{ℓ−j}c_{j}, ℓ ≥ 2.
□

In the special case ℓ = 1 we obtain

(6.49)  b(n) = \frac{1}{θ} ∫_{ℝ} λξ_{θ}(λ, n)dλ − \frac{θ}{2}.

Next, we want to investigate the case where the coupling constant θ tends to ∞. It is clear that H_{θ,n}f will only converge if f(n) = 0. On the other hand,

(6.50)  (H_{n,θ} − z)^{−1} → (H − z)^{−1} − \frac{1}{g(z, n)}⟨(H − \overline{z})^{−1}δ_{n}, .⟩(H − z)^{−1}δ_{n}

converges (in norm) as θ → ∞. Comparison with (1.106) shows (H_{n,θ} − z)^{−1} → (H_{n}^{∞} − z)^{−1} if we embed (H_{n}^{∞} − z)^{−1} into ℓ^{2}(ℤ) using (H_{n}^{∞} − z)^{−1}δ_{n} = 0, that is, (H_{n}^{∞} − z)^{−1} = (1l − P_{n}^{∞})(H − z)^{−1}(1l − P_{n}^{∞}) (cf. Remark 1.11). In particular, (6.40) is still valid

(6.51)  tr((H_{n}^{∞} − z)^{−1} − (H − z)^{−1}) = − ∫_{ℝ} \frac{ξ(λ, n)}{(λ − z)^{2}} dλ.

Here the xi function ξ(λ, n) is defined by

(6.52) g(z, n) = |g(i, n)| exp (∫_{ℝ} (\frac{1}{λ − z} − \frac{λ}{1 + λ^{2}} )ξ(λ, n)dλ) , z ∈ ℂ\σ(H),

respectively by

(6.53)  ξ(λ, n) = \frac{1}{π} lim_{ε↓0} arg g(λ + iε, n),  arg(.) ∈ (−π, π].

Since ξ(., n) is not compactly supported, the asymptotic expansions require a somewhat more careful analysis. We abbreviate

(6.54)  E_{0} = inf σ(H),  E_{∞} = sup σ(H),

and note that g(λ, n) > 0 for λ < E_{0}, which follows from (H − λ) > 0 (implying (H − λ)^{−1} > 0). Similarly, g(λ, n) < 0 for λ > E_{∞} follows from (H − λ) < 0. Thus ξ(λ, n) satisfies 0 ≤ ξ(λ, n) ≤ 1,

(6.55)  ∫_{ℝ} \frac{ξ(λ, n)}{1 + λ^{2}}dλ = arg g(i, n), and ξ(λ, n) = \{ \begin{matrix}0 for & z < E_{0} \\ 1 for & z > E_{∞} \end{matrix} .

Using (6.55) together with the asymptotic behavior of g(., n) we infer

(6.56)  g(z, n) = \frac{1}{E_{∞} − z} exp (∫_{E_{0}}^{E_{∞}} \frac{ξ(λ, n)dλ}{λ − z}) .

As before we set H_{n}^{∞}δ_{n} = 0 implying

tr(f(H_{n}^{∞}) − f(H)) = ∫_{ℝ} f′(λ)ξ(λ, n)dλ
(6.57)  = ∫_{E_{0}}^{E_{∞}} f′(λ)ξ(λ, n)dλ − f(E_{∞})

for continuously differentiable f with compact support (again compact support is no restriction since only values in [E_{0}, E_{∞}] contribute).

Theorem 6.10. Let ξ(λ, n) be defined as above. Then we have the following trace formula

(6.58)  b^{(ℓ)}(n) = tr(H^{ℓ} − (H_{n}^{∞})^{ℓ}) = E_{∞}^{ℓ} − ℓ ∫_{E_{0}}^{E_{∞}} λ^{ℓ−1}ξ(λ, n)dλ,

where

b^{(1)}(n) = b(n),
(6.59)  b^{(ℓ)}(n) = ℓ g_{ℓ}(n) − ∑_{j=1}^{ℓ−1} g_{ℓ−j}(n)b^{(j)}(n), ℓ > 1.

Proof. The claim follows after expanding both sides of

(6.60)  ln ((E_{∞} − z)g(z, n)) = ∫_{E_{0}}^{E_{∞}} \frac{ξ(λ, n)dλ}{λ − z}

and comparing coefficients as before. □

The special case ℓ = 1 of equation (6.58) reads

(6.61) b(n) = E_{∞} − ∫_{E_{0}}^{E_{∞}} ξ(λ, n)dλ = \frac{E_{0} + E_{∞}}{2} + \frac{1}{2} ∫_{E_{0}}^{E_{∞}}(1 − 2ξ(λ, n))dλ.

Similarly, H_{n_{0}}^{β} can be obtained as the limit of the operator H + θ⟨δ_{n}^{β}, .⟩δ_{n}^{β} as θ → ∞. We have

(6.62) γ^{β}(z, n) = −\frac{β}{a(n)} exp ( ∫_{ℝ} \frac{ξ^{β}(λ, n)dλ}{λ − z}), z ∈ ℂ\σ(H_{n}^{β}), β ∈ ℝ\\{0\},

where

(6.63) ξ^{β}(λ, n) = \frac{1}{π} lim_{ε↓0} arg (γ^{β}(λ + iε, n)) − δ^{β}, δ^{β} = \{ \begin{matrix}0, & βa(n) < 0 \\ 1, & βa(n) > 0 \end{matrix}

and 0 ≤ sgn(−a(n)β) ξ^{β}(λ, n) ≤ 1. The function ξ^{β}(λ, n) is compactly supported and we get as before

Theorem 6.11. Let ξ^{β}(λ, n) be defined as above. Then we have

(6.64)  b^{β,(ℓ)}(n) = (ℓ + 1)\frac{β}{a(n)} ∫_{ℝ} λ^{ℓ}ξ^{β}(λ, n)dλ, ℓ ∈ ℕ,

where

b^{β,(0)}(n) = 1 + β^{2},
(6.65) b^{β,(ℓ)}(n) = (ℓ + 1)γ_{ℓ}^{β}(n) − \frac{β}{a(n)} ∑_{j=1}^{ℓ} γ_{ℓ−j}^{β}(n)b^{β,(j−1)}(n), ℓ ∈ ℕ.

Specializing to ℓ = 0 in (6.64) we obtain

(6.66)  a(n) = \frac{1}{β + β^{−1}} ∫_{ℝ} ξ^{β}(λ, n)dλ.

Finally, observe that ξ^{β}(λ, n_{0}) for two values of β and one fixed n_{0} determines the sequences a(n), b(n) for all n ∈ ℤ.

Lemma 6.12. Let β_{1} ≠ β_{2} ∈ ℝ ∪ {∞} be given. Then (β_{j}, ξ^{β_{j}}(., n_{0})), j = 1, 2, for one fixed n_{0} ∈ ℤ uniquely determines a(n)^{2}, b(n) for all n ∈ ℤ.

Proof. If we know β and ξ^{β}(z, n_{0}), we know a(n_{0}) by (6.66) and γ^{β}(z, n_{0}) by (6.62). Hence the result follows from Theorem 2.30. □

Jacobi operators with periodic coefficients

Some of the most interesting Jacobi operators are those with periodic coefficients a, b. In this chapter we will develop Floquet theory to investigate this case. This will allow us to give a complete characterization of the qualitative behavior of solutions and hence also a complete spectral characterization. Especially, we will see that all important quantities associated with periodic sequences have a simple polynomial dependence with respect to the spectral parameter z and can be computed explicitly. In fact, this feature is shared by a much larger class of Jacobi operators to be considered in the next chapter. In addition, this chapter also provides the motivation for definitions being made there.

7.1. Floquet theory

For this entire chapter we will assume a, b to be periodic with period N (for N = 1 see Sections 1.3 and 7.4).

Hypothesis H. 7.1. Suppose there is an N ∈ ℕ such that

(7.1)  a(n + N) = a(n),  b(n + N) = b(n).

In addition, we agree to abbreviate

(7.2)  A = ∏_{j=1}^{N} a(n_{0} + j) = ∏_{j=1}^{N} a(j),  B = ∑_{j=1}^{N} b(n_{0} + j) = ∑_{j=1}^{N} b(j).

We start by recalling the transfer matrix Φ(z, n, n_{0}) (cf. (1.30)) which transfers initial conditions u(n_{0}), u(n_{0}+1) at n_{0} of a solution u into the corresponding values u(n), u(n + 1) at n. It suggests itself to investigate what happens if we move on N steps, that is, to look at the monodromy matrix

(7.3)  M(z, n_{0}) = Φ(z, n_{0} + N, n_{0}).

A first naive guess would be that all initial conditions return to their starting values after N steps (i.e., M(z, n_{0}) = 1l) and hence all solutions are periodic. However, this is too much to hope for since it already fails for N = 1 (see Section 1.3).

On the other hand, since it does not matter whether we start our N steps at n_{0}, at n_{0} + N, or even n_{0} + ℓN, ℓ ∈ ℤ, we infer that M(z, n_{0}) is periodic, that is, M(z, n_{0} + N) = M(z, n_{0}), n_{0} ∈ ℤ. Moreover, we even have Φ(z, n_{0} + ℓN, n_{0}) = M(z, n_{0})^{ℓ}. Thus Φ(z, n, n_{0}) exhibits an exponential behavior if we move on N points in each step. If we factor out this exponential term, the remainder should be periodic.

For this purpose we rewrite M(z, n_{0}) a little bit. Using periodicity (7.1) we obtain from (1.33)

(7.4)  det M(z, n_{0}) = 1

and hence we can find a periodic matrix Q(z, n_{0}) such that

(7.5)  M(z, n_{0}) = exp(iNQ(z, n_{0})), tr Q(z, n_{0}) = 0.

Now we can write

(7.6)  Φ(z, n, n_{0}) = P (z, n, n_{0}) exp(i(n − n_{0})Q(z, n_{0})), P (z, n_{0}, n_{0}) = 1l.

A simple calculation

P (z, n + N, n_{0}) = Φ(z, n + N, n_{0})M(z, n_{0})^{−1} exp( − i(n − n_{0})Q(z, n_{0}))
= Φ(z, n + N, n_{0} + N) exp( − i(n − n_{0})Q(z, n_{0}))
(7.7)  = Φ(z, n, n_{0}) exp( − i(n − n_{0})Q(z, n_{0})) = P (z, n, n_{0})

shows that P (z, n, n_{0}) is indeed periodic as anticipated.

Next we want to turn this result into a corresponding result for the solutions of τu = zu. The key is clearly to investigate the Jordan canonical form of M(z, n_{0}) (resp. Q(z, n_{0})).

From (1.32) we see

M(z, n_{1}) = Φ(z, n_{1} + N, n_{0} + N)M(z, n_{0})Φ(z, n_{0}, n_{1})
(7.8)  = Φ(z, n_{1}, n_{0})M(z, n_{0})Φ(z, n_{1}, n_{0})^{−1}

and hence the discriminant

(7.9)  Δ(z) = \frac{1}{2}tr M(z, n_{0}) = \frac{1}{2}(c(z, n_{0} + N, n_{0}) + s(z, n_{0} + N + 1, n_{0})),

plus the eigenvalues m^{±}(z) (resp. ±q(z)) of M(z, n_{0}) (resp. Q(z, n_{0}))

(7.10) m^{±}(z) = exp(±iNq(z)) = Δ(z) ± (Δ(z)^{2} − 1)^{1/2}, m^{+}(z)m^{−}(z) = 1,

are independent of n_{0}. The branch in the above root is fixed as follows

(7.11)  (Δ(z)^{2} − 1)^{1/2} = \frac{−1}{2A} ∏_{j=0}^{2N−1} \sqrt{z − E_{j}},

where (E_{j})_{j=0}^{2N−1} are the zeros of Δ(z)^{2} − 1 (cf. Appendix A.7). This definition implies the following expansion for large z

(7.12)  m^{±}(z) = (2Δ(z))^{∓1}(1 + O(\frac{1}{z^{2N}})) = (\frac{A}{z^{N}})^{±1}(1 ± \frac{B}{z} + O(\frac{1}{z^{2}} ))

and will ensure |m^{+}(z)| ≤ 1, |m^{−}(z)| ≥ 1 later on.

The eigenvalues m^{±}(z) are called Floquet multipliers and q(z) is called Floquet momentum.

For later use we note

m^{+}(z) − m^{−}(z) = 2i sin(Nq(z)) = 2(Δ(z)^{2} − 1)^{1/2},
(7.13)  m^{+}(z) + m^{−}(z) = 2 cos(Nq(z)) = 2Δ(z).

Remark 7.2. It is natural to look at (Δ(z)^{2}−1)^{1/2} as meromorphic function on the corresponding Riemann surface. The two functions associated with ±(Δ(z)^{2}−1)^{1/2} can then be viewed as the two branches of this single function on the Riemann surface. This will be exploited in Chapter 8 and Chapter 9.

Now we are able to compute the Jordan canonical form of M(z, n_{0}). We need to distinguish three cases.

Case 1). Δ(z)^{2} ≠ 1 (and hence m^{+}(z) ≠ m^{−}(z)). If s(z, n_{0} + N, n_{0}) ≠ 0 we set

(7.14)  e^{±}(z, n_{0}) = ( \begin{matrix}1 \\ ϕ_{±}(z, n_{0}) \end{matrix} ) ,

where

(7.15) ϕ_{±}(z, n) = \frac{m^{±}(z) − c(z, n_{0} + N, n_{0})}{s(z, n_{0} + N, n_{0})} = \frac{c(z, n_{0} + N + 1, n_{0})}{m^{±}(z) − s(z, n_{0} + N + 1, n_{0})}.

If s(μ, n_{0} + N, n_{0}) = 0, then s(μ, n, n_{0}) is periodic with respect to n and s(μ, n_{0} + N, n_{0}) = m^{σ}(μ) for some σ ∈ {±}. Moreover, det M(μ, n_{0}) = 1 says c(μ, n_{0} + N, n_{0}) = s(μ, n_{0} + N + 1, n_{0})^{−1} = m^{−σ}(μ) and hence ϕ_{−σ}(z, n_{0}) tends to a finite limit ϕ_{−σ}(μ, n_{0}) as z → μ. Thus we can set e^{σ}(μ, n_{0}) = (0, 1) and e^{−σ}(μ, n_{0}) = (1, ϕ_{−σ}(μ, n_{0})).

Now the matrix

(7.16)  U(z, n_{0}) = (e^{+}(z, n_{0}), e^{−}(z, n_{0}))

will transform M(z, n_{0}) into

(7.17)  U(z, n_{0})^{−1}M(z, n_{0})U(z, n_{0}) = ( \begin{matrix}m^{+}(z) & 0 \\ 0 & m^{−}(z) \end{matrix} ) .

Moreover, we have two corresponding solutions (Floquet solutions)

(7.18)  u_{±}(n, z) = e_{1}^{±}(z, n_{0})c(z, n, n_{0}) + e_{2}^{±}(z, n_{0})s(z, n, n_{0}),

satisfying

(7.19)  u_{±}(z, n + N) = m^{±}(z)u_{±}(z, n).

They are linearly independent and unique up to multiples.

Case 2). Δ(z) = ±1 (and hence m^{+}(z) = m^{−}(z) = ±1) and M(z, n_{0}) has two linearly independent eigenvectors. We necessarily have

(7.20)  M(z, n_{0}) = ±1l

and there is nothing to be done. All solutions satisfy

(7.21)  u(z, n + N) = ±u(z, n).

Case 3). Δ(z) = ±1 (and hence m^{+}(z) = m^{−}(z) = ±1) and M(z, n_{0}) has only one eigenvector. If s(z, n_{0} + N, n_{0}) ≠ 0 we set

(7.22)  e(z, n_{0}) = ( s(z, n_{0} + N, n_{0})_{\frac{s(z,n_{0}+N+1,n_{0})−c(z,n_{0}+N,n_{0})}{2}} ) , ê(z, n_{0}) = ( \begin{matrix}0 \\ 1 \end{matrix} )

or, if c(z, n_{0} + N + 1, n_{0}) ≠ 0 we set

(7.23)  e(z, n_{0}) = ( −c(z, n_{0} + N + 1, n_{0})^{\frac{c(z,n_{0}+N,n_{0})−s(z,n_{0}+N+1,n_{0})}{2}} ) , ê(z, n_{0}) = ( \begin{matrix}1 \\ 0 \end{matrix} ) .

If both are zero we have Case 2.

Then U(z, n_{0}) = (e(z, n_{0}), ê(z, n_{0})) will transform M(z, n_{0}) into

(7.24)  U(z, n_{0})^{−1}M(z, n_{0})U(z, n_{0}) = ( \begin{matrix}±1 & 1 \\ 0 & ±1 \end{matrix} ) .

Moreover, there are solutions u(z), û(z) such that

(7.25)  u(z, n + N) = ±u(z, n), û(z, n + N) = ±û(z, n) + u(z, n).

Summarizing these results we obtain Floquet’s theorem.

Theorem 7.3. The solutions of τu = zu can be characterized as follows. (i). If Δ(z)^{2} ≠ 1 there exist two solutions satisfying

(7.26)  u_{±}(z, n) = p_{±}(z, n)e^{±iq(z)n}, p_{±}(z, n + N) = p_{±}(z, n).

(ii). If Δ(z) = ±1, then either all solutions satisfy

(7.27)  u(z, n + N) = ±u(z, n)

or there are two solutions satisfying

(7.28)  u(z, n) = p(z, n), û(z, n) = p̂(z, n) + n p(z, n)

with p(z, n + N) = ±p(z, n), p̂(z, n + N) = ±p̂(z, n).

If we normalize the Floquet solutions by u_{±}(z, n_{0}) = 1 they are called Floquet functions ψ_{±}(z, n, n_{0}) and a straightforward calculation yields

ψ_{±}(z, n, n_{0}) = c(z, n, n_{0}) + ϕ_{±}(z, n_{0})s(z, n, n_{0}),
(7.29)  ϕ_{±}(z, n) = \frac{ψ_{±}(z, n + 1, n_{0})}{ψ_{±}(z, n, n_{0})}.

Observe that ϕ_{±}(z, n) is periodic, ϕ_{±}(z, n+N) = ϕ_{±}(z, n), and satisfies the Riccati equation (1.52).

Moreover, using our expansion (7.12) we get for large z

ϕ_{−}(z, n) = \frac{s(z, n + N + 1, n)}{s(z, n + N, n)} + O(z^{1−2N}),
(7.30)  ϕ_{+}(z, n) = −\frac{c(z, n + N, n)}{s(z, n + N, n)} + O(z^{1−2N}).

7.2. Connections with the spectra of finite Jacobi operators

In this section we want to gain further insight by considering finite Jacobi matrices. The results will also shed some light on our trace formulas.

First, we look at the operators H_{n_{1},n_{2}}^{β_{1},β_{2}}. We are mainly interested in the special case Ĥ_{n_{0}}^{β} = H_{n_{0},n_{0}+N}^{β,β} associated with the following Jacobi matrices (cf. (1.64))

Ĥ_{n_{0}}^{∞} = J_{n_{0},n_{0}+N},  Ĥ_{n_{0}}^{0} = J_{n_{0}+1,n_{0}+N+1},

(7.31)  Ĥ_{n_{0}}^{β} = ( \begin{matrix}b̂_{n_{0}+1} & a_{n_{0}+1} \\ a_{n_{0}+1} & b_{n_{0}+2} & ... \\ & ... & ... & ... \\ & & ... & b_{n_{0}+N−1} & a_{n_{0}+N−1} \\ & & & a_{n_{0}+N−1} & b̂_{n_{0}+N} \end{matrix} ) ,

(abbreviating a(n), b(n) as a_{n}, b_{n}) with b̂_{n_{0}+1} = b_{n_{0}+1} − a_{n_{0}}β^{−1}, b̂_{n_{0}+N} = b_{n_{0}} − a_{n_{0}}β, and β ∈ ℝ\{0}.

Second, we look at the operators

(7.32)  H̃_{n_{0}}^{θ} = ( \begin{matrix}b_{n_{0}+1} & a_{n_{0}+1} & & & e^{−iθ}a_{n_{0}} \\ a_{n_{0}+1} & b_{n_{0}+2} & ... \\ & ... & ... & ... \\ & & ... & b_{n_{0}+N−1} & a_{n_{0}+N−1} \\ e^{iθ}a_{n_{0}} & & & a_{n_{0}+N−1} & b_{n_{0}+N} \end{matrix} ) ,

with θ ∈ [0, 2π). These latter operators are associated with the following coupled boundary conditions (θ ∈ [0, 2π))

(7.33)

u(n_{0}) exp(iθ) − u(n_{0} + N) = 0,
u(n_{0} + 1) exp(iθ) − u(n_{0} + 1 + N) = 0.

The spectrum of Ĥ_{n_{0}}^{β} is given by the zeros {λ_{j}^{β}(n_{0})}Ñ ; j=1 (where Ñ = N − 1 for β = 0, ∞ and Ñ = N otherwise) of the characteristic equation

(7.34)  det(z − Ĥ_{n_{0}}^{β}) = ∏_{j=1}^{Ñ}(z − λ_{j}^{β}(n_{0})).

In the special case β = ∞ we set λ_{j}^{∞}(n_{0}) = μ_{j}(n_{0}) and consequently λ_{j}^{0}(n_{0}) = μ_{j}(n_{0} + 1) by (7.31).

As in (1.65) we can give an alternate characterization of the eigenvalues λ_{j}^{β}(n_{0}). Recall from Remark 1.9 that Ĥ_{n_{0}}^{β} can be obtained from H by imposing boundary conditions at n_{0}, n_{0}+N. Hence any eigenfunction must satisfy these boundary conditions. The solution satisfying the boundary condition at n_{0} is (up to a constant) given by

(7.35)  s_{β}(z, n, n_{0}) = sin(α)c(z, n, n_{0}) − cos(α)s(z, n, n_{0}), β = cot(α).

It is an eigenfunction of Ĥ_{n_{0}}^{β} if and only if it also satisfies the boundary condition at n_{0} + N. Thus we infer for β ∈ ℝ\{0}

cos(α)s_{β}(z, n_{0} + N, n_{0}) + sin(α)s_{β}(z, n_{0} + N + 1, n_{0})

= \frac{−(β + β^{−1})^{−1}}{A} ∏_{j=1}^{N}(z − λ_{j}^{β}(n_{0})), (7.36)

where the leading coefficient follows from (1.68). For β = 0, ∞ we have

c(z, n_{0} + N + 1, n_{0}) = \frac{−a(n_{0})}{A} ∏_{j=1}^{N−1}(z − μ_{j}(n_{0} + 1)),

(7.37)  s(z, n_{0} + N, n_{0}) = \frac{a(n_{0})}{A} ∏_{j=1}^{N−1}(z − μ_{j}(n_{0})).

Since the function satisfying the boundary condition at one side is unique (up to a constant), the spectrum σ(Ĥ_{n_{0}}^{β}) is simple (cf. Remark 1.10).

For later use note that standard linear algebra immediately yields the following trace relations (recall (7.2))

∑_{j=1}^{N−1} μ_{j}(n_{0}) = trĤ_{n_{0}}^{∞} = B − b(n_{0}),

(7.38)  ∑_{j=1}^{N} λ_{j}^{β}(n_{0}) = trĤ_{n_{0}}^{β} = B − (β + β^{−1})a(n_{0}).

Higher powers can be computed from (1.67).

Next, we turn to H̃_{n_{0}}^{θ}. Again the spectrum is given by the zeros {E_{j}^{θ}}_{j=1}^{N} of the characteristic equation

(7.39)  det(z − H̃_{n_{0}}^{θ}) = ∏_{j=1}^{N}(z − E_{j}^{θ}).

Our notation suggests that σ(H̃_{n_{0}}^{θ}) is independent of n_{0}. In fact, extending the eigenfunctions of H̃_{n_{0}}^{θ} to a solution of (1.19) on ℤ, we get a solution with the property u(n + N) = e^{iθ}u(n). Conversely, any solution of this kind gives rise to an eigenfunction of H̃_{n_{0}}^{θ}. In other words, the spectrum of H̃_{n_{0}}^{θ} can be characterized by

(7.40) λ ∈ σ(H̃_{n_{0}}^{θ}) ⇔ (m^{+}(λ) − e^{iθ})(m^{−}(λ) − e^{iθ}) = 0 ⇔ Δ(λ) = cos(θ),

where the independence of n_{0} is evident. Moreover, we have σ(H̃_{n_{0}}^{θ}) = σ(H̃_{n_{0}}^{−θ}) (−θ is to be understood mod2π) since the corresponding operators are anti-unitarily equivalent.

In the special case θ = 0, π we set H̃_{n_{0}}^{0} = H̃_{n_{0}}^{+}, H̃_{n_{0}}^{π} = H̃_{n_{0}}^{−} and similarly for the eigenvalues E_{j}^{0} = E_{j}^{+}, E_{j}^{π} = E_{j}^{−}.

By the above analysis, E_{j}^{±} are the zeros of Δ(z) ∓ 1 and we may write

(7.41)  Δ(z) ∓ 1 = \frac{1}{2A} ∏_{j=1}^{N}(z − E_{j}^{±}).

Since the N − 1 zeros of dΔ(z)/dz must interlace the N zeros of both Δ(z) ∓ 1 we infer

(7.42)  E_{1}^{±} < E_{1}^{∓} ≤ E_{2}^{∓} < E_{2}^{±} ≤ E_{3}^{±} < ⋯ ≤ E_{N}^{±(−1)^{N−1}} < E_{N}^{±(−1)^{N}}

for sgn(A) = ±(−1)^{N}. The numbers in the above sequence are denoted by E_{j}, 0 ≤ j ≤ 2N − 1. A typical discriminant is depicted below.

[figure: E_{0}; E_{1}; E_{2}; E_{3}; E_{4}; E_{5}; Δ(λ); −1; +1; λ]^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

This also shows (using (7.42)) that for θ ∈ (0, π)∪(π, 2π) the spectrum is simple. The case when the spectrum of H̃_{n}^{±} is degenerated can be easily characterized.

Lemma 7.4. The following conditions for E ∈ ℝ are equivalent.

(i). E is a double eigenvalue of H̃_{n}^{±}. (ii). All solutions of τu = Eu satisfy u(E, n + N) = ±u(E, n). (iii). M(E, n) = ±1l. (iv). E = E_{2j−1} = E_{2j} ∈ σ(Ĥ_{n}^{β}) for all β ∈ ℝ ∪ {∞}.

In particular, if one of these conditions holds for one n ∈ ℤ, it automatically holds for all.

The trace relation for H̃_{n}^{θ} reads

(7.43)  ∑_{j=1}^{N} E_{j}^{θ} = trH̃_{n}^{θ} = B, or ∑_{j=0}^{2N−1} E_{j} = 2B.

It can be used to eliminate B in the previous two trace relations (7.38) which produces the same trace relations already found in Section 6.2. This will become clear in Chapter 8 (see (8.11)).

Now, let us use this information to investigate H. Our choice for the branch in (7.11) implies |m^{+}(z)| < 1 for |z| large (see also (7.12) and Appendix A.7). Thus ψ_{+} is square summable near +∞ for |z| large. But since ψ_{+} is holomorphic for z ∈ ℂ\ℝ it is square summable for all z ∈ ℂ\ℝ (see Lemma 2.2). So we must have |m^{+}(z)| < 1 for z ∈ ℂ\ℝ and hence by continuity |m^{+}(z)| ≤ 1 for z ∈ ℂ.

In particular, we can use ψ_{±} to compute g(z, n). For the Wronskian of the Floquet functions we obtain

(7.44) W (ψ_{−}, ψ_{+}) = \frac{2a(n_{0})(Δ(z)^{2} − 1)^{1/2}}{s(z, n_{0} + N, n_{0})} = a(n_{0})(ϕ_{+}(z, n_{0}) − ϕ_{−}(z, n_{0}))

and if we evaluate (1.99) at m = n = n_{0} (u_{±} = ψ_{±}) we get from (7.37)

(7.45)  g(z, n_{0}) = \frac{a(n_{0})^{−1}}{ϕ_{+}(z, n_{0}) − ϕ_{−}(z, n_{0})} = \frac{∏_{j=1}^{N−1}(z − μ_{j}(n_{0}))}{− ∏_{j=0}^{2N−1} \sqrt{z − E_{j}}} .

Since n_{0} is arbitrary we must have

(7.46) ψ_{+}(z, n, n_{0})ψ_{−}(z, n, n_{0}) = \frac{a(n_{0})}{a(n)} \frac{s(z, n + N, n)}{s(z, n_{0} + N, n_{0})} = ∏_{j=1}^{N−1} \frac{z − μ_{j}(n)}{z − μ_{j}(n_{0})}.

Similarly, we compute

γ^{β}(z, n) = \frac{(ϕ_{+}(z, n) + β)(ϕ_{−}(z, n) + β)}{a(n)(ϕ_{+}(z, n) − ϕ_{−}(z, n))}
= \frac{(m^{+}(z) − sin^{−1}(α)s_{β}(z, n + N, n))(m^{−}(z) − sin^{−1}(α)s_{β}(z, n + N, n))}{2a(n)s(z, n + N, n)(Δ(z)^{2} − 1)^{1/2}}
= \frac{cos(α)s_{β}(z, n + N, n) + sin(α)s_{β}(z, n + N + 1, n)}{−2 sin^{2}(α)a(n)(Δ(z)^{2} − 1)^{1/2}}

(7.47) = −\frac{β}{a(n)} \frac{∏_{j=1}^{N}(z − λ_{j}^{β}(n))}{∏_{j=0}^{2N−1} \sqrt{z − E_{j}}},

where we have used (7.15), det M(z, n) = 1, and m^{+}(z) + m^{−}(z) = 2Δ(z). By (2.141) and (7.45) the spectrum of H is given by

(7.48)  σ(H) = {λ ∈ ℝ||Δ(λ)| ≤ 1} = ⋃_{j=0}^{N−1}[E_{2j}, E_{2j+1}]

and it is purely absolutely continuous (σ_{p}(H) = σ_{sc}(H) = ∅).

The sets

(7.49)  \overline{ρ}_{0} = (−∞, E_{0}], \overline{ρ}_{j} = [E_{2j−1}, E_{2j}], \overline{ρ}_{N} = [E_{2N−1}, ∞),

1 ≤ j ≤ N − 1, are called spectral gaps. In addition, since g(z, n) and γ^{β}(z, n) are Herglotz functions we infer (compare the argument in Section 8.1)

(7.50)  λ_{j}^{β}(n) ∈ \overline{ρ}_{j},  1 ≤ j ≤ N − 1, β ∈ ℝ ∪ \{∞\},

and

(7.51)  λ_{N}^{β}(n) ∈ \overline{ρ}_{N} for a(n)β < 0,  λ_{N}^{β}(n) ∈ \overline{ρ}_{0} for a(n)β > 0.

The most important case is β = ∞, λ_{j}^{∞}(n) = μ_{j}(n) and the typical situation looks as follows.

E_{0}  ^{E_{1}} •^{μ_{1}(n)}^{E_{2}}  ^{E_{3}} •^{μ_{2}(n)}  ^{E_{4}} ^{E_{5}}

The fact that s_{β}(λ_{j}^{β}(n_{0}), n, n_{0}) satisfies the same boundary condition at n_{0} and n_{0} + N implies that s_{β}(λ_{j}^{β}(n_{0}), n, n_{0}) is a Floquet solution. Hence we can associate a sign σ_{j}^{β}(n) with λ_{j}^{β}(n) such that

(7.52)  M(n, λ_{j}^{β}(n)) ( \begin{matrix}sin(α) \\ − cos(α) \end{matrix} ) = m^{σ_{j}^{β}(n)}(λ_{j}^{β}(n)) ( \begin{matrix}sin(α) \\ − cos(α) \end{matrix} ) .

If β = ∞ we abbreviate σ_{j}^{∞}(n) = σ_{j}(n). Moreover, if λ_{j}^{β}(n) does not coincide with one of the band edges E_{j}, then the corresponding Floquet solution is an eigenfunction of H_{σ_{j}^{β}(n),n}^{β} and consequently

(7.53)  σ(H_{n}^{β}) = σ(H) ∪ {λ_{j}^{β}(n)}Ñ ; j=1.

This characterization of the pairs (λ_{j}^{β}(n), σ_{j}^{β}(n)) will be useful later on because it does not rely on periodicity.

Remark 7.5. Another way to investigate periodic operators would be a (constant fiber) direct integral decomposition of H

(7.54)  ℓ^{2}(ℤ) ≅ ∫_{[0,2π)}^{⊕} ℓ^{2}(n_{0}, n_{0} + N + 1)\frac{dθ}{2π} ,  H ≅ ∫_{[0,2π)}^{⊕} H̃_{n_{0}}^{θ} \frac{dθ}{2π} ,

compare [195], Chapter XIII.16. However, since this approach will not reveal any new information for us, we will not investigate it further.

7.3. Polynomial identities

In the previous section we have seen that several objects exhibit a simple polynomial structure with respect to z. The present section further pursues this observation.

We first define the following polynomials

G(z, n) = \frac{A}{a(n)}s(z, n + N, n) = \frac{−A}{a(n − 1)}c(z, n + N, n − 1)

= ∏_{j=1}^{N−1}(z − μ_{j}(n)),
H(z, n) = A(s(z, n + N + 1, n) − c(z, n + N, n)),
(7.55)  R(z) = 4A^{2}(Δ(z)^{2} − 1) = ∏_{j=0}^{2N−1}(z − E_{j}).

In terms of these polynomials the monodromy matrix reads

(7.56)  M(z, n) = \frac{1}{A} ( \begin{matrix}AΔ(z) − \frac{1}{2}H(z, n) & a(n)G(z, n) \\ −a(n)G(z, n + 1) & AΔ(z) + \frac{1}{2}H(z, n) \end{matrix} ) .

Using (7.56) and (7.52) we get

(7.57) H(n, μ_{j}(n + 1)) = −R^{1/2}(μ̂_{j}(n + 1)),  H(n, μ_{j}(n)) = R^{1/2}(μ̂_{j}(n)),

where we have set R^{1/2}(z) = 2A(Δ(z)^{2}−1)^{1/2} and used the convenient abbreviation

(7.58)  R^{1/2}(λ̂_{j}^{β}(n)) = σ_{j}^{β}(n)R^{1/2}(λ_{j}^{β}(n)), λ̂_{j}^{β}(n) = (λ_{j}^{β}(n), σ_{j}^{β}(n)).

Thus Newton’s interpolation formula yields

H(z, n) = ∑_{j=1}^{N−1} R^{1/2}(μ̂_{j}(n)) ∏_{k≠j}^{N−1} \frac{z − μ_{k}(n)}{μ_{j}(n) − μ_{k}(n)} + (z − b(n))G(z, n)

= − ∑_{j=1}^{N−1} R^{1/2}(μ̂_{j}(n + 1)) ∏_{k≠j}^{N−1} \frac{z − μ_{k}(n + 1)}{μ_{j}(n + 1) − μ_{k}(n + 1)}

+ (z − b(n + 1))G(z, n + 1), (7.59)

where the factor in front of G(z, n), G(z, n + 1) follows after expanding both sides and considering the highest two coefficients. In addition, we obtain from (7.59)

H(z, n) + H(z, n − 1) = 2(z − b(n))G(z, n),
(7.60) H(z, n) − H(z, n − 1) = 2 ∑_{j=1}^{N−1} R^{1/2}(μ̂_{j}(n)) ∏_{k≠j}^{N−1} \frac{z − μ_{k}(n)}{μ_{j}(n) − μ_{k}(n)}.

Next, from (7.56) and (7.60) we obtain

\frac{1}{A} ∑_{j=1}^{N−1} R^{1/2}(μ̂_{j}(n)) ∏_{k≠j}^{N−1} \frac{z − μ_{k}(n)}{μ_{j}(n) − μ_{k}(n)}

= c(z, n + N − 1, n − 1) − c(z, n + N, n)

= s(z, n + N + 1, n) − s(z, n + N, n − 1). (7.61)

Finally, we have

(7.62)  ϕ_{±}(z, n) = \frac{H(z, n) ± R^{1/2}(z)}{2a(n)G(z, n)} = \frac{2a(n)G(z, n + 1)}{H(z, n) ∓ R^{1/2}(z)},

and thus

(7.63)  4a(n)^{2}G(z, n)G(z, n + 1) = H(z, n)^{2} − R(z),

which is equivalent to det M(z, n) = 1.

Remark 7.6. In the case of closed gaps the factor

(7.64)  Q(z) = ∏_{j∈Γ′}(z − E_{2j})

cancels from all equations since μ_{j}(n) = E_{2j−1} = E_{2j} if j ∈ Γ′. Here

(7.65) Γ′ = {1 ≤ j ≤ N − 1|E_{2j−1} = E_{2j}}, Γ = {1 ≤ j ≤ N − 1|E_{2j−1} < E_{2j}}.

In particular, observe \sqrt{z − E_{2j−1}}\sqrt{z − E_{2j}} = z − μ_{j}(n) for j ∈ Γ′.

7.4. Two examples: period one and two

In this section we consider the two simplest cases when the period is one or two.

We start with period one. We assume a(n) = \frac{1}{2}, b(n) = 0 and consider them to be periodic with period N. By inspection, the Floquet functions (n_{0} = 0) are given by

ψ_{±}(z, n) = u_{±}(z, n) = (z ± R_{2}^{1/2}(z))^{n},
(7.66)  ϕ_{±}(z) = z ± R_{2}^{1/2}(z).

Hence the discriminant and the Floquet multipliers read

Δ(z) = \frac{1}{2}(s(z, N + 1) − s(z, N − 1)) = \frac{1}{2}(m^{+}(z) + m^{−}(z)),
(7.67) m^{±}(z) = Δ(z) ± (Δ(z)^{2} − 1)^{1/2} = (z ± R_{2}^{1/2}(z))^{N}.

Next we calculate

Δ(z)^{2} − 1 = \frac{1}{4}(m^{+}(z) − m^{−}(z))^{2}
= \frac{1}{4}((z + R_{2}^{1/2}(z))^{N} − (z − R_{2}^{1/2}(z))^{N})^{2}
(7.68)  = (z^{2} − 1)s(z, N)^{2}.

Therefore we have

(7.69)  E_{0} = −1, E_{2j−1} = E_{2j} = μ_{j} = λ_{j}^{β} = − cos(\frac{jπ}{N}), E_{2N−1} = 1,

1 ≤ j ≤ N − 1, and all spectral gaps \overline{ρ}_{j}, 1 ≤ j ≤ N − 1 are closed. We can now verify the first relation in (7.38) directly and from the second we obtain

(7.70)  λ_{N}^{β} = −\frac{β + β^{−1}}{2} .

Which shows that λ_{N}^{β} ∈ \overline{ρ}_{N} for β < 0 and λ_{N}^{β} ∈ \overline{ρ}_{0} for 0 < β.

Now let us consider the case of period N = 2. In this case A = a(0)a(1) and B = b(0) + b(1). Moreover, one computes

(7.71)  Δ(z) = \frac{(z − \frac{B}{2} )^{2} − C^{2}}{2A} , C^{2} = a(0)^{2} + a(1)^{2} + \frac{(b(0) − b(1))^{2}}{4}

and hence

R(z) = 4A^{2}(Δ(z)^{2} − 1) = ((z − \frac{B}{2} )^{2} − C^{2})^{2} − 4A^{2}
(7.72)  = (z − E_{0})(z − E_{1})(z − E_{2})(z − E_{3}),

where

(7.73)  E_{0,1} = \frac{B}{2} − \sqrt{C^{2} ± 2|A|}, E_{2,3} = \frac{B}{2} + \sqrt{C^{2} ± 2|A|}.

Conversely, we have B = E_{0} + E_{2} = E_{1} + E_{3}, C^{2} = ((E_{0} − E_{2})^{2} + (E_{1} − E_{3})^{2})/8, and |A| = ((E_{0} − E_{2})^{2} − (E_{1} − E_{3})^{2})/16. Our trace formulas (7.38) produce

(7.74)  μ_{1}(0) = b(1), μ_{1}(1) = b(0), λ_{1}^{β}(n) = B − (β + β^{−1})a(n)

and we observe Δ(μ_{1}(0)) = Δ(μ_{1}(1)) = \frac{(μ_{1}(0)−μ_{1}(1))^{2}−4C^{2}}{8A} = Δ(μ_{1}). A short computation shows

(7.75)  m^{σ_{1}(0)}(μ_{1}) = −\frac{a(1)}{a(0)}, m^{σ_{1}(1)}(μ_{1}) = −\frac{a(0)}{a(1)},

where m^{±}(μ_{1}) = m^{±}(μ_{1}(0)) = m^{±}(μ_{1}(1)). Moreover, we have σ_{1}(0) = −σ_{1}(1) = ±sgn(|a(0)| − |a(1)|) and a(0) = a(1) implies Δ(μ_{1})^{2} = 1. Solving m^{±}(μ_{1}) for a(n) shows

a(0)^{2} = −A m^{−σ_{1}(0)}(μ_{1}) = −A(Δ(μ_{1}) − σ_{1}(0)(Δ(μ_{1})^{2} − 1)^{1/2}),
(7.76) a(1)^{2} = −A m^{σ_{1}(0)}(μ_{1}) = −A(Δ(μ_{1}) + σ_{1}(0)(Δ(μ_{1})^{2} − 1)^{1/2}).

In particular, this shows that the coefficients a(n), b(n) are expressible in terms of E_{j}, 0 ≤ j ≤ 3, and (e.g.) μ_{1}(0), σ_{1}(0). This result will be generalized in Section 8.1.

7.5. Perturbations of periodic operators

In this section we are going to study short-range perturbations H of periodic operators H_{p} associated with sequences a, b satisfying a(n) → a_{p}(n) and b(n) → b_{p}(n) as |n| → ∞. Our main hypothesis for this section reads

Hypothesis H . 7.7. Suppose a_{p}, b_{p} are given periodic sequences and H_{p} is the corresponding Jacobi operator. Let H be a perturbation of H_{p} such that

(7.77)  ∑_{n∈ℤ} |n(a(n) − a_{p}(n))| < ∞, ∑_{n∈ℤ} |n(b(n) − b_{p}(n))| < ∞.

We start with two preliminary lemmas.

Lemma 7.8. Consider the Volterra sum equation

(7.78)  f(n) = g(n) + ∑_{m=n+1}^{∞} K(n, m)f(m).

Suppose there is a sequence K̂(n, m) such that

(7.79) |K(n, m)| ≤ K̂(n, m), K̂(n + 1, m) ≤ K̂(n, m), K̂(n, .) ∈ ℓ^{1}(0, ∞).

Then, for given g ∈ ℓ^{∞}(0, ∞), there is a unique solution f ∈ ℓ^{∞}(0, ∞), fulfilling the estimate

(7.80)  |f(n)| ≤ ( sup_{m>n} |g(m)|) exp ( ∑_{m=n+1}^{∞} K̂(n, m)).

If g(n) and K(n, m) depend continuously (resp. holomorphically) on a parameter and if K̂ does not depend on this parameter, then the same is true for f(n).

Proof. Using the standard iteration trick

(7.81)  f_{0}(n) = g(n), f_{j+1}(n) = ∑_{m=n+1}^{∞} K(n, m)f_{j}(m),

we see that the solution is formally given by

(7.82)  f(n) = ∑_{j=0}^{∞} f_{j}(n).

To prove uniqueness and existence it remains to show that this iteration converges. We claim

(7.83)  |f_{j}(n)| ≤ \frac{sup_{m>n} |g(m)|}{j!}( ∑_{m=n+1}^{∞} K̂(n, m))^{j}, j ≥ 1,

which follows from

|f_{j+1}(n)| ≤ ∑_{m=n+1}^{∞} K̂(n, m)\frac{sup_{l>m} |g(l)|}{j!}( ∑_{l=m+1}^{∞} K̂(m, l))^{j}

≤ \frac{sup_{m>n} |g(m)|}{j!} ∑_{m=n+1}^{∞} K̂(n, m)( ∑_{l=m+1}^{∞} K̂(n, l))^{j}

≤ \frac{sup_{m>n} |g(m)|}{(j + 1)!} ∑_{m=n+1}^{∞} (( ∑_{l=m}^{∞} K̂(n, l))^{j+1}

− ( ∑_{l=m+1}^{∞} K̂(n, l))^{j+1})

= \frac{sup_{m>n} |g(m)|}{(j + 1)!}( ∑_{l=n+1}^{∞} K̂(n, l))^{j+1},

where we have used (s = K̂(n, m) ≥ 0, S = ∑_{ℓ=m+1}^{∞} K̂(n, ℓ) ≥ 0)

(7.84)  (S + s)^{j+1} − S^{j+1} = sS^{j} ∑_{ℓ=0}^{j}(1 + \frac{s}{S})^{ℓ} ≥ (j + 1)sS^{j}, j ∈ ℕ.

This settles the iteration and the estimate (7.80). The rest follows from uniform convergence of the series (7.82). □

Remark 7.9. A similar result holds for equations of the type

(7.85)  f(n) = g(n) + ∑_{m=−∞}^{n−1} K(n, m)f(m).

Lemma 7.10. Assume (H.7.7). Then there exist solutions u_{±}(z, .), z ∈ ℂ, of τu = zu satisfying

(7.86)  lim_{n→±∞} |m_{p}^{∓}(z)^{n/N}(u_{±}(z, n) − u_{p,±}(z, n))| = 0,

where u_{p,±}(z, .), m_{p}^{±}(z), and N are the Floquet solutions, Floquet multipliers, and period of H_{p}, respectively (cf. (7.19)). In addition, u_{±}(z, .) can be assumed continuous (resp. holomorphic) with respect to z whenever u_{p,±}(z, .) are and they satisfy the equations

(7.87) u_{±}(z, n) = \frac{a_{p}(n − {0 \atop 1})}{a(n − {0 \atop 1})}u_{p,±}(z, n) ± ∑_{m= {n+1 \atop −∞}}^{{∞ \atop n−1}} \frac{a_{p}(n − {0 \atop 1})}{a(n − {0 \atop 1})}K(z, n, m)u_{±}(z, m),

where

K(z, n, m) = \frac{((τ − τ_{p})u_{p,−}(z))(m)u_{p,+}(z, n) − u_{p,−}(z, n)((τ − τ_{p})u_{p,+}(z))(m)}{W_{p}(u_{p,−}(z), u_{p,+}(z))}
= \frac{s_{p}(z, n, m + 1)}{a_{p}(m + 1)}(a(m) − a_{p}(m)) + \frac{s_{p}(z, n, m)}{a_{p}(m)}(b(m) − b_{p}(m))

+ \frac{s_{p}(z, n, m − 1)}{a_{p}(m − 1)}(a(m − 1) − a_{p}(m − 1)) (7.88)

(W_{p}(., ..) denotes the Wronskian formed with a_{p} rather than a).

Proof. We only prove the claim for u_{+}(z, .), the one for u_{−}(z, .) being similar. Using the transformation ũ_{+}(z, .) = m_{p}^{−}(z)^{n/N}u_{+}(z, .) we get a sequence which is bounded near +∞.

Suppose ũ_{+}(z, .) satisfies (disregarding summability for a moment)

ũ_{+}(z, n) = \frac{a_{p}(n)}{a(n)}ũ_{p,+}(z, n)

+ ∑_{m=n+1}^{∞} \frac{a_{p}(n)}{a(n)}m_{p}^{−}(z)^{(n−m)/N}K(z, n, m)ũ_{+}(z, m). (7.89)

Then u_{+}(z, .) fulfills τu = zu and (7.86). Hence, if we can apply Lemma 7.8, we are done. To do this, we need an estimate for K(z, n, m) or, equivalently, for the growth rate of the transfer matrix Φ(z, n, m). By (7.6) it suffices to show

(7.90)  ∥(m_{p}^{−}(z)M_{p}(z, m))^{n}∥ ≤ const(z, m) n, n ∈ ℕ.

Abbreviate M(z) = m_{p}^{−}(z)M_{p}(z, m) and note that the eigenvalues of M(z) are 1 and m_{p}^{−}(z)^{2}. After performing the unitary transformation U(z) = (e(z), e^{⊥}(z)), where e(z) is the normalized eigenvector of M(z) corresponding to the eigenvalue 1 and e^{⊥}(z) is its orthogonal complement, we have

(7.91)  M̃(z) = U(z)^{−1}M(z)U(z) = ( \begin{matrix}1 & α(z) \\ 0 & m_{p}^{−}(z)^{2} \end{matrix} ) .

Using

(7.92)  M̃(z)^{n} = ( \begin{matrix}1 & α(z) ∑_{j=0}^{n−1} m_{p}^{−}(z)^{2j} \\ 0 & m_{p}^{−}(z)^{2n} \end{matrix} ) , n ∈ ℕ,

and |m_{p}^{−}(z)| ≤ 1 we see ∥M̃(z)^{n}∥ ≤ max{1, |α(z)|n} ≤ (1 + |α(z)|)n, which is the desired estimate. Since e(z) and α(z) are continuous with respect to z, the constant in (7.90) can be chosen independent of z as long as z varies in compacts. □

Theorem 7.11. Suppose (H.7.7) holds. Then we have σ_{ess}(H) = σ(H_{p}), the point spectrum of H is finite and confined to the spectral gaps of H_{p}, that is, σ_{p}(H) ⊂ ℝ\σ(H_{p}). Furthermore, the essential spectrum of H is purely absolutely continuous.

Proof. That σ_{ess}(H) = σ_{ess}(H_{p}) follows from Lemma 3.9. To prove the remaining claims we use the solutions u_{±}(λ, .) of τu = λu for λ ∈ σ(H_{p}) found in Lemma 7.10. Since u_{±}(λ, .), λ ∈ σ(H_{p}) are bounded and do not vanish near ±∞, there are no eigenvalues in the essential spectrum of H and invoking Theorem 3.18 shows that the essential spectrum of H is purely absolutely continuous. Moreover, (7.86) with λ = E_{0} implies that H − E_{0} is non-oscillatory since we can assume (perhaps after flipping signs) u_{p,±}(E_{0}, n) ≥ ε > 0, n ∈ ℤ, and by Corollary 4.11 there are only finitely many eigenvalues below E_{0}. Similarly, (using Remark 4.12) there are only finitely many eigenvalues above E_{2N+1}. Applying Corollary 4.20 in each gap (E_{2j−1}, E_{2j}), 1 ≤ j ≤ N, shows that the number of eigenvalues in each gap is finite as well. □

These results enable us to define what is known as scattering theory for the pair (H, H_{p}), where H is a Jacobi operator satisfying (H.7.7).

Since we are most of the time interested in the case z ∈ σ(H_{p}), we shall normalize u_{p,±}(λ, 0) = 1 for λ ∈ σ(H_{p}). In particular, note that we have \overline{u_{p,±}(λ)} = u_{p,∓}(λ), where the bar denotes complex conjugation. Since one computes

(7.93) W (u_{±}(λ), \overline{u_{±}(λ)}) = W_{p}(u_{p,±}(λ), u_{p,∓}(λ)) = ∓\frac{2i sin(q(λ)N)}{s_{p}(λ, N)}, λ ∈ σ(H_{p}),

(s_{p}(λ, n) is the solution of τ_{p}u = zu corresponding to the initial condition s_{p}(λ, 0) = 0, s_{p}(λ, 1) = 1) we conclude that u_{±}(λ), \overline{u_{±}(λ)} are linearly independent for λ in the interior of σ(H_{p}) (if two bands collide at E, the numerator and denominator of (7.93) both approach zero when λ → E and both have a nonzero limit). Hence we might set

(7.94)  u_{±}(λ, n) = α(λ)\overline{u_{∓}(λ, n)} + β_{∓}(λ)u_{∓}(λ, n), λ ∈ σ(H_{p}),

where

α(λ) = \frac{W (u_{∓}(λ), u_{±}(λ))}{W (u_{∓}(λ), \overline{u_{∓}(λ)})} = \frac{s_{p}(λ, N)}{2i sin(q(λ)N)}W (u_{−}(λ), u_{+}(λ)),

(7.95) β_{±}(z) = \frac{W (u_{∓}(λ), \overline{u_{±}(λ))}}{W (u_{±}(λ), \overline{u_{±}(λ)})} = ±\frac{s_{p}(λ, N)}{2i sin(q(λ)N)}W (u_{∓}(λ), \overline{u_{±}(λ)}).

The function α(λ) can be defined for all λ ∈ ℂ\{E_{p,j}}. Moreover, the Plücker identity (2.169) with f_{1} = u_{∓}, f_{2} = u_{±}, f_{3} = \overline{u_{∓}}, f_{4} = \overline{u_{±}}, implies

(7.96)  |α(λ)|^{2} = 1 + |β_{±}(λ)|^{2} and \overline{β_{±}(λ)} = −β_{∓}(λ).

Using (7.87) one can also show (7.97)

W (u_{−}(λ), u_{+}(λ)) = W_{p}(u_{p,−}(λ), u_{p,+}(λ)) + ∑_{n∈ℤ} u_{±}(λ, n)((τ − τ_{p})u_{p,∓}(λ))(n)

and

(7.98)  W (u_{∓}(λ), \overline{u_{±}(λ)}) = ∓ ∑_{n∈ℤ} u_{±}(λ, n)((τ − τ_{p})u_{p,±}(λ))(n).

We now define the scattering matrix

(7.99)  S(λ) = ( \begin{matrix}T (λ) & R_{−}(λ) \\ R_{+}(λ) & T (λ) \end{matrix} ) , λ ∈ σ(H_{p}),

of the pair (H, H_{p}), where T (λ) = α(λ)^{−1} and R_{±}(λ) = α(λ)^{−1}β_{±}(λ). The matrix S(λ) is easily seen to be unitary since by (7.96) we have |T (λ)|^{2} + |R_{±}(λ)|^{2} = 1 and T (λ)\overline{R_{+}(λ)} = −\overline{T (λ)}R_{−}(λ).

The quantities T (λ) and R_{±}(λ) are called transmission and reflection coefficients respectively. The following equation further explains this notation: (7.100)

T (λ)u_{±}(λ, n) = \{ \begin{matrix}T (λ)u_{p,±}(λ, n), & n → ±∞ \\ u_{p,±}(λ, n) + R_{∓}(λ)u_{p,∓}(λ, n), & n → ∓∞ \end{matrix} , λ ∈ σ(H_{p}).

If we regard u_{p,±}(λ, n) as incoming plain wave packet, then T (λ)u_{p,±}(λ, n) and R_{∓}(λ)u_{p,∓}(λ, n) are the transmitted and reflected packets respectively.

The quantities T (λ) and R_{±}(λ) can be expressed in terms of m̃_{±}(z) = m̃_{±}(z, 0) as follows

T (λ) = \frac{\overline{u_{±}(λ, 0)}}{u_{∓}(λ, 0)} \frac{2iIm(m̃_{±}(λ))}{m̃_{−}(λ) + m̃_{+}(λ)},
(7.101)  R_{±}(λ) = −\frac{\overline{u_{±}(λ, 0)}}{u_{±}(λ, 0)} \frac{m̃_{∓}(λ) + \overline{m̃_{±}(λ)}}{m̃_{−}(λ) + m̃_{+}(λ)}, λ ∈ σ(H_{p}).

Here we have set m̃_{±}(λ) = lim_{ε↓0} m̃_{±}(λ + iε), λ ∈ ℝ as usual. In addition, one verifies (λ ∈ σ(H_{p}))

g(λ + i0, n) = \frac{u_{−}(λ, n)u_{+}(λ, n)}{W (u_{−}(λ), u_{+}(λ))} = T (λ)\frac{s_{p}(λ, N)}{2i sin(q(λ)N)}u_{−}(λ, n)u_{+}(λ, n)
(7.102)  = \frac{s_{p}(λ, N)}{2i sin(q(λ)N)}|u_{±}(λ, n)|^{2}(1 + R_{±}(λ)\frac{u_{±}(λ, n)}{\overline{u_{±}(λ, n)}}).

We can now give a first application of the trace formulas derived in Section 6.2. Denote by E_{0} ≤ E_{1} < E_{2} ≤ ⋯ < E_{2M} ≤ E_{2M+1} the band edges of H, where equality holds if and only if E_{2j} = E_{2j+1} is an eigenvalue of H. That is,

(7.103)  σ(H) = ⋃_{j=0}^{M−1}[E_{2j}, E_{2j+1}].

Furthermore, define the number μ_{j}(n) associated with each spectral gap ρ_{j} = (E_{2j−1}, E_{2j}) by (compare Section 8.1 for further details)

(7.104)  μ_{j}(n) = sup\{E_{2j−1}\} ∪ \{λ ∈ ρ_{j}|g(λ, n) < 0\} ∈ \overline{ρ_{j}}, 1 ≤ j ≤ M.

Then we infer

ξ(λ, n) = \frac{1}{2}χ_{(E_{0},E_{∞})}(λ) + \frac{1}{2} ∑_{j=1}^{M} (χ_{(E_{2j−1},μ_{j}(n))}(λ) − χ_{(μ_{j}(n),E_{2j})}(λ))

+ χ_{(E_{∞},∞)}(λ) + \frac{1}{π} arg (1 + R_{±}(λ)\frac{u_{±}(λ, n)}{\overline{u_{±}(λ, n)}})χ_{σ(H_{p})}(λ) (7.105)

since we have

(7.106)  ξ(λ, n) = \frac{1}{2} + \frac{1}{π} arg (1 + R_{±}(λ)\frac{u_{±}(λ, n)}{\overline{u_{±}(λ, n)}}), λ ∈ σ(H_{p}).

Hence we obtain from (6.58)

b^{(ℓ)}(n) = \frac{1}{2} ∑_{j=0}^{2M+1} E_{j}^{ℓ} − ∑_{j=1}^{M−1} μ_{j}(n)^{ℓ}

+ \frac{ℓ}{π} ∫_{σ(H_{p})} λ^{ℓ−1} arg (1 + R_{±}(λ)\frac{u_{±}(λ, n)}{\overline{u_{±}(λ, n)}})dλ. (7.107)

Remark 7.12. If H is reflectionless, that is R_{±}(λ) = 0, then H can be obtained from H_{p} by inserting the corresponding number of eigenvalues using the double commutation method provided in Section 11.6 since this transformation preserves the reflectionless property (cf. Remark 11.22).

Finally, we will write down the eigenfunction expansions for this case. We set (cf. (2.123))

(7.108)  \underline{U}(λ, n) = ( \begin{matrix}u_{+}(λ, n) \\ u_{−}(λ, n) \end{matrix} ) = U(λ) ( \begin{matrix}c(λ, n) \\ s(λ, n) \end{matrix} ) ,

where

(7.109)  U(λ) = ( \begin{matrix}u_{+}(λ, 0) & −u_{+}(λ, 0)m̃_{+}(λ) \\ u_{−}(λ, 0) & u_{−}(λ, 0)m̃_{−}(λ) \end{matrix} ) .

By the considerations at the end of Section 2.5 this choice of basis will diagonalize the matrix measure in the eigenfunction expansion. A short calculation (using (7.93)) shows

(7.110)  |u_{±}(λ, 0)|^{2} = \frac{− sin(q(λ)N)}{a(0)^{2}s_{p}(λ, N)Im(m̃_{±}(λ))}, λ ∈ σ(H_{p}),

that u_{±}(λ, n) are not correctly normalized. However, this can be taken into account easily. Transforming the spectral measure to this new basis yields (use (7.101))

dρ̃_{ac}(λ) = (\overline{U^{−1}(λ)})^{⊤} dρ_{ac}(λ) U^{−1}(λ)
(7.111)  = \frac{−s_{p}(λ, N)}{4 sin(q(λ)N)}|T (λ)|^{2} ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) χ_{σ(H_{p})}(λ)dλ.

For the pure point part we can choose (e.g.)

(7.112)  dρ̃_{pp}(λ) = ∑_{E_{j}∈σ_{p}(H)} ( \begin{matrix}γ_{+}(E_{j}) & 0 \\ 0 & 0 \end{matrix} ) dΘ(λ − E_{j}),

where γ_{+}(E_{j}) is the norming constant corresponding to the eigenvalue E_{j}, that is,

(7.113)  γ_{+}(E_{j}) = ( ∑_{n∈ℤ} |u_{+}(E_{j}, n)|^{2})^{−1}

and Θ(λ) = 0 for λ ≤ 0 respectively Θ(λ) = 1 for λ > 0.

Transforming (2.131) shows that \underline{U}(λ, n) are orthogonal with respect to dρ̃ = dρ̃_{ac} + dρ̃_{pp},

⟨\underline{U}(λ, m), \underline{U}(λ, n)⟩_{L^{2}} = ∑_{i,j=0}^{1} ∫_{ℝ} \overline{U_{i}(λ, m)}U_{j}(λ, n)dρ̃_{i,j}(λ)

(7.114)  ≡ ∫_{ℝ} \overline{\underline{U}(λ, m)}\underline{U}(λ, n)dρ̃(λ) = δ_{m,n}.

Moreover, we obtain a unitary transformation Ũ : ℓ^{2}(ℤ) → L^{2}(ℝ, dρ̃) defined by

(Ũf)(λ) = ∑_{n∈ℤ} f(n)\underline{U}(λ, n),

(7.115)  (Ũ^{−1}\underline{F})(n) = ∫_{ℝ} \overline{\underline{U}(λ, n)}\underline{F} (λ)dρ̃(λ),

which maps the operator H to the multiplication operator by λ,

(7.116)  ŨHŨ^{−1} = H̃,

where^{(7.117)  H̃\underline{F} (λ) = λ\underline{F} (λ), \underline{F} (λ) ∈ L^{2}(ℝ, dρ̃).}

Reflectionless Jacobi operators

In this chapter we are going to look at a class of operators for which the trace formulas of Section 6.2 become particularly simple. Based on this fact we are able to give a detailed spectral and inverse spectral analysis.

8.1. Spectral analysis and trace formulas

In this section we discuss the direct spectral problem for a certain class of reflectionless bounded Jacobi operators.

A Jacobi operator H is called reflectionless if for all n ∈ ℤ,

(8.1)  ξ(λ, n) = \frac{1}{2} for a.e. λ ∈ σ_{ess}(H),

where ξ(λ, n) is the xi-function of H introduced in (6.53).

For instance, periodic operators and operators with purely discrete spectrum are special cases of reflectionless operators. The following result further illustrates the reflectionless condition (8.1).

Lemma 8.1. Suppose Λ ⊂ σ(H). Then the following conditions are equivalent. (i). For all n ∈ ℤ, ξ(λ, n) = \frac{1}{2} for a.e. λ ∈ Λ. (ii). For some n_{0} ∈ ℤ, n_{1} ∈ ℤ\{n_{0}, n_{0} + 1},

ξ(λ, n_{0}) = ξ(λ, n_{0} + 1) = ξ(λ, n_{1}) = \frac{1}{2} for a.e λ ∈ Λ.

(iii). For some n_{0} ∈ ℤ,

m̃_{+}(λ + i0, n_{0}) = −\overline{m̃_{−}(λ + i0, n_{0})} for a.e. λ ∈ Λ,

or equivalently a(n_{0})^{2}m_{+}(λ + i0, n_{0}) = −a(n_{0} − 1)^{2}\overline{m_{−}(λ + i0, n_{0})} − λ + b(n_{0}). (iv) For some n_{0} ∈ ℤ,

Re(g(λ, n_{0})) = Re(h(λ, n_{0})) = 0 for a.e. λ ∈ Λ.

(v) For some n_{0} ∈ ℤ,

m̃_{+}^{β}(λ + i0, n_{0}) = −\overline{m̃_{−}^{β}(λ + i0, n_{0})} for a.e. λ ∈ Λ.

(vi). For some n_{0} ∈ ℤ and u_{±}(z, n) as in (2.20),

u_{+}(λ + i0, n_{0}) = \overline{u_{−}(λ + i0, n_{0})} for a.e. λ ∈ Λ.

Proof. Without restriction we choose n_{0} = 0. Moreover, for a.e. λ ∈ Λ we can assume that both m̃_{±}(λ + i0) exist and that |m̃_{−}(λ + i0)|^{2} + |m̃_{+}(λ + i0)|^{2} > 0, s(λ, n_{1})c(λ, n_{1}) ≠ 0. Hence we fix such a λ and abbreviate m̃_{±} = m̃_{±}(λ+i0). Using (2.20) we see that the requirement ξ(λ, n) = \frac{1}{2}, that is, g(λ + i0, n) = \overline{−g(λ + i0, n)}, is equivalent to

Re(\overline{(m̃_{−} + m̃_{+})}(c(λ, n)^{2} + a(n)(m̃_{−} − m̃_{+})s(λ, n)c(λ, n))

− a(n)^{2}(m̃_{−}m̃_{+})s(λ, n)^{2}) = 0. (8.2)

Clearly (i) ⇒ (ii). In order to prove that (ii) ⇒ (iii) we first pick n = 0 in (8.2) yielding Re(m̃_{−}) = −Re(m̃_{+}). Furthermore choosing n = 1 shows Im(m̃_{−}) = Im(m̃_{+}) or Re(m̃_{−}) = Re(m̃_{+}) = 0. In the latter case, Re(m̃_{−}) = Re(m̃_{+}) = 0, we obtain for n = n_{1} that Im(m̃_{−}+m̃_{+})Im(m̃_{−}−m̃_{+}) = 0 and since Im(m̃_{±}) > 0 again Im(m̃_{−}) = Im(m̃_{+}). This completes (ii) ⇒ (iii). The case (iii) ⇒ (i) is obvious from (8.2) and the rest is evident from (2.20) and (2.92). □

The last condition (vi) implies vanishing of the reflection coefficients R_{±}(λ) (cf. (7.99)), explaining the term reflectionless (see also (7.106)). Moreover, condition (iii) allows us to strengthen Theorem 2.29 for reflectionless Jacobi operators.

Lemma 8.2. Suppose H is reflectionless and σ_{ess}(H) is of positive Lebesgue measure, then H is uniquely determined by one of the Weyl m̃-functions m̃_{+}(z, n_{0}) or m̃_{−}(z, n_{0}). The result still holds if the reflectionless condition only holds for a set of positive Lebesgue measure.

Proof. Combine Lemma 8.1 (iii) and Theorem B.8 (iii). □

Since the spectrum of H is closed, it can be written as the complement of a countable union of disjoint open intervals, that is,

(8.3)  σ(H) = Σ = ℝ\ ⋃_{j∈J_{0}∪{∞}} ρ_{j},

where J ⊆ ℕ, J_{0} = J ∪ {0},

ρ_{0} = (−∞, E_{0}), ρ_{∞} = (E_{∞}, ∞),

E_{0} ≤ E_{2j−1} < E_{2j} ≤ E_{∞}, ρ_{j} = (E_{2j−1}, E_{2j}), j ∈ J, (8.4)

−∞ < E_{0} < E_{∞} < ∞, ρ_{j} ∩ ρ_{k} = ∅ for j ≠ k.

We emphasize that the notation employed in (8.4) implies that E_{2j} = E_{2k+1} for some k ∈ J_{0} ∪ {∞} whenever E_{2j} ∈ σ_{d}(H).

Next, we turn to Dirichlet eigenvalues associated with each spectral gap ρ_{j}. Formally μ_{j}(n) is the zero of g(z, n) in ρ_{j}. Unfortunately, this definition causes trouble since μ_{j} could lie at a boundary point of ρ_{j} where g(., n) is no longer holomorphic. Hence we use the more sophisticated one

(8.5)  μ_{j}(n) = sup\{E_{2j−1}\} ∪ \{λ ∈ ρ_{j}|g(λ, n) < 0\} ∈ \overline{ρ_{j}}, j ∈ J.

Strict monotonicity of g(λ, n) with respect to λ ∈ ρ_{j} (cf. (2.37)) then yields

(8.6)  \begin{matrix}g(λ, n) < 0, & λ ∈ (E_{2j−1}, μ_{j}(n)), \\ g(λ, n) > 0, & λ ∈ (μ_{j}(n), E_{2j}), \end{matrix} j ∈ J.

Moreover, let μ ∈ ℝ\σ_{ess}(H), then g(μ, n) = 0 implies u_{−}(μ, n) = 0 or u_{+}(μ, n) = 0 (if μ ∈ σ_{d}(H) we have u_{−}(μ, n) = u_{+}(μ, n) = 0). Thus μ ∈ σ_{d}(H_{n}^{∞}) and hence we infer

(8.7)  σ(H_{n}^{∞}) = σ_{ess}(H) ∪ {μ_{j}(n)}_{j∈J}.

However, observe that μ_{j}(n) is not necessarily an eigenvalue of H_{n}^{∞} unless μ_{j}(n) ∉ σ_{ess}(H).

Equation (8.6) shows that ξ(λ, n) = 1, λ ∈ (E_{2j−1}, μ_{j}(n)), and ξ(λ, n) = 0, λ ∈ (μ_{j}(n), E_{2j}), j ∈ J. Thus we know ξ(λ, n) for λ ∈ ℝ\σ_{ess}(H) and if we assume ξ(λ, n) = 1/2 for λ ∈ σ_{ess}(H) we obtain (see also Section 6.2)

Lemma 8.3. Let H be reflectionless. Then

g(z, n) = \frac{−1}{z − E_{∞}} exp (∫_{E_{0}}^{E_{∞}} \frac{ξ(λ, n)dλ}{λ − z})

(8.8)  = \frac{−1}{\sqrt{z − E_{0}}\sqrt{z − E_{∞}}} ∏_{j∈J} \frac{z − μ_{j}(n)}{\sqrt{z − E_{2j−1}}\sqrt{z − E_{2j}}} .

In particular, denoting by χ_{Λ}(.) the characteristic function of the set Λ ⊂ ℝ, one can represent ξ(λ, n) by

ξ(λ, n) = \frac{1}{2}(χ_{(E_{0},∞)}(λ) + χ_{(E_{∞},∞)}(λ))

+ \frac{1}{2} ∑_{j∈J} (χ_{(E_{2j−1},∞)}(λ) + χ_{(E_{2j},∞)}(λ) − 2χ_{(μ_{j}(n),∞)}(λ))

= \frac{1}{2}χ_{(E_{0},E_{∞})}(λ) + \frac{1}{2} ∑_{j∈J} (χ_{(E_{2j−1},μ_{j}(n))}(λ) − χ_{(μ_{j}(n),E_{2j})}(λ))

+ χ_{(E_{∞},∞)}(λ) for a.e. λ ∈ ℝ. (8.9)

In the case of two gaps the situation is depicted below.

[figure: E_{0}; E_{1}; •; μ_{1}(n); E_{2}; E_{3}; •; μ_{2}(n); E_{4}; E_{5}; ξ(λ,n); 1/2; 1; λ]

For later purpose we observe that evaluation of (6.58) shows

(8.10)  b^{(ℓ)}(n) = \frac{1}{2}(E_{0}^{ℓ} + E_{∞}^{ℓ} + ∑_{j∈J}(E_{2j−1}^{ℓ} + E_{2j}^{ℓ} − 2μ_{j}(n)^{ℓ}))

and in the special case ℓ = 1

(8.11)  b(n) = \frac{1}{2}(E_{0} + E_{∞} + ∑_{j∈J}(E_{2j−1} + E_{2j} − 2μ_{j}(n))).

Now we consider the inverse spectral problem associated with reflectionless operators more closely. To reconstruct H we need (e.g.) m̃_{±}(z, n). However, given E_{j}, μ_{j}(n) we only know g(z, n) and thus only a(n)^{2}(m̃_{−}(z, n)+m̃_{+}(z, n)) = −g(z, n)^{−1}. Thus it suffices to split −g(z, n)^{−1} into a(n)^{2}m̃_{+}(z, n) and a(n)^{2}m̃_{−}(z, n). Rather than treating −g(z, n)^{−1} directly, our strategy is to consider the Herglotz representation

(8.12)  − g(z, n)^{−1} = z − b(n) + ∫_{ℝ} \frac{dρ̃_{n}(λ)}{λ − z}

and to seek the corresponding split up of the measure dρ̃_{n} = dρ̃_{−,n} + dρ̃_{+,n}. For if we have dρ̃_{±,n}, we also know (see (2.13))

(8.13)  a(n)^{2} = ∫_{ℝ} dρ̃_{+,n},  a(n − 1)^{2} = ∫_{ℝ} dρ̃_{−,n}

and

(8.14)  m_{±}(z, n) = a(n − {0 \atop 1})^{−2} ∫_{ℝ} \frac{dρ̃_{±,n}(λ)}{λ − z}.

Invoking Theorem 2.29 will then show that H is uniquely determined.

The process of splitting up dρ̃_{n} can be reduced to splitting up the pure point, absolutely continuous, and singularly continuous part, respectively.

We start with the pure point part and introduce

(8.15)  σ_{p}(H_{n}^{∞}) = {μ̃_{j}(n)}_{j∈J̃} = {μ ∈ ℝ|ρ̃_{n}({μ}) > 0}, J̃ ⊆ ℕ.

Note that σ_{p}(H_{n}^{∞}) is known from dρ̃_{n}. Even though the two sets {μ̃_{j}(n)}_{j∈J̃} and {μ_{j}(n)}_{j∈J} are closely related, unfortunately neither {μ̃_{j}(n)}_{j∈J̃} ⊆ {μ_{j}(n)}_{j∈J} nor {μ_{j}(n)}_{j∈J} ⊆ {μ̃_{j}(n)}_{j∈J̃} is true in general. We have observed before that μ_{j} might not be an eigenvalue if it lies at the boundary of its gap. On the other hand, an accumulation point of the μ_{j}’s might be an eigenvalue (see Remark 8.11 (i) below).

Moreover, we introduce the numbers

(8.16)  R̃_{j}(n) = lim_{ε↓0} iεg(μ̃_{j}(n) + iε, n)^{−1} = ρ̃_{n}({μ̃_{j}(n)}) > 0,

and

(8.17) σ̃_{j}(n) = lim_{ε↓0} h(μ̃_{j}(n) + iε, n) = \frac{ρ̃_{+,n}(\{μ̃_{j}(n)\}) − ρ̃_{−,n}(\{μ̃_{j}(n)\})}{ρ̃_{+,n}(\{μ̃_{j}(n)\}) + ρ̃_{−,n}(\{μ̃_{j}(n)\})} ∈ [−1, 1].

Equivalently, we have (use (2.201))

(8.18)  m̃_{±}(z, n) = −\frac{1 ∓ σ̃_{j}(n)}{2} \frac{R̃_{j}(n)a(n)^{−2}}{z − μ̃_{j}(n)} + O(z − μ̃_{j}(n))^{0}.

In particular, σ̃_{j}(n) is either ±1 (depending on whether μ̃_{j} is an eigenvalue of H_{+,n} or H_{−,n}) or in (−1, +1) (if μ̃_{j} is an eigenvalue of both H_{±,n} and hence also of H).

The numbers R̃_{j}(n) can be evaluated using (8.8)

R̃_{j}(n)^{2} = (μ̃_{j}(n) − E_{0})(μ̃_{j}(n) − E_{∞})(μ̃_{j}(n) − E_{2j−1})(μ̃_{j}(n) − E_{2j}) ×

∏_{k∈J\\{j\}} \frac{(μ̃_{j}(n) − E_{2k−1})(μ̃_{j}(n) − E_{2k})}{(μ̃_{j}(n) − μ_{k}(n))^{2}}. (8.19)

If μ̃_{j} = μ_{k} = E_{2k} = E_{2j−1} for some k (resp. μ̃_{j} = μ_{k} = E_{2k−1} = E_{2j}) the vanishing factors (μ̃_{j} − μ_{k})^{2} in the denominator and (μ̃_{j} − E_{2j−1})(μ̃_{j} − E_{2k}) (resp. (μ̃_{j} − E_{2k−1})(μ̃_{j} − E_{2j})) in the numerator have to be omitted.

By now we know how to split up the pure point and the absolutely continuous (which is being taken care of by the reflectionless condition) part of dρ̃_{n}, that is,

(8.20) dρ̃_{n,±,pp}(\{μ̃_{j}(n)\}) = \frac{1 ± σ̃_{j}(n)}{2}dρ̃_{n,pp}(\{μ̃_{j}(n)\}), dρ̃_{n,±,ac} = \frac{1}{2}dρ̃_{n,ac}.

Since we do not want to bother with the singularly continuous part, we will assume σ_{sc}(H_{n}^{∞}) = ∅, that is, ρ̃_{n,sc} = 0, for simplicity. In this case the spectral data E_{j}, j ∈ J ∪ {0, ∞}, plus μ_{j}(n_{0}), j ∈ J, plus σ̃_{j}(n_{0}), j ∈ J̃, for one fixed n_{0} ∈ ℤ are minimal and uniquely determine a(n)^{2}, b(n). Without this assumption we would need to introduce σ(n, .) : σ_{sc}(H_{n}^{∞}) → [−1, 1] such that dρ̃_{n,±,sc}(λ) = \frac{1±σ(n,λ)}{2}dρ̃_{n,sc}(λ). The corresponding modifications are straightforward.

Now let us focus on the actual reconstruction of a(n)^{2}, b(n) from given spectral data as above and present an explicit expression of a(n)^{2}, b(n) in terms of the spectral data.

As a preparation we observe that dρ̃_{n,+} − dρ̃_{n,−} consists only of a pure point part. In fact, due to the reflectionless condition, the absolutely continuous part cancels and the singularly continuous part is absent by assumption. As a consequence a(n)^{2}(m̃_{−}(z, n) − m̃_{+}(z, n)) is meromorphic and we obtain an interesting expression for h(z, n) (cf. (2.201)).

Lemma 8.4. Suppose H is reflectionless and σ_{sc}(H_{n}^{∞}) = ∅. Then we have

(8.21)  h(z, n) = g(z, n)(z − b(n) + ∑_{j∈J̃} \frac{σ̃_{j}(n)R̃_{j}(n)}{z − μ̃_{j}(n)}).

Proof. Since the (signed) measure dρ̃_{n,+} − dρ̃_{n,−} is pure point, the integral in the corresponding Herglotz representation can be easily computed. □

Our idea for reconstructing a(n), b(n) is to express the moments of m_{±}(z, n) in terms of the spectral data. Once this is done, the rest follows from (2.118). Since we already know g(z, n) and h(z, n) in terms of the spectral data, it suffices to relate these quantities to m_{±}(z, n). But this is the content of (2.202) from which we infer

a(n)^{2}m_{+}(z, n) ± a(n − 1)^{2}m_{−}(z, n) = ∓z ± b(n) − \{ \frac{1}{g(z,n)} ; \frac{h(z,n)}{g(z,n)}

(8.22)  = − ∑_{j=0}^{∞} \frac{c_{±,j}(n)}{z^{j+1}},

where the coefficients c_{±,j}(n) are to be determined. From (8.21) we infer

(8.23)  c_{−,ℓ}(n) = ∑_{j∈J̃} σ̃_{j}(n)R̃_{j}(n)μ̃_{j}(n)^{ℓ}, ℓ ∈ ℕ_{0}.

Moreover, c_{+,ℓ}(n) can be determined from (cf. (6.56))

(8.24)  \frac{1}{g(z, n)} = −z exp ( − ∑_{ℓ=1}^{∞} \frac{b^{(ℓ)}(n)}{ℓz^{ℓ}}),

which implies

c_{+,−2}(n) = 1,
(8.25)  c_{+,ℓ−2}(n) = \frac{1}{ℓ} ∑_{j=1}^{ℓ} c_{+,ℓ−j−2}(n)b^{(j)}(n), ℓ ∈ ℕ.

Thus c_{+,ℓ} are expressed in terms of E_{j}, μ_{j}(n) (by (8.10)). Here c_{+,−2}(n) and c_{+,−1}(n) have been introduced for notational convenience only.

In particular, combining the case ℓ = 0 with our previous results we obtain

(8.26)  a(n − {0 \atop 1})^{2} = \frac{b^{(2)}(n) − b(n)^{2}}{4} ± ∑_{j∈J̃} \frac{σ̃_{j}(n)}{2}R̃_{j}(n).

Similarly, for ℓ = 1,

b(n ± 1) = \frac{1}{a(n − {0 \atop 1})^{2}} (\frac{2b^{(3)}(n) − 3b(n)b^{(2)}(n) + b(n)^{3}}{12}

± ∑_{j∈J̃} \frac{σ̃_{j}(n)}{2}R̃_{j}(n)μ̃_{j}(n)). (8.27)

However, these formulas are only the tip of the iceberg. Combining

c_{±,ℓ}(n) = ∫_{ℝ} λ^{ℓ}(a(n)^{2}dρ_{+,n}(λ) ± a(n − 1)^{2}dρ_{−,n}(λ))
(8.28)  = a(n)^{2}m_{+,ℓ}(n) ± a(n − 1)^{2}m_{−,ℓ}(n), ℓ ∈ ℕ_{0},

with (2.118) we obtain the main result of this section.

Theorem 8.5. Let H be a given bounded reflectionless Jacobi operator. Suppose the singular continuous spectrum of H_{n}^{∞} is empty and the spectral data (E_{j}), (μ_{j}), and (σ̃_{j}) corresponding to H (as above) are given for one fixed n ∈ ℤ. Then the sequences a^{2}, b can be expressed explicitly in terms of the spectral data as follows

a(n ± k − {0 \atop 1})^{2} = \frac{C_{±,n}(k + 1)C_{±,n}(k − 1)}{C_{±,n}(k)^{2}},
(8.29)  b(n ± k) = \frac{D_{±,n}(k)}{C_{±,n}(k)} − \frac{D_{±,n}(k − 1)}{C_{±,n}(k − 1)}, k ∈ ℕ,

where C_{±,n}(k) and D_{±,n}(k) are defined as in (2.109) and (2.113) using m_{±,ℓ}(n) = \frac{c_{+,ℓ}(n)±c_{−,ℓ}(n)}{2a(n− {0 \atop 1} )}. The quantities a(n)^{2}, a(n − 1)^{2}, and c_{±,ℓ}(n) have to be expressed in terms of the spectral data using (8.26), (8.25), and (8.23), (8.10), respectively.

In addition, we get the neat Corollary 8.6. Let H be a reflectionless Jacobi operator with spectrum consisting of only one band, that is σ(H) = [E_{0}, E_{∞}]. Then the sequences a(n)^{2}, b(n) are necessarily constant

(8.30)  a(n)^{2} = \frac{(E_{∞} − E_{0})^{2}}{16} , b(n) = \frac{E_{0} + E_{∞}}{2} .

If J is finite, that is, H has only finitely many spectral gaps, then {μ̃_{j}(n)}_{j∈J̃} ⊆ {μ_{j}(n)}_{j∈J} and we can forget about the μ̃’s. This case will be investigated more closely in Section 8.3.

Finally, we turn to general eigenvalues associated with H_{n}^{β}. Associated with each spectral gap ρ_{j} we set

(8.31)  λ_{j}^{β}(n) = sup\{E_{2j−1}\} ∪ \{λ ∈ ρ_{j}|γ^{β}(λ, n) < 0\} ∈ \overline{ρ_{j}}, j ∈ J.

The strict monotonicity of γ^{β}(λ, n) with respect to λ ∈ ρ_{j}, j ∈ J_{0} ∪ {∞}, that is,

(8.32)  \frac{d}{dλ}γ^{β}(λ, n) = (1 + β^{2})⟨δ_{n}^{β}, (H − λ)^{−2}δ_{n}^{β}⟩, λ ∈ ρ_{j},

then yields

(8.33)  \begin{matrix}γ^{β}(λ, n) < 0, & λ ∈ (E_{2j−1}, λ_{j}^{β}(n)), \\ γ^{β}(λ, n) > 0, & λ ∈ (λ_{j}^{β}(n), E_{2j}), \end{matrix} j ∈ J.

Since γ^{β}(λ, n) is positive (resp. negative) for a(n)β < 0 (resp. a(n)β > 0) as |λ| → ∞, there must be an additional zero

(8.34)  λ_{∞}^{β} = \{ \begin{matrix}sup\{E_{∞}\} ∪ \{λ ∈ ρ_{∞}|γ^{β}(λ, n) < 0\}, & a(n)β < 0 \\ sup\{λ ∈ ρ_{0}|γ^{β}(λ, n) < 0\}, & a(n)β > 0 \end{matrix}

for λ ≥ E_{∞} (resp. λ ≤ E_{0}). Summarizing, ξ^{β}(λ, n) is given by

ξ^{β}(λ, n) = \frac{1}{2}χ_{(E_{0},E_{∞})}(λ) + \frac{1}{2} ∑_{j∈J} (χ_{(E_{2j−1},λ_{j}^{β}(n))}(λ) − χ_{(λ_{j}^{β}(n),E_{2j})}(λ))

+ χ_{(E_{∞},λ_{∞}^{β})}(λ), a(n)β < 0, (8.35)

and

ξ^{β}(λ, n) = −\frac{1}{2}χ_{(E_{0},E_{∞})}(λ) + \frac{1}{2} ∑_{j∈J} (χ_{(E_{2j−1},λ_{j}^{β}(n))}(λ) − χ_{(λ_{j}^{β}(n),E_{2j})}(λ))

− χ_{(λ_{∞}^{β},E_{0})}(λ), a(n)β > 0. (8.36)

Thus we have for β ≠ 0, ∞,

(8.37)  γ^{β}(z, n) = −\frac{β}{a(n)} \frac{z − λ_{∞}^{β}(n)}{\sqrt{z − E_{0}}\sqrt{z − E_{∞}}} ∏_{j∈J} \frac{z − λ_{j}^{β}(n)}{\sqrt{z − E_{2j−1}}\sqrt{z − E_{2j}}} ,

and we remark that the numbers λ_{j}^{β}(n) are related to the spectrum of H_{n}^{β} as follows

(8.38)  σ(H_{n}^{β}) = σ_{ess}(H) ∪ {λ_{j}^{β}(n)}_{j∈J∪{∞}}.

Again observe that λ_{j}^{β}(n) is not necessarily an eigenvalue of H_{n}^{β} unless λ_{j}^{β}(n) ∉ σ_{ess}(H).

Evaluation of (6.58) shows

b^{β,(ℓ)}(n) = \frac{−β}{2a(n)}(E_{0}^{ℓ+1} + E_{∞}^{ℓ+1} − 2λ_{∞}^{β}(n)^{ℓ+1}

+ ∑_{j∈J}(E_{2j−1}^{ℓ+1} + E_{2j}^{ℓ+1} − 2λ_{j}^{β}(n)^{ℓ+1})) (8.39)

and in the special case ℓ = 0,

(8.40) a(n) = \frac{1}{2(β + β^{−1})}(E_{0} + E_{∞} − 2λ_{∞}^{β}(n) + ∑_{j∈J}(E_{2j−1} + E_{2j} − 2λ_{j}^{β}(n))).

8.2. Isospectral operators

In this section we will use the results of the previous section to characterize the isospectral class of certain reflectionless Jacobi operators.

Before we can do this, we need a condition which ensures σ_{sc}(H_{n}^{∞}) = ∅.

Hypothesis H. 8.7. Suppose Σ is infinite and the set of all accumulation points of the set of band edges {E_{2j−1}, E_{2j}}_{j∈J} is countable.

We will denote the set of all accumulation points of the set of band edges by

(8.41)  {Ê_{j}}_{j∈Ĵ},  Ĵ ⊆ ℕ.

Let us first note some consequences of this hypothesis.

Lemma 8.8. Condition (H.8.7) holds if and only if σ(H) is a countable union of disjoint closed intervals (which may degenerate to points).

Explicitly, we have

(8.42)  σ(H) = ( ⋃_{j∈J_{0}} Σ_{j}) ∪ ( ⋃_{j∈Ĵ} Σ̂_{j}),

where

(8.43)  Σ_{j} = [E_{2j}, E_{2j}^{(r)}], j ∈ J_{0}, Σ̂_{j} = [Ê_{j}, Ê_{j}^{(r)}], j ∈ Ĵ,

with

(8.44)  x^{(r)} = inf{E_{j}|x < E_{j} or j = ∞} ∈ {E_{j}}_{j∈J∪{∞}} ∪ {Ê_{j}}_{j∈Ĵ}.

Proof. Denote the right hand side of (8.42) by Σ. Since x^{(r)} = inf{λ ∈ ρ(H)|x < λ} we see Σ ⊆ σ(H). To show ℝ\Σ ⊆ ρ(H) pick x ∉ Σ. There are three cases x^{(r)} = E_{2j}, x^{(r)} = E_{2j−1}, or x^{(r)} = Ê_{j}. In the first case we have x ∈ (E_{2j−1}, E_{2j}) ⊂ ρ(H) which is what we want. In the second case we set E = sup{E_{2k}|E_{2k} ≤ E_{2j−1}}. By minimality of x^{(r)} = E_{2j−1} there cannot be an E_{2k} in [x, E_{2j−1}) and we either have E_{2j−1} = E_{2k}, reducing it to the first case, or E ≤ x and hence x ∈ [E, E_{2j−1}] ⊆ Σ contradicting case two. Case three is similar to case two. Thus σ(H) = Σ. The intervals in (8.42) are disjoint unless E_{2j} = Ê_{k} in which case the corresponding intervals are equal. Hence after throwing out the superfluous intervals we obtain a disjoint union.

Conversely, let σ(H) = ∪_{j∈J̃}[Ẽ_{2j}, Ẽ_{2j+1}] be a countable union of disjoint closed intervals. Then we have {E_{j}} ⊆ {Ẽ_{j}}. and the claim follows since {Ẽ_{j}} is closed.

□

Since the spectral measure of −g(z, n)^{−1} is purely absolutely continuous on the interior of each of the intervals Σ_{j}, Σ̂_{j} the singularly continuous part must be zero (since it is supported on a countable set). The considerations of the previous section indicate that we can choose a number μ_{j}(n) for each spectral gap and a number σ̃_{j} for each Dirichlet eigenvalue. Since the set of all Dirichlet eigenvalues is contained in the set {μ_{j}(n)}_{j∈J} ∪ {Ê_{j}}_{j∈J̃} we will specify σ_{j}(n), σ̂_{j}(n) for each μ_{j}(n), Ê_{j}, j ∈ J, Ĵ, respectively. However, the previous section shows that σ_{j}(n) (resp. σ̂_{j}(n)) is not needed if μ_{j}(n) (resp. Ê_{j}) is not in σ_{p}(H_{n}^{∞}), we will get rid of all superfluous σ’s by an equivalence relation later on.

We first introduce the following hypothesis.

Hypothesis H.8.9. Suppose the set {((μ_{j}, σ_{j}))_{j∈J}, (σ̂_{j})_{j∈Ĵ}} satisfies the following requirements. (i). (μ_{j}, σ_{j}) ∈ [E_{2j−1}, E_{2j}] × [−1, 1], j ∈ J, and σ̂_{j} ∈ [−1, 1], j ∈ Ĵ. (ii). If μ_{j} ∈ (E_{2j−1}, E_{2j}) then σ_{j} ∈ {±1}. (iii). If μ_{j} = E_{2j−1} = E_{2k} (resp. μ_{j} = E_{2j} = E_{2k−1}) for some j, k ∈ J then (μ_{j}, σ_{j}) = (μ_{k}, σ_{k}) and σ_{j} ∈ (−1, 1). (iv). If μ_{j} = Ê_{k} for some j ∈ J, k ∈ J̃ we have σ_{j} = σ̂_{k}. (v). For both ±, at least one of the sets {j ∈ J| ± σ_{j} < 1} or {j ∈ Ĵ| ± σ̂_{j} < 1} is infinite.

Given Hypothesis (H.8.9) we define the set

(8.45)  D_{0}(Σ) = {{((μ_{j}, σ_{j}))_{j∈J}, (σ̂_{j})_{j∈Ĵ}}|(H.8.9) holds} .

Furthermore, for each element in D_{0}(Σ) we can define g(z) as in (8.8) and R_{j}, R̂_{j}, j ∈ J, Ĵ as in (8.16) with μ̃_{j}(n) replaced by μ_{j}, Ê_{j}, respectively. We will set

(8.46)  {((μ_{j}, σ_{1,j}))_{j∈J}, (σ̂_{1,j})_{j∈Ĵ}} ≅ {((μ_{j}, σ_{2,j}))_{j∈J}, (σ̂_{2,j})_{j∈Ĵ}}

if σ_{1,j} = σ_{2,j} for all j ∈ {j ∈ J|R_{j} > 0} and if σ̂_{1,j} = σ̂_{2,j} for all j ∈ {j ∈ Ĵ|R̂_{j} > 0}. The set D_{0}(Σ) modulo this equivalence relation will be denoted by D(Σ).

For any set Σ satisfying (H.8.7) the isospectral set of reflectionless Jacobi operators H in ℓ^{2}(ℤ) is denoted by

(8.47) Iso_{R}(Σ) = {reflectionless Jacobi operators H on ℓ^{2}(ℤ)|σ(H) = Σ, a > 0}.

With each H ∈ Iso_{R}(Σ) we can associate an element in D(Σ) by fixing a base point n ∈ ℤ. In fact, defining μ_{j}(n) as in (8.5) and σ_{j}(n), σ̂_{j}(n) as in (8.17) with μ̃_{j}(n) replaced by μ_{j}, Ê_{j}, respectively, we obtain a mapping

(8.48)  I_{n} : Iso_{R}(Σ) → D(Σ) .

Theorem 8.10. Suppose Σ satisfies (H.8.7) and fix n ∈ ℤ. Then the map I_{n} is a bijection.

Proof. It suffices to show that I_{n} is surjective (since Dirichlet data uniquely determine an operator). Let an element of D(Σ) be given. Define g(z, n), dρ̃_{n}, R_{j}(n), R̂_{j}(n) as in (8.8), (8.12), (8.13), respectively. Since dρ̃_{n,pp} is supported on {μ_{j}}_{j∈J} ∪ {Ê_{j}}_{j∈Ĵ} (as noted before), we can split it up according to

dρ̃_{n,±,pp}(\{μ_{j}\}) = \frac{1 ± σ_{j}}{2} dρ̃_{n,pp}(\{μ_{j}\}),
(8.49)  dρ̃_{n,±,pp}(\{Ê_{j}(n)\}) = \frac{1 ± σ̂_{j}}{2} dρ̃_{n,pp}(\{Ê_{j}\}).

Similarly, the absolutely continuous part splits up according to dρ̃_{n,±,ac} = \frac{1}{2}dρ̃_{n,ac}. Due to (H.8.9) this splitting is well-defined and by virtue of (8.14) and Theorem 2.13 this defines an operator H. It remains to show σ(H) = Σ. By Remark 3.5(ii) it suffices to consider all points μ_{j} for which both m̃_{±}(z, n) have a pole (i.e., σ_{j} ∈ (−1, 1) and R_{j} > 0). Near such a point we infer using (8.18)

(8.50)  g(z, n + 1) = −(1 − σ_{j}^{2})\frac{R_{j}}{z − μ_{j}} + O(z − μ_{j})^{0}

showing μ_{j} ∈ σ(H). This proves σ(H) = Σ. □

We conclude this section with a simple example illustrating an explicit construction to the effect that an accumulation point of eigenvalues of H may or may not be an eigenvalue of H.

Remark 8.11. (i). Suppose H is reflectionless and has pure point spectrum only. Pick an accumulation point Ê of eigenvalues and define

(8.51)  R̂ = lim_{ε↓0} iεg(Ê + iε, 0)^{−1}

and

(8.52)  g_{δ}(z, 0) = − (−g(z, 0)^{−1} − (δ − R̂)(z − Ê)^{−1})^{−1} , δ ≥ 0.

Then g_{δ} is a Herglotz function corresponding to a pure point measure. Computing the zeros μ_{δ,j} of g_{δ}(z, 0) and choosing σ_{δ,j}, σ̂_{δ,j} ∈ [−1, 1] according to our requirements yields a corresponding Jacobi operator H_{δ} by Theorem 8.10. Since

(8.53)  R̂_{δ} = lim_{ε↓0} iεg_{δ}(Ẽ + iε, 0)^{−1} = δ,

one obtains the following case distinctions (σ̂ the sigma corresponding to Ê). (1). δ = 0, then Ẽ ∉ σ_{p}(H_{δ,±}).

(2). δ > 0, σ̂ ∈ {±1}, then Ẽ ∈ σ_{p}(H_{δ,σ̂}), Ẽ_{j_{0}} ∉ σ_{p}(H_{δ}).
(3). δ > 0, σ̂ ∈ (−1, 1), then Ẽ ∈ σ_{p}(H_{δ,±}) ∩ σ_{p}(H_{δ}).

(ii). Let us further comment on (H.8.9). If (ii) should not hold, then μ_{j} would be an eigenvalue of the operator obtained from I_{n}^{−1} (by (8.50)). Similarly, if we drop condition (iii), then E_{2j} = μ_{j} might not be an eigenvalue.

8.3. The finite-gap case

In this section we will prove some additional results in the case of reflectionless Jacobi operators with finitely many gaps. This class is closely connected to the Toda hierarchy (to be studied in Chapter 12) and hence deserves special attention. Given (E_{j})_{j=0}^{2r+1} and ((μ_{j}(n_{0}), σ_{j}(n_{0})))_{j=1}^{r} for one fixed n_{0} we will reconstruct the sequences a, b recursively. We will derive all results without using information from the previous two sections.

Our approach will be modeled after the information gained in the previous chapter about periodic operators (in particular Section 7.3). However, note that for periodic sequences (with no closed gaps) (E_{j})_{j=0}^{2r+1} is already too much ((E_{j}^{θ})_{j=1}^{r+1} for some θ plus A would be enough). If H has no eigenvalues, that is E_{2j} < E_{2j+1}, the constructed sequences will turn out to be quasi-periodic and to be expressible in terms of Riemann theta functions. This case will be investigated more closely in Section 9.2.

Let r ∈ ℕ_{0} and

(8.54)  E_{0} ≤ E_{1} < E_{2} ≤ E_{3} < E_{4} ≤ ⋯ < E_{2r−1} ≤ E_{2r} < E_{2r+1}

be given real numbers. Here we have excluded the case E_{2j} = E_{2j+1} since the corresponding factors would cancel from all equations. In addition, we will assume that E_{2j} < E_{2j+1} for at least one j.

Define R_{2r+2}^{1/2}(z) as

(8.55)  R_{2r+2}^{1/2}(z) = − ∏_{j=0}^{2r+1} \sqrt{z − E_{j}},

where \sqrt{.} is defined as in (1.116). It is most natural to consider R_{2r+2}^{1/2}(z) as a function on the associated Riemann surface M which is the same as the Riemann surface of

(8.56)  R̃_{2g+2}^{1/2}(z) = − ∏_{j∈Γ} \sqrt{z − E_{2j}}\sqrt{z − E_{2j+1}},

where Γ = {j|0 ≤ j ≤ r, E_{2j} < E_{2j+1}}, g = |Γ| − 1 ≥ 0. As a set, M is given by all pairs

(8.57)  p = (z, ±R̃_{2g+2}^{1/2}(z)), z ∈ ℂ,

plus two points at infinity, ∞_{±}. We will denote by π the projection on M and by R̃_{2g+2}^{1/2} the evaluation map, that is,

(8.58)  π(p) = z, R̃_{2g+2}^{1/2}(p) = ±R̃_{2g+2}^{1/2}(z), p = (z, ±R̃_{2g+2}^{1/2}(z)).

The points {(E_{2j}, 0), (E_{2j+1}, 0)|j ∈ Γ} ⊆ M are called branch points and g is called the genus of M. In addition, we define

(8.59)  R_{2r+2}^{1/2}(p) = R̃_{2g+2}^{1/2}(p) ∏_{j∉Γ}(π(p) − E_{2j})

and p^{∗} = (z, ±R̃_{2g+2}^{1/2}(z))^{∗} = (z, ∓R̃_{2g+2}^{1/2}(z)), ∞_{±}^{∗} = ∞_{∓}. Clearly, any polynomial P (z) gives rise to a function P (p) = P (π(p)) on M. For additional information on Riemann surfaces we refer to Section 9.1 and Appendix A.

To set the stage, let

(8.60)  (μ̂_{j}(n_{0}))_{j=1}^{r} = (μ_{j}(n_{0}), σ_{j}(n_{0}))_{j=1}^{r}

be a list of pairs satisfying (cf. (H.8.9))

Hypothesis H. 8.12. Suppose μ̂_{j}(n_{0}), 1 ≤ j ≤ r, satisfy (i).

μ_{j}(n_{0}) ∈ (E_{2j−1}, E_{2j}) and σ_{j}(n_{0}) ∈ {±1} or

μ_{j}(n_{0}) ∈ {E_{2j−1}, E_{2j}} and σ_{j}(n_{0}) ∈ (−1, 1). (8.61)

If E_{2j−2} < E_{2j−1} = μ_{j}(n_{0}) (resp. E_{2j} < E_{2j+1} = μ_{j}(n_{0})), then σ_{j}(n_{0}) is superfluous and we will set σ_{j}(n_{0}) = 0 in this case. (ii).

(8.62)  \begin{aligned}E_{2j−2} = E_{2j−1} = μ_{j}(n_{0}) & ⇒ & μ̂_{j−1}(n_{0}) = μ̂_{j}(n_{0}) \\ E_{2j} = E_{2j+1} = μ_{j}(n_{0}) & ⇒ & μ̂_{j+1}(n_{0}) = μ̂_{j}(n_{0}) \end{aligned} ,

for 1 ≤ j ≤ r. In particular, μ_{1}(n_{0}) ≠ E_{0} and μ_{r}(n_{0}) ≠ E_{2r+1}.

In the notation of the previous section we have

(8.63)  D(Σ) = {(μ̂_{j}(n_{0}))_{j=1}^{r}|(H.8.12) holds}

with

(8.64)  Σ = ⋃_{j=0}^{r}[E_{2j}, E_{2j+1}]

and Theorem 8.10 is valid.

We require (H.8.12) (ii) without loss of generality since otherwise the construction below would give μ_{j}(n) = μ_{j}(n_{0}) for all n and the factor z − μ_{j}(n) would cancel with the factor \sqrt{z − E_{2j−2}}\sqrt{z − E_{2j−1}} (resp. \sqrt{z − E_{2j}}\sqrt{z − E_{2j+1}}) from all equations. For convenience we set

(8.65)  R_{2r+2}^{1/2}(μ̂_{j}(n_{0})) = σ_{j}(n_{0})R_{2r+2}^{1/2}(μ_{j}(n_{0})), R_{2r+2}(z) = (R_{2r+2}^{1/2}(z))^{2}.

To begin with, we define the polynomial associated with the Dirichlet eigenvalues

(8.66)  G_{r}(z, n_{0}) = ∏_{j=1}^{r}(z − μ_{j}(n_{0}))

and the residues

(8.67)  R_{j}(n_{0}) = lim_{z→μ_{j}(n_{0})}(z − μ_{j}(n_{0}))\frac{R_{2r+2}^{1/2}(z)}{G_{r}(z, n_{0})}.

If μ_{j}(n_{0}) ≠ μ_{j−1}(n_{0}), μ_{j+1}(n_{0}) we have

(8.68)  R_{j}(n_{0}) = \frac{R_{2r+2}^{1/2}(μ_{j}(n_{0}))}{∏_{k≠j}^{r}(μ_{j}(n_{0}) − μ_{k}(n_{0}))} ≥ 0.

If μ_{j}(n_{0}) = μ_{j−1}(n_{0}) or μ_{j}(n_{0}) = μ_{j+1}(n_{0}) the vanishing factors in the numerator and denominator have to be omitted. For notational convenience we set in addition

(8.69)  R̂_{j}(n_{0}) = \{ \begin{matrix}σ_{j}(n_{0})R_{j}(n_{0}), & μ_{j}(n_{0}) ≠ μ_{j−1}(n_{0}), μ_{j+1}(n_{0}) \\ \frac{σ_{j}(n_{0})}{2}R_{j}(n_{0}), & μ_{j}(n_{0}) = μ_{j−1}(n_{0}) or μ_{j+1}(n_{0}) \end{matrix} .

The factor 1/2 in the case where two Dirichlet eigenvalues coincide will ensure that we do not count the corresponding residue twice.

Using this notation we define a second polynomial

H_{r+1}(z, n_{0}) = ∑_{j=1}^{r} R̂_{j}(n_{0}) ∏_{{k=1 \atop k≠j}}^{r} (z − μ_{k}(n_{0})) + (z − b(n_{0}))G_{r}(z, n_{0})

(8.70)  = G_{r}(z, n_{0})(z − b(n_{0}) + ∑_{j=1}^{r} \frac{R̂_{j}(n_{0})}{z − μ_{j}(n_{0})}),

where

(8.71)  b(n_{0}) = \frac{1}{2} ∑_{j=0}^{2r+1} E_{j} − ∑_{j=1}^{r} μ_{j}(n_{0}).

This definition implies

(8.72)  lim_{z→μ_{j}(n_{0})} \frac{H_{r+1}(z, n_{0})}{R_{2r+2}^{1/2}(z)} = σ_{j}(n_{0}), 1 ≤ j ≤ r.

Our next task is to define μ̂_{j}(n_{0} + 1). This will be done using a procedure motivated by the approach of Section 2.7. The last equation (8.72) shows that we can define a polynomial G_{r}(z, n_{0} + 1) by (compare (2.187))

(8.73)  4a(n_{0})^{2}G_{r}(z, n_{0})G_{r}(z, n_{0} + 1) = H_{r+1}(z, n_{0})^{2} − R_{2r+2}(z).

As we want the highest coefficient of G_{r}(z, n_{0} + 1) to be one, we have introduced a nonzero constant a(n_{0})^{2} ≠ 0.

First of all, note that G_{r}(z, n_{0} + 1) does not vanish identically (H_{r+1}^{2} is nonnegative and R_{2r+2} is not since E_{2j} < E_{2j+1} for at least one j) and that it is a polynomial of degree at most r (since the highest two powers of z on the right cancel due to the definition of b(n_{0})).

We claim that there are at least two zeros of G_{r}(., n_{0})G_{r}(., n_{0}+1) in the interval [E_{2j−1}, E_{2j}]. Since

4a(n_{0})^{2}G_{r}(E_{2j−1}, n_{0})G_{r}(E_{2j−1}, n_{0} + 1) ≥ H_{r+1}(E_{2j−1}, n_{0})^{2} ≥ 0 and
(8.74)  4a(n_{0})^{2}G_{r}(E_{2j}, n_{0})G_{r}(E_{2j}, n_{0} + 1) ≥ H_{r+1}(E_{2j}, n_{0})^{2} ≥ 0,

this is clear if μ_{j}(n_{0}) ∈ (E_{2j−1}, E_{2j}). If μ_{j}(n_{0}) ∈ {E_{2j−1}, E_{2j}}, say μ_{j}(n_{0}) = E_{2j−1}, we have to consider two cases. First, let E_{2j−2} < E_{2j−1}, then R_{2r+2}(.) has a first order zero near E_{2j−1} and hence H_{r+1}(z, n_{0})^{2} − R_{2r+2}(z) < 0 for z ∈ (E_{2j−1}, E_{2j−1} + ε) and some ε > 0 sufficiently small. Implying one zero in (E_{2j−1}, E_{2j}]. Second, let E_{2j−2} = E_{2j−1}, then the same conclusion applies because of (8.72) and σ_{j}(n_{0}) ∈ (−1, 1).

Since G_{r}(z, n_{0} + 1) is of degree at most r we conclude that it must be of the form

(8.75)  G_{r}(z, n_{0} + 1) = ∏_{j=1}^{r}(z − μ_{j}(n_{0} + 1)),

with

(8.76)  μ_{j}(n_{0} + 1) ∈ [E_{2j−1}, E_{2j}].

Observe that E_{2j−2} = E_{2j−1} = μ_{j}(n_{0}) (resp. E_{2j} = E_{2j+1} = μ_{j}(n_{0})) implies E_{2j−1} ≠ μ_{j}(n_{0} + 1) (resp. E_{2j} ≠ μ_{j}(n_{0} + 1)). Moreover, if E_{2j−2} = E_{2j−1} = μ_{j}(n_{0} + 1), then μ_{j}(n_{0} + 1) = μ_{j−1}(n_{0} + 1) and if E_{2j} = E_{2j+1} = μ_{j}(n_{0} + 1), then μ_{j}(n_{0} + 1) = μ_{j+1}(n_{0} + 1) (since the left-hand side of (8.73) has a double zero in this case). In addition, μ_{1}(n_{0} + 1) ≠ E_{0} (resp. μ_{r}(n_{0} + 1) ≠ E_{2r+1}) since otherwise we must have E_{0} = E_{1} implying that the left-hand side of (8.73) has a single zero and the right-hand side a double zero. It remains to define

(8.77)  σ_{j}(n_{0} + 1) = − lim_{z→μ_{j}(n_{0}+1)} \frac{H_{r+1}(z, n_{0})}{R_{2r+2}^{1/2}(z)}, 1 ≤ j ≤ r.

Clearly, if H_{r+1}(μ_{j}(n_{0} + 1), n_{0}) ≠ 0 we have σ_{j}(n_{0} + 1) ∈ {±1} by (8.73). If H_{r+1}(μ_{j}(n_{0} + 1), n_{0}) = 0 we have μ_{j}(n_{0} + 1) ∈ {E_{2j−1}, E_{2j}}, say E_{2j−1}. Then, if E_{2j−2} < E_{2j−1} we get σ_{j}(n_{0} + 1) = 0 and if E_{2j−2} = E_{2j−1} we obtain σ_{j}(n_{0} + 1) ∈ (−1, 1) since (8.73) is negative near E_{2j−1}.

Thus we have constructed the list (μ̂_{j}(n_{0} + 1))_{j=1}^{r} from the list (μ̂_{j}(n_{0}))_{j=1}^{r} such that (μ̂_{j}(n_{0} + 1))_{j=1}^{r} satisfies (H.8.12).

The coefficient a(n_{0})^{2} can be calculated by expanding both sides of (8.73) (using (8.66), (8.70), and (8.55)). Comparing the coefficient of z^{2g} shows

(8.78)  a(n_{0})^{2} = \frac{1}{2} ∑_{j=1}^{r} R̂_{j}(n_{0}) + \frac{b^{(2)}(n_{0}) − b(n_{0})^{2}}{4} > 0,

where

(8.79)  b^{(2)}(n) = \frac{1}{2} ∑_{j=0}^{2r+1} E_{j}^{2} − ∑_{j=1}^{r} μ_{j}(n)^{2}.

That a(n_{0})^{2} is positive follows from a(n_{0})^{2}G_{r}(E_{2r+1}, n_{0})G_{r}(E_{2r+1}, n_{0} + 1) ≥ 0. In addition, we obtain

H_{r+1}(z, n_{0}) = − ∑_{j=1}^{r} R̂_{j}(n_{0} + 1) ∏_{k≠j}^{r} (z − μ_{k}(n_{0} + 1))

+ (z − b(n_{0} + 1))G_{r}(z, n_{0} + 1), (8.80)

implying

(8.81)  a(n_{0})^{2} = −\frac{1}{2} ∑_{j=1}^{r} R̂_{j}(n_{0} + 1) + \frac{b^{(2)}(n_{0} + 1) − b(n_{0} + 1)^{2}}{4} > 0,

and

(8.82)  H_{r+1}(z, n_{0} + 1) = −H_{r+1}(z, n_{0}) + 2(z − b(n_{0} + 1))G_{r}(z, n_{0} + 1).

As (8.70) is symmetric in G_{r}(z, n_{0} + 1) and G_{r}(z, n_{0}) we can even reverse this process using (8.80) as definition for H_{r+1} (which only differs from (8.70) by the sign in front of R_{2r+2}^{1/2}(.)) and defining σ_{j}(n_{0}) with reversed sign, that is, μ̂_{j}(n_{0}) = (μ(n_{0}), lim_{z→μ_{j}(n_{0})} H_{r+1}(z, n_{0})/R_{2r+2}^{1/2}(z)). Thus given (μ̂_{j}(n_{0}))_{j=1}^{r} we can recursively define the set (μ̂_{j}(n))_{j=1}^{r} for all n ∈ ℤ. In summary we have proven the following theorem.

Theorem 8.13. Given the list (μ̂_{j}(n_{0}))_{j=1}^{r} we can define the list (μ̂_{j}(n))_{j=1}^{r} for all n ∈ ℤ, which again fulfills (H.8.12). In addition, we get two sequences a(n), b(n) defined as

a(n)^{2} = \frac{1}{2} ∑_{j=1}^{r} R̂_{j}(n) + \frac{1}{8} ∑_{j=0}^{2r+1} E_{j}^{2} − \frac{1}{4} ∑_{j=1}^{r} μ_{j}(n)^{2} − \frac{b(n)^{2}}{4} > 0,

(8.83)  b(n) = \frac{1}{2} ∑_{j=0}^{2r+1} E_{j} − ∑_{j=1}^{r} μ_{j}(n),

(cf (8.69)). The sign of a(n) can be chosen freely. The sequences a(n), b(n) are real and bounded.

Proof. It remains to show boundedness of a(n) and b(n). This is clear for b(n) by virtue of (8.83). Combining (8.78) and (8.81)

(8.84)  a(n)^{2} + a(n − 1)^{2} = \frac{b^{(2)}(n) − b(n)^{2}}{2}

we see that a(n) is bounded as well. □

Remark 8.14. (i). The question when a(n)^{2}, b(n) are periodic will be answered in Theorem 9.6. (ii). Suppose μ_{j−1} ≠ μ_{j} ≠ μ_{j+1} for a moment. Using the residue theorem we may write

(8.85)  μ_{j}(n + 1) + μ_{j}(n) = \frac{1}{2πi} ∫_{Γ_{j}} z \frac{2H_{r+1}(z, n)H′_{r+1}(z, n) − R′_{2r+2}(z)}{H_{r+1}(z, n)^{2} − R_{2r+2}(z)}dz,

(here the prime denotes the derivative with respect to z) where Γ_{j} is a closed path in the complex plane encircling (once) the interval [E_{2j−1}, E_{2j}] counter-clock-wise. Further we may also write

(8.86)  μ_{j}(n + 1) − μ_{j}(n) = \frac{1}{2πi} ∫_{γ_{j}} zd ln \frac{G_{r}(z, n + 1)}{G_{r}(z, n)}.

Let me point out another interesting observation. The procedure for calculating μ_{j}(n) from μ_{j}(n_{0}) with n > n_{0} differs from that with n < n_{0} only by two signs! One in the definition of H_{r+1} and one in the definition of μ̂_{j}. These cancel in each intermediate step but not in the first and not in the last. This means calculating μ̂_{j}(n_{0} + k) from μ̂_{j}(n_{0}) yields the same result as calculating μ̂_{j}(n_{0} − k) from μ̂_{j}(n_{0})^{∗} after switching signs in the last result. Here μ̂_{j}(n_{0})^{∗} = (μ_{j}(n_{0}), σ_{j}(n_{0}))^{∗} = (μ_{j}(n_{0}), −σ_{j}(n_{0})).

Lemma 8.15. Denote the sequences calculated from {μ̂_{j}(n_{0})^{∗}}_{j=1}^{r} by a_{R}, b_{R}. Then we have a_{R}(n_{0} + k) = a(n_{0} − k − 1), b_{R}(n_{0} + k) = b(n_{0} − k) (cf. Lemma 1.7).

In addition, if all μ_{j}(n_{0}) coincide with band edges, that is if {μ_{j}(n_{0})}_{j=1}^{r} ⊂ {E_{2j−1}, E_{2j}}_{j=1}^{r}, then we have μ̂_{j}(n_{0} + k) = μ̂_{j}(n_{0} − k)^{∗}, 1 ≤ j ≤ r, and hence a(n_{0} + k) = a(n_{0} − k − 1), b(n_{0} + k) = b(n_{0} − k).

Proof. Our observation says in formulas that μ̂_{R,j}(n_{0} + k) = μ̂_{j}(n_{0} − k)^{∗}. From this the result for b_{R} is clear from (8.83). For the corresponding result for a_{R} use (8.78) and (8.81). □

Our next task is to investigate the spectrum of the Jacobi operator H associated with the sequences a, b. To do this we need solutions of the Jacobi and Riccati equations first.

Using our polynomials we define two meromorphic functions on M,

(8.87)  ϕ(p, n) = \frac{H_{r+1}(p, n) + R_{2r+2}^{1/2}(p)}{2a(n)G_{r}(p, n)} = \frac{2a(n)G_{r}(p, n + 1)}{H_{r+1}(p, n) − R_{2r+2}^{1/2}(p)}

and, the Baker-Akhiezer function,

(8.88)  ψ(p, n, n_{0}) = \begin{matrix}n−1 \\ ∏ & ∗ \\ j=n_{0} \end{matrix} ϕ(p, j).

Notice

(8.89)  ϕ(p, n)^{2} = \frac{G_{r}(p, n + 1)}{G_{r}(p, n)} \frac{H_{r+1}(p, n) + R_{2r+2}^{1/2}(p)}{H_{r+1}(p, n) − R_{2r+2}^{1/2}(p)}

and hence

(8.90)  ψ(p, n, n_{0})^{2} = \frac{G_{r}(p, n)}{G_{r}(p, n_{0})} \begin{matrix}n−1 \\ ∏ & ∗ \\ j=n_{0} \end{matrix} \frac{H_{r+1}(p, j) + R_{2r+2}^{1/2}(p)}{H_{r+1}(p, j) − R_{2r+2}^{1/2}(p)}.

Theorem 8.16. The function ϕ(p, n) fulfills

(8.91)  a(n)ϕ(p, n) + \frac{a(n − 1)}{ϕ(p, n − 1)} = π(p) − b(n),

and hence ψ(p, n, n_{0}) satisfies

a(n)ψ(p, n + 1, n_{0}) + a(n − 1)ψ(p, n − 1, n_{0}) + b(n)ψ(p, n, n_{0})

= π(p)ψ(p, n, n_{0}). (8.92)

Proof. A straightforward calculation (using (8.82)) yields

a(n)ϕ(p, n) + \frac{a(n − 1)}{ϕ(p, n − 1)} = \frac{H_{r+1}(p, n) + R_{2r+2}^{1/2}(p)}{2G_{r}(p, n)}

+ \frac{H_{r+1}(p, n − 1) − R_{2r+2}^{1/2}(p)}{2G_{r}(p, n)} = π(p) − b(n), (8.93)

and the rest follows from

(8.94)  ϕ(p, n) = \frac{ψ(p, n + 1, n_{0})}{ψ(p, n, n_{0})}.
□

With ψ_{±}, ϕ_{±} we mean the chart expression of ψ, ϕ in the charts (Π_{±}, π) (cf. Appendix A.7), that is,

(8.95)  ϕ_{±}(z, n) = \frac{H_{r+1}(z, n) ± R_{2r+2}^{1/2}(z)}{2a(n)G_{r}(z, n)}, etc..

Now Lemma 6.7 together with ϕ_{±}(z, n) = (a(n)/z)^{±1}(1 + O(z^{−1})) implies Theorem 8.17. The functions ψ_{±}(z, n, n_{0}) are square summable near ±∞ and the Green function of H is given by (recall the notation from (1.100))

g(z, n) = \frac{G_{r}(z, n)}{R_{2r+2}^{1/2}(z)},  z ∈ ρ(H),

(8.96)  h(z, n) = \frac{H_{r+1}(z, n)}{R_{2r+2}^{1/2}(z)},  z ∈ ρ(H).

Hence the spectrum of H reads

σ(H) = ⋃_{j=0}^{r}[E_{2j}, E_{2j+1}],

σ_{ac}(H) = ⋃_{j∈Γ}[E_{j}, E_{j+1}], σ_{sc}(H) = ∅, σ_{p}(H) = {E_{j}|j ∉ Γ}. (8.97)

In addition, if μ_{j}(n) ≠ μ_{j}(n_{0}), we have ψ_{σ_{j}(n)}(μ_{j}(n), n, n_{0}) = 0 if σ_{j}(n)^{2} = 1 and ψ_{−}(μ_{j}(n), n, n_{0}) = ψ_{+}(μ_{j}(n), n, n_{0}) = 0 if σ_{j}(n)^{2} < 1.

The facts concerning the spectrum of H can be read off from the following lemma:

Lemma 8.18. The Weyl M-matrix is given by

(8.98)  M(z) = \frac{1}{2a(0)R_{2r+2}^{1/2}(z)} ( \begin{matrix}2a(0)G_{r}(z, 0) & H_{r+1}(z, 0) \\ H_{r+1}(z, 0) & 2a(0)G_{r}(z, 1) \end{matrix} ) ,

and the associated matrix valued measure dρ by

dρ_{0,0}(λ) = \frac{G_{r}(λ, 0)}{πiR_{2r+2}^{1/2}(λ)}χ_{σ_{ac}(H)}(λ)dλ + ∑_{E_{j}∈σ_{p}(H)} P (E_{j}, 0, 0)dΘ(λ − E_{j}),

dρ_{0,1}(λ) = \frac{H_{r+1}(λ, 0)}{2πia(0)R_{2r+2}^{1/2}(λ)}χ_{σ_{ac}(H)}(λ)dλ + ∑_{E_{j}∈σ_{p}(H)} P (E_{j}, 0, 1)dΘ(λ − E_{j}),

dρ_{1,1}(λ) = \frac{G_{r}(λ, 1)}{πiR_{2r+2}^{1/2}(λ)}χ_{σ_{ac}(H)}(λ)dλ + ∑_{E_{j}∈σ_{p}(H)} P (E_{j}, 1, 1)dΘ(λ − E_{j}),

(8.99)

where −P (E_{j}, n, m) is the residue of G(z, n, m) at z = E_{j} (cf. (2.34)). Here Θ(λ) = 0 for λ ≤ 0 and Θ(λ) = 1 for λ > 0.

Proof. The M-matrix easily follows from (8.96) and dρ_{0,0}, dρ_{0,1}, dρ_{1,1} are immediate from (B.31) and (B.41) (cf. Remark 3.5(i)). □

We remark that σ_{j}(n_{0}) in the case of μ_{j}(n) = E_{2j−2} = E_{2j−1} can be alternatively interpreted in terms of Weyl m-functions

(8.100)  \frac{a(n − 1)^{2}}{a(n)^{2}} lim_{z→μ_{j}(n)} \frac{m_{+,n}(z)}{m_{−,n}(z)} = lim_{z→μ_{j}(n)} \frac{ϕ_{+}(z, n)}{ϕ_{−}(z, n)} = \frac{1 + σ_{j}(n)}{1 − σ_{j}(n)} > 0.

Last, we collect some useful relations. First note

a(n)^{2}G_{r}(z, n + 1) − a(n − 1)^{2}G_{r}(z, n − 1) + (z − b(n))^{2}G_{r}(z, n)

= (z − b(n))H_{r+1}(z, n). (8.101)

Further, from (8.87) we get

(8.102)  ϕ(p, n)ϕ(p^{∗}, n) = \frac{G_{r}(p, n + 1)}{G_{r}(p, n)},

and hence

(8.103)  ψ(p, n, n_{0})ψ(p^{∗}, n, n_{0}) = \frac{G_{r}(p, n)}{G_{r}(p, n_{0})}.

Moreover,

ϕ(p, n) − ϕ(p^{∗}, n) = \frac{R_{2r+2}^{1/2}(p)}{a(n)G_{r}(p, n)},
(8.104)  ϕ(p, n) + ϕ(p^{∗}, n) = \frac{H_{r+1}(p, n)}{a(n)G_{r}(p, n)}.

8.4. Further spectral interpretation

In this section we will try to relate (μ̂_{j}(n))_{j=1}^{r} to the spectra of the operators H_{n}^{∞} associated with a, b. In fact we will even be a bit more general.

We first define a polynomial K_{r+1}^{β}(z, n) for β ∈ ℝ by

ϕ(p, n) + β = \frac{H_{r+1}(p, n) + 2a(n)βG_{r}(p, n) + R_{2r+2}^{1/2}(p)}{2a(n)G_{r}(p, n)}

(8.105)  = \frac{2a(n)K_{r+1}^{β}(p, n)}{H_{r+1}(p, n) + 2a(n)βG_{r}(p, n) − R_{2r+2}^{1/2}(p)}.

Thus we have

K_{r+1}^{β}(z, n) = G_{r}(z, n + 1) + \frac{β}{a(n)}H_{r+1}(z, n) + β^{2}G_{r}(z, n)

(8.106)  = \frac{β}{a(n)} ∏_{j=1}^{r+1}(z − λ_{j}^{β}(n)), β ∈ ℝ\\{0\},

where (λ_{j}^{β}(n))r̃ ; j=1, with r̃ = r if β = 0 and r̃ = r + 1 otherwise, are the zeros of K_{r+1}^{β}(z, n). Since we have K_{r+1}^{β}(z, n) → G_{r}(z, n + 1) as β → 0 we conclude λ_{j}^{β}(n) = μ_{j}(n + 1) + O(β) for 1 ≤ j ≤ r and λ_{r+1}^{β}(n) = a(n)β^{−1} + b(n + 1) + O(β). Also note that the analogue of (8.96) reads (z ∈ ρ(H))

\frac{K_{r+1}^{β}(z, n)}{R_{2r+2}^{1/2}(z)} = \frac{(ψ_{+}(z, n + 1) + βψ_{+}(z, n))(ψ_{−}(z, n + 1) + βψ_{−}(z, n))}{W (ψ_{−}(z), ψ_{+}(z))}
(8.107)  = γ^{β}(z, n),

where we have omitted the dependence of ψ_{±} on n_{0} (which cancels).

As in the last section we can use

(8.108) (H_{r+1}(z, n) + 2a(n)βG_{r}(z, n))^{2} − R_{2r+2}(z) = 4a(n)^{2}K_{r+1}^{β}(z, n)G_{r}(z, n)

to show

(8.109)  λ_{j}^{β}(n) ∈ [E_{2j−1}, E_{2j}],  1 ≤ j ≤ r,

and (β ∈ ℝ\{0})

(8.110) λ_{r+1}^{β}(n) ≥ E_{2r+1} for a(n)β < 0,  λ_{r+1}^{β}(n) ≤ E_{0} for a(n)β > 0,

(whether λ_{r+1}^{β}(n) ≥ E_{2r+1} or λ_{r+1}^{β}(n) ≤ E_{0} follows from the behavior as β → 0 together with a continuity argument). Evaluating (8.108) at z = λ_{j}^{β}(n) yields

(8.111)  R_{2r+2}(λ_{j}^{β}(n)) = (H_{r+1}(λ_{j}^{β}(n), n) + 2a(n)βG_{r}(λ_{j}^{β}(n), n))^{2},

and hence we define

(8.112)  σ_{j}^{β}(n) = − lim_{z→λ_{j}^{β}(n)} \frac{H_{r+1}(z, n) + 2a(n)βG_{r}(z, n)}{R_{2r+2}^{1/2}(z)}.

This implies (use (8.106))

R_{2r+2}^{1/2}(λ̂_{j}^{β}(n)) = −H_{r+1}(λ_{j}^{β}(n), n) − 2a(n)βG_{r}(λ_{j}^{β}(n), n)
(8.113)  = H_{r+1}(λ_{j}^{β}(n), n) + 2a(n)β^{−1}G_{r}(λ_{j}^{β}(n), n + 1).

For the sake of completeness we also state the trace formulas in the finite-gap case. Lemma 8.19. The zeros λ_{j}^{β}(n) of K_{r+1}^{β}(z, n) fulfill for β ∈ ℝ (cf. (6.64))

(8.114)  b^{β,(ℓ)}(n) = \frac{β}{2a(n)}( ∑_{j=0}^{2r+1} E_{j}^{ℓ+1} − 2 ∑_{j=1}^{r+1} λ_{j}^{β}(n)^{ℓ+1}).

Especially for ℓ = 0

(8.115)  a(n) = \frac{1}{2(β + β^{−1})}( ∑_{j=0}^{2r+1} E_{j} − 2 ∑_{j=1}^{r+1} λ_{j}^{β}(n)).

We further compute

(8.116)  (ϕ(p, n) + β)(ϕ(n, p^{∗}) + β) = \frac{K_{r+1}^{β}(p, n)}{G_{r}(p, n)},

and

a(n + 1)^{2}K_{r+1}^{β}(z, n + 1) = a(n)^{2}G_{r}(z, n) + b_{β}(n + 1)H_{r+1}(z, n)

+ b_{β}(n + 1)^{2}G_{r}(z, n + 1), (8.117) where b_{β}(n) = b(n) − βa(n).

The next theorem characterizes the spectrum of H_{n}^{β}. Theorem 8.20. Suppose β ∈ ℝ ∪ {∞}. Then

σ(H_{n}^{β}) = σ(H) ∪ {λ_{j}^{β}(n)}r̃ ; j=1,
(8.118)  σ_{ac}(H_{n}^{β}) = σ_{ac}(H), σ_{sc}(H_{n}^{β}) = ∅,

where r̃ = r for β = 0, ∞ and r̃ = r + 1 otherwise. Whether λ_{j}^{β}(n) is an eigenvalue of H_{+,n}^{β} or H_{−,n}^{β} depends on whether σ_{j}^{β}(n) = +1 or σ_{j}^{β}(n) = −1, respectively. If σ_{j}^{β}(n) ∈ (−1, 1), then λ_{j}^{β}(n_{0}) is an eigenvalue of both H_{±,n_{0}}^{β} if λ_{j}^{β}(n) ∈ σ_{p}(H) and no eigenvalue for both otherwise. Moreover, for all β_{1,2} ∈ ℝ ∪ {∞} we have λ̂_{j}^{β_{1}}(n) ≠ λ̂_{j}^{β_{2}}(n) provided β_{1} ≠ β_{2}.

Proof. The claims concerning the spectra are immediate by looking at Weyl mfunctions and applying Lemma B.7 and the following considerations. It remains to show λ̂_{j}^{β_{1}}(n) ≠ λ̂_{j}^{β_{2}}(n) provided β_{1} ≠ β_{2}. Suppose the contrary. Then we have β_{1} = ϕ(λ̂_{j}^{β_{2}}(n), n) = ϕ(λ̂_{j}^{β_{1}}(n), n) = β_{2} a contradiction. Nevertheless we remark that the case λ_{j}^{β_{1}}(n) = λ_{j}^{β_{2}}(n) (but σ_{j}^{β_{1}}(n) ≠ σ_{j}^{β_{2}}(n)) can occur. □

Next, we will investigate how λ_{j}^{β}(n) varies with β. Therefore we use (8.106) to calculate

(8.119)  \frac{∂}{∂β} K_{r+1}^{β}(z, n)|_{z=λ_{j}^{β}(n)}

in two ways

\frac{−β}{a(n)} ∏_{k≠j}^{r+1} (λ_{j}^{β}(n) − λ_{k}^{β}(n))\frac{∂}{∂β} λ_{j}^{β}(n)

= \frac{1}{a(n)}(H_{r+1}(λ_{j}^{β}(n), n) + 2a(n)βG_{r}(λ_{j}^{β}(n), n)). (8.120)

This yields

Lemma 8.21. The variation of the zeros λ_{j}^{β}(n) of K_{r+1}^{β}(z, n) with respect to β ∈ ℝ\{0} is described by

(8.121)  \frac{∂}{∂β} λ_{j}^{β}(n) = \frac{R_{2r+2}^{1/2}(λ̂_{j}^{β}(n))}{β ∏_{k≠j}^{r+1}(λ_{j}^{β}(n) − λ_{k}^{β}(n))}, 1 ≤ j ≤ r + 1.

If λ_{j}^{β}(n) = λ_{j+1}^{β}(n) the vanishing factors in the numerator and denominator have to be omitted.

Finally, we note that the connection with (8.23) is given by

(8.122)  c_{−,ℓ}(n) = ∑_{j=1}^{r} μ_{j}(n)^{ℓ}R̂_{j}(n), ℓ ∈ ℕ.

Quasi-periodic Jacobi operators and Riemann theta functions

In the previous chapter we have seen how Riemann surfaces arise naturally in the investigation of reflectionless finite-gap sequences. However, there we have only used them as a convenient abbreviation. In this chapter we want to use Riemann surfaces as an independent tool which will help us to gain further insight. For simplicity we will only consider the case of nonsingular surfaces, that is, r = g in (8.59) (i.e., R̃_{2g+2}(z) = R_{2r+2}(z)). The corresponding sequences will turn out to be expressible in terms of Riemann theta functions and are hence quasi-periodic.

The present chapter is entirely independent of Chapter 8 in the sense that one could take Theorem 9.2 as a definition for the Baker-Akhiezer function. But without our knowledge from Chapter 7 and 8 there is no reason why this object should be of any interest. Nevertheless we will show how all major results from Chapter 8 can be obtained using only methods from this chapter.

9.1. Riemann surfaces

The purpose of this section is mainly to clarify notation. For further information and references we refer the reader to Appendix A.

We consider the Riemann surface M associated with the following function

(9.1)  R_{2g+2}^{1/2}(z),  R_{2g+2}(z) = ∏_{j=0}^{2g+1}(z − E_{j}), E_{0} < E_{1} < ⋯ < E_{2g+1},

g ∈ ℕ. M is a compact, hyperelliptic Riemann surface of genus g. Again we will choose R_{2g+2}^{1/2}(z) as the fixed branch

(9.2)  R_{2g+2}^{1/2}(z) = − ∏_{j=0}^{2g+1} \sqrt{z − E_{j}},

where \sqrt{.} is defied as in (1.116).

A point of M is denoted by

(9.3)  p = (z, ±R_{2g+2}^{1/2}(z)), z ∈ ℂ, or p = ∞_{±}

and M comes with three holomorphic maps on it. The involution map

(9.4)  \begin{aligned}∗ : & M & → & M \\ & (z, ±R_{2g+2}^{1/2}(z)) & ↦ & (z, ±R_{2g+2}^{1/2}(z))^{∗} = (z, ∓R_{2g+2}^{1/2}(z)) \\ & ∞_{±} & ↦ & ∞_{∓} \end{aligned} ,

the projection

(9.5)  \begin{aligned}π : & M & → & ℂ ∪ \{∞\} \\ & (z, ±R_{2g+2}^{1/2}(z)) & ↦ & z \\ & ∞_{±} & ↦ & ∞ \end{aligned} ,

and the evaluation map

(9.6)  \begin{aligned}R_{2g+2}^{1/2} : & M & → & ℂ ∪ \{∞\} \\ & (z, ±R_{2g+2}^{1/2}(z)) & ↦ & ±R_{2g+2}^{1/2}(z) \\ & ∞_{±} & ↦ & ∞ \end{aligned} .

The sets Π_{±} = {(z, ±R_{2g+2}^{1/2}(z))|z ∈ ℂ\σ(H)} are called upper, lower sheet, respectively.

Clearly, the eigenvalues λ̂_{j}^{β}(n) can be viewed as points on our Riemann surface

λ̂_{j}^{β}(n) = (λ_{j}^{β}(n), σ_{j}^{β}(n)R_{2g+2}^{1/2}(λ_{j}^{β}(n))) ∈ M,

R_{2g+2}^{1/2}(λ̂_{j}^{β}(n)) = σ_{j}^{β}(n)R_{2g+2}^{1/2}(λ_{j}^{β}(n)). (9.7)

The divisor associated with the eigenvalues λ̂_{j}^{β}(n) is defined by

(9.8)  D_{\underline{λ̂}^{β}(n)}(λ̂_{j}^{β}(n)) = 1,  1 ≤ j ≤ g̃,

and 0 otherwise, where g̃ = g + 1 if β ∈ ℝ\{0} and g̃ = g if β ∈ {0, ∞}. We have

(9.9)  deg(D_{\underline{λ̂}^{β}(n)}) = g̃

and since σ_{p}(H_{n}^{β}) is simple, D_{\underline{λ̂}^{β}(n)} is nonspecial,

(9.10)  i(D_{\underline{λ̂}^{β}(n)}) = 0,

by Lemma A.20. We set D\underline{λ̂}_{β} = D_{\underline{λ̂}^{β}(0)}.

We proceed with considering the function ϕ(p, n)+β with β ∈ ℝ\{0}. Equation (8.105) shows that there is a simple pole at ∞_{−} and that all further poles are also simple and can only be at μ̂_{j}(n), 1 ≤ j ≤ g. A closer investigation using (8.105), (8.113) shows that the divisor of ϕ(p, n) + β is given by

(9.11)  (ϕ(p, n) + β) = D_{\underline{λ̂}^{β}(n)} − D_{\underline{μ̂}(n)} − D_{∞_{−}}, β ∈ ℝ\\{0\}.

If we want to include the case β = 0 we have to add an ’additional’ eigenvalue at ∞_{+} (the eigenvalue λ_{g+1}^{β}(n) disappears at infinity)

(9.12)  (ϕ(p, n)) = D_{\underline{μ̂}(n+1)} − D_{\underline{μ̂}(n)} − D_{∞_{−}} + D_{∞_{+}}.

In principal, some of the poles and zeros might still cancel, but Theorem 8.20 shows that this does not happen.

By taking Abel’s map \underline{α}_{p_{0}} on both sides we get (β ∈ ℝ\\{0\})

\underline{α}_{p_{0}}(D_{\underline{λ̂}^{β}(n)}) = \underline{α}_{p_{0}}(D_{\underline{μ̂}(n)} + D_{∞_{−}}),
(9.13)  \underline{α}_{p_{0}}(D_{\underline{μ̂}(n+1)} + D_{∞_{+}}) = \underline{α}_{p_{0}}(D_{\underline{μ̂}(n)} + D_{∞_{−}}).

This shows that \underline{α}_{p_{0}}(D_{\underline{λ̂}^{β}(n)}) is independent of β ∈ ℝ\\{0\} which can also be shown using (8.121) (compare (13.52)).

Combining the last two equations yields for arbitrary β

(9.14)  \underline{α}_{p_{0}}(D_{\underline{λ̂}^{β}(n+1)} + D_{∞_{+}}) = \underline{α}_{p_{0}}(D_{\underline{λ̂}^{β}(n)} + D_{∞_{−}}).

A result we will soon investigate further.

Next, we are interested in the poles and zeros of the Baker-Akhiezer function. Because of (8.88) the divisor of ψ(p, n) = ψ(p, n, 0) reads

(9.15)  (ψ(p, n)) = D_{\underline{μ̂}(n)} − D\underline{μ̂} + n(D_{∞_{+}} − D_{∞_{−}})

and by taking Abel’s map on both sides we obtain

(9.16)  \underline{α}_{p_{0}}(D_{\underline{μ̂}(n)}) = \underline{α}_{p_{0}}(D\underline{μ̂}) − n\underline{A}_{∞_{−}}(∞_{+}).

9.2. Solutions in terms of theta functions

We first fix the representatives of a canonical homology basis {a_{j}, b_{j}}_{j=1}^{g} on M. We require a_{j} to surround the points E_{2j−1}, E_{2j} (changing sheets twice) and b_{j} to surround E_{0}, E_{2j−1} on the upper sheet. The corresponding canonical basis {ζ_{j}}_{j=1}^{g} for H^{1}(M) (the space of holomorphic differentials) has the following representation

(9.17)  \underline{ζ} = ∑_{j=1}^{g} \underline{c}(j)\frac{π^{j−1} dπ}{R_{2g+2}^{1/2}}

and we have

(9.18)  ∫_{a_{j}} ζ_{k} = δ_{j,k},  ∫_{b_{j}} ζ_{k} = τ_{j,k}.

To every compact Riemann surface of genus g ∈ ℕ belongs a Riemann theta function (cf. Sections A.5 and A.6)

(9.19)  θ(\underline{z}) = ∑_{\underline{m}∈ℤ^{g}} exp 2πi (⟨\underline{m}, \underline{z}⟩ + \frac{⟨\underline{m}, \underline{τ} \underline{m}⟩}{2}) ,  \underline{z} ∈ ℂ^{g},

with ⟨ˆ\underline{z}, \underline{z}⟩ = ∑_{j=1}^{g} ẑ_{j}z_{j} and \underline{τ} the matrix of b-periods of the differentials (9.17). It has the following fundamental properties

θ(−\underline{z}) = θ(\underline{z}),
(9.20) θ(\underline{z} + \underline{m} + \underline{τ} \underline{n}) = exp 2πi (−⟨\underline{n}, \underline{z}⟩ − \frac{⟨\underline{n}, \underline{τ} \underline{n}⟩}{2}) θ(\underline{z}),  \underline{n}, \underline{m} ∈ ℤ^{g}.

We introduce

(9.21)  \underline{z}(p, n) = \underline{Â}_{p_{0}}(p) − \underline{α̂}_{p_{0}}(D_{\underline{μ̂}(n)}) − \underline{Ξ̂}_{p_{0}} ∈ ℂ^{g}, \underline{z}(n) = \underline{z}(∞_{+}, n),

where \underline{Ξ}_{p_{0}} is the vector of Riemann constants

(9.22)  Ξ̂_{p_{0},j} = \frac{1 − ∑_{k=1}^{g} τ_{j,k}}{2} , p_{0} = (E_{0}, 0),

and \underline{A}_{p_{0}} (\underline{α}_{p_{0}}) is Abel’s map (for divisors). The hat indicates that we regard it as a (single-valued) map from M̂ (the fundamental polygon associated with M) to ℂ^{g}. \underline{z}(p, n) is independent of the base point p_{0} and we note that \underline{z}(∞_{+}, n) = \underline{z}(∞_{−}, n+1) mod ℤ^{g}. We recall that the function θ(\underline{z}(p, n)) has zeros precisely at the points μ̂_{j}(n), 1 ≤ j ≤ g, that is,

(9.23)  θ(\underline{z}(p, n)) = 0 ⇔ p ∈ \{μ̂_{j}(n)\}_{j=1}^{g}.

Since divisors of the type D\underline{μ̂} with μ_{j} ∈ [E_{2j−1}, E_{2j}] are of special importance to us, we want to tell whether a given divisor D ∈ M_{g} (M_{g}, the g-th symmetric power of M) is of the above type or not by looking at \underline{α}_{p_{0}}(D).

Lemma 9.1. Introduce two manifolds

M^{D} = ⊗_{j=1}^{g}π^{−1}([E_{2j−1}, E_{2j}]) ≃ ⊗_{1}^{g}S^{1},
(9.24)  J^{D} = \{[\underline{A}] ∈ J(M)| [i Im(\underline{A} + \underline{Ξ̂}_{p_{0}})] = [0]\} ≃ ⊗_{1}^{g}S^{1},

where [.] denotes the equivalence classes in the Jacobian variety J(M). Then the mapping

(9.25)  \begin{aligned}\underline{α}_{p_{0}} : & M^{D} & → & J^{D} \\ & \underline{μ̂} & ↦ & \underline{α}_{p_{0}}(D\underline{μ̂}) \end{aligned}

is an isomorphism.

Proof. Let μ_{j} ∈ [E_{2j−1}, E_{2j}] and observe

(9.26)  i Im(Â_{p_{0},k}(μ̂_{j})) = \frac{τ_{j,k}}{2} .

This can be shown as in Section A.7 by taking all integrals as limits along the real axis (we may intersect some b-cycles though we are required to stay in M̂, but we only add a-cycles which are real). Thus \underline{α}_{p_{0}}(M^{D}) ⊆ J^{D}. Since divisors in M^{D} are nonspecial by Lemma A.20 we see that \underline{α}_{p_{0}} is an immersion by (A.86) and injective by Lemma A.9. Moreover, \underline{α}_{p_{0}}(M^{D}) is open (since holomorphic maps are open) and compact (since M^{D} is). So \underline{α}_{p_{0}}(M^{D}) is both open and closed, and hence must be the whole of J^{D} by connectedness. □

As a consequence we note that when n changes continuously to n + 1 in (9.16), μ_{j}(n) changes continuously to μ_{j}(n + 1), always remaining in its gap. Thus we obtain

(9.27)  θ(\underline{z}(p, m))θ(\underline{z}(p, n)) > 0

for π(p) ∈ (−∞, E_{0}) ∪ (E_{2g+1}, ∞) ∪ {∞} and m, n ∈ ℤ. Moreover, since the set of all possible Dirichlet eigenvalues M^{D} is compact, we infer that there are positive constants C_{1} ≤ C_{2} such that

(9.28)  C_{1} ≤ |θ(\underline{z}(n))| ≤ C_{2}.

Finally, we recall ω_{∞_{+},∞_{−}}, the normalized abelian differential of the third kind with simple poles at ∞_{±} and residues ±1, respectively (cf. (A.20)). We can make the following ansatz

(9.29)  ω_{∞_{+},∞_{−}} = \frac{∏_{j=1}^{g}(π − λ_{j})}{R_{2g+2}^{1/2}}dπ,

where the constants λ_{j} have to be determined from the normalization

(9.30)  ∫_{a_{j}} ω_{∞_{+},∞_{−}} = 2 ∫_{E_{2j−1}}^{E_{2j}} \frac{∏_{k=1}^{g}(z − λ_{k})}{R_{2g+2}^{1/2}(z)}dz = 0,

which shows λ_{j} ∈ (E_{2j−1}, E_{2j}). With these preparations we are now able to express the Baker-Akhiezer function ψ(p, n) in terms of theta functions.

Theorem 9.2. The Baker-Akhiezer function ψ is given by

(9.31)  ψ(p, n) = C(n, 0)\frac{θ(\underline{z}(p, n))}{θ(\underline{z}(p, 0))} exp (n ∫_{p_{0}}^{p} ω̂_{∞_{+},∞_{−}}),

and ϕ is given by

(9.32)  ϕ(p, n) = C(n)\frac{θ(\underline{z}(p, n + 1))}{θ(\underline{z}(p, n))} exp ( ∫_{p_{0}}^{p} ω̂_{∞_{+},∞_{−}}).

The hat indicates that we require the path of integration to lie in M̂. The base point p_{0} has been chosen to be (E_{0}, 0) (any other branch point would do as well). The constants C(n) and C(n, n_{0}) are real and read

(9.33)  C(n)^{2} = \frac{θ(\underline{z}(n − 1))}{θ(\underline{z}(n + 1))}, C(n, m) = \begin{matrix}n−1 \\ ∏ & ∗ \\ j=m \end{matrix} C(j).

Proof. The function ψ is at least well-defined on M̂ since changing the path in M̂ can only change the values of the integral in the exponential by 2πi. But by

(9.34)  ∫_{a_{j}} ω_{∞_{+},∞_{−}} = 0,

the b-periods are given by (cf. (A.21))

(9.35)  ∫_{b_{j}} ω_{∞_{+},∞_{−}} = 2πiÂ_{∞_{−},j}(∞_{+}),

which shows together with (9.20) that it extends to a meromorphic function on M. Since ψ is fixed by its poles and zeros up to a constant (depending on n) it remains to determine this constant. This can be done using ψ(∞_{+}, n)ψ(∞_{−}, n) = 1, which follows from (8.103). From Section A.7 we recall \underline{Â}_{p_{0}}(∞_{+}) = −\underline{Â}_{p_{0}}(∞_{−}) mod ℤ^{g}

and similarly we obtain ∫_{p_{0}}^{p} ω̂_{∞_{+},∞_{−}} = − ∫_{p_{0}}^{p^{∗}} ω̂_{∞_{+},∞_{−}} for π(p) < E_{0}. This justifies our choice for C(n, 0)^{2}. By (9.27), C(n, m)^{2} is positive and the formula for ϕ now follows from ϕ(p, n) = ψ(n + 1, p)/ψ(n, p). □

The sign of C(n) is related to that of a(n) as will be shown in Theorem 9.4 below. Next, let us prove some basic properties of ϕ and ψ.

Lemma 9.3. Set σ(H) = ⋃_{j=0}^{g}[E_{2j}, E_{2j+1}]. The functions ϕ and ψ have the following properties (i). ϕ and hence ψ is real for π(p) ∈ ℝ\σ(H). (ii). Let z ∈ ℝ, then we have

(9.36)  |ψ_{±}(z, n)| ≤ Mk(z)^{±n}, M > 0,

where

(9.37)  k(z) = exp (Re(∫_{p_{0}}^{p} ω̂_{∞_{+},∞_{−}})) > 0, p = (z, R_{2g+2}^{1/2}(z))

and k(z) = 1 for z ∈ σ(H) and k(z) < 1 for z ∈ ℝ\σ(H). Furthermore, k(z) has minima precisely at the points λ_{j}, 1 ≤ j ≤ g, and tends to 0 as |z| → ∞.

Proof. We begin with realvaluedness of ϕ (which implies realvaluedness of ψ). Let μ̂_{k} be an arbitrary point with π(μ̂_{k}) ∈ [E_{2k−1}, E_{2k}]. Proceeding as usual we get from (9.35) for the differential of the third kind

(9.38)  i Im( ∫ μ̂_{k} ; p_{0} ω̂_{∞_{+},∞_{−}}) = −πiÂ_{∞_{−},k}(∞_{+})

and, similarly, i Im(z_{j}(μ̂_{k}, n)) = \frac{τ_{j,k}}{2} (cf. (9.26)). Now we recall theta functions with integer characteristics (cf. [81], VI.1.5)

(9.39) θ \left[{\underline{n} \atop \underline{n̂}}\right] (\underline{z}) = ∑_{\underline{m}∈ℤ^{g}} exp 2πi (⟨\underline{m} + \frac{1}{2}\underline{n̂}, \underline{z} + \frac{1}{2}\underline{n}⟩ + \frac{1}{2}⟨\underline{m} + \frac{1}{2}\underline{n̂}, \underline{τ} (\underline{m} + \frac{1}{2}\underline{n̂})⟩) ,

\underline{z} ∈ ℂ^{g}, which are clearly real for \underline{z} ∈ ℝ^{g}. Inserting

θ(\underline{z}(μ̂_{k}, n)) = exp 2πi(\frac{1}{8}⟨\underline{δ}(k), \underline{τ} \underline{δ}(k)⟩ + \frac{1}{2}⟨\underline{δ}(k), Re(\underline{z}(μ̂_{k}, n))⟩)

×θ \left[{0 \atop \underline{δ}(k)}\right] (Re(\underline{z}(μ̂_{k}, n))) (9.40)

into ϕ shows that the complex factors cancel since C(n) ∈ ℝ and z_{j}(μ̂_{k}, n + 1) = z_{j}(μ̂_{k}, n) + Â_{∞_{−},j}(∞_{+}). The argument for μ > E_{2g+1} and μ < E_{0} is even easier since Im(\underline{z}(μ̂, n)) = Im(∫ μ̂ ; p_{0} ω̂_{∞_{+},∞_{−}}) = 0 in this case.

Now we turn to the second claim. Since the theta functions are quasi-periodic we only have to investigate the exponential factor. For the asymptotic behavior we need to know Re(∫_{p_{0}}^{p} ω̂_{∞_{+},∞_{−}}). We first assume p ∈ Π_{+}, π(p) = λ ∈ ℝ, and choose as integration path the lift of the straight line from E_{0} + iε to λ + iε and take the limit ε ↓ 0 (we may intersect as many b-cycles as we like since all a-periods are zero). Hence

(9.41)  ∫_{p_{0}}^{p} ω̂_{∞_{+},∞_{−}} = ∫_{E_{0}}^{λ} \frac{∏_{j=1}^{g}(π − λ_{j})dπ}{R_{2g+2}^{1/2}} .

Clearly, (9.41) is negative for λ < E_{0}. For λ ∈ [E_{0}, E_{1}], (9.41) is purely imaginary and hence the real part is 0. At E_{1} our real part starts to decrease from zero until it hits its minimum at λ_{1}. Then it increases until it reaches 0, which must be at E_{2} by (9.30). Proceeding like this we get the desired behavior of k(z). For p ∈ Π_{−} we have to change the sign of Re(∫_{p_{0}}^{p} ω_{∞_{+},∞_{−}}). □

Next we are interested in expanding ϕ near ∞_{±} (in the coordinates (Π_{±}, z) induced by the projection on the upper/lower sheet Π_{±}). We start with

(9.42)  exp ( ∫_{p_{0}}^{p} ω̂_{∞_{+},∞_{−}}) = −(\frac{ã}{z})^{±1}(1 + \frac{b̃}{z} + \frac{c̃}{z^{2}} + O(\frac{1}{z^{3}} ))^{±1}, p ∈ Π_{±},

where ã, b̃, c̃ ∈ ℝ are constants depending only on the Riemann surface (i.e., on E_{j}). We have

(9.43)  ã = lim_{λ→∞} λ exp ( − ∫_{−λ}^{E_{0}} \frac{∏_{j=1}^{g}(z − λ_{j})}{R_{2g+2}^{1/2}(z)}dz) > 0

and for b̃ we get from (9.30)

(9.44)  b̃ = \frac{1}{2} ∑_{j=0}^{2g+1} E_{j} − ∑_{j=1}^{g} λ_{j}.

For the theta functions we get near ∞_{+} (cf. (A.119) and don’t confuse z and \underline{z})

\frac{θ(\underline{z}(p, n + 1))}{θ(\underline{z}(p, n))} = \frac{θ(\underline{z}(n + 1))}{θ(\underline{z}(n))}(1 + \frac{1}{z} ∑_{j=1}^{g} B_{j}(n + 1)

+ \frac{1}{2z^{2}} (( ∑_{j=1}^{g} B_{j}(n + 1))^{2}

+ ∑_{j=1}^{g} ( ∑_{k=0}^{2g+1} \frac{E_{k}}{2} + \frac{c_{j}(g − 1)}{c_{j}(g)})B_{j}(n + 1)

+ ∑_{j=1}^{g} ∑_{k=1}^{g} BB_{j,k}(n + 1)) + O(\frac{1}{z^{3}} )), (9.45)

with

B_{j}(n) = c_{j}(g)\frac{∂}{∂w_{j}} ln (\frac{θ(\underline{w} + \underline{z}(n))}{θ(\underline{w} + \underline{z}(n − 1))})|_{\underline{w}=0},
(9.46)  BB_{j,k}(n) = c_{j}(g)c_{k}(g)\frac{∂^{2}}{∂w_{j}∂w_{k}} ln (\frac{θ(\underline{w} + \underline{z}(n))}{θ(\underline{w} + \underline{z}(n − 1))})|_{\underline{w}=0}.

Near ∞_{−} one has to make the substitutions \underline{c}(n) → −\underline{c}(n) and n → n − 1. The constants \underline{c}(g) come from (9.17). In summary,

Theorem 9.4. The functions ϕ_{±} have the following expansions

(9.47)  ϕ_{±}(z, n) = (\frac{a(n)}{z})^{±1}(1 ± \frac{b(n + {1 \atop 0})}{z} + O(\frac{1}{z^{2}} )),

where the sequences a(n), b(n) are given by

a(n)^{2} = ã^{2} \frac{θ(\underline{z}(n + 1))θ(\underline{z}(n − 1))}{θ(\underline{z}(n))^{2}},

(9.48)  b(n) = b̃ + ∑_{j=1}^{g} c_{j}(g)\frac{∂}{∂w_{j}} ln (\frac{θ(\underline{w} + \underline{z}(n))}{θ(\underline{w} + \underline{z}(n − 1))})|_{\underline{w}=0}.

The sings of a(n) and C(n) must be opposite

(9.49)  a(n) = −ã C(n)\frac{θ(\underline{z}(n + 1))}{θ(\underline{z}(n))}.

The sequences a(n), b(n) are obviously quasi-periodic with g periods (i.e., their Fourier exponents possess a g-term integral basis).

We remark that we have

(9.50) \{μ_{j}(n_{0})\}_{j=1}^{g} ⊂ \{E_{2j−1}, E_{2j}\}_{j=1}^{g} ⇔ [\underline{z}(p, n_{0} + k)] = −[\underline{z}(p^{∗}, n_{0} − k)].

Inserting this into (9.48) yields again a(n_{0} +k) = a(n_{0} −k−1), b(n_{0} +k) = b(n_{0} −k) (compare Lemma 8.15).

The expansion coefficients ã, b̃ can be interpreted as averages of our coefficients a(n) and b(n) as follows.

Lemma 9.5. Suppose a(n) > 0 and π(p) ∈ (−∞, E_{0}), then

(9.51)  lim_{N→∞} \sqrt[N]{ψ(p, N)} = − exp ( ∫_{p_{0}}^{p} ω̂_{∞_{+},∞_{−}}).

In particular, we obtain

(9.52)  ã = lim_{N→∞} \sqrt[N]{∏_{j=1}^{N} a(j)},  b̃ = lim_{N→∞} \frac{1}{N} ∑_{j=1}^{N} b(j),

that is, ã is the geometric mean of the a’s and b̃ is the arithmetic mean of the b’s.

Proof. The first claim follows from lim_{N→∞} \sqrt[N]{|θ(\underline{z}(N))|} = 1 (cf. (9.28)). The second follows by comparing asymptotic coefficients with (9.47) respectively (6.27).

□

Now we want to know when our sequences are periodic.

Theorem 9.6. A necessary and sufficient condition for a(n), b(n) to be periodic is that R_{2g+2}(z) is of the form

(9.53)  \frac{1}{4A^{2}} R_{2g+2}(z)Q(z)^{2} = Δ(z)^{2} − 1,

where Q(z) and Δ(z) are polynomials, Q(z) has leading coefficient one, and A^{2} > 0 (compare (7.64)). The period N is given by

(9.54)  N = deg(Q) + g + 1.

Proof. Necessity has been shown in Chapter 7 (see (7.55) and Remark 7.6). For the remaining direction note that in the above case ω_{∞_{+},∞_{−}} is given by

(9.55)  ω_{∞_{+},∞_{−}} = \frac{1}{N} \frac{2AΔ′ dπ}{R_{2g+2}^{1/2}Q} ,

where N = deg(Q) + g + 1 and the prime denotes the derivative with respect to z. We only have to check that the a-periods are zero

∫_{a_{j}} ω_{∞_{+},∞_{−}} = \frac{2}{N} ∫_{E_{2j−1}}^{E_{2j}} \frac{Δ′(z)dz}{(Δ(z)^{2} − 1)^{1/2}}

(9.56)  = \frac{2}{N} ln |Δ(z) + (Δ(z)^{2} − 1)^{1/2}||_{E_{2j−1}}^{E_{2j}} = 0.

For the b-periods we get

∫_{b_{j}} ω_{∞_{+},∞_{−}} = \frac{−2}{N} ∑_{k=0}^{j−1} ∫_{E_{2k}}^{E_{2k+1}} \frac{Δ′(z) dz}{(Δ(z)^{2} − 1)^{1/2}}

= \frac{−2i}{N} ∑_{k=0}^{j−1}(−1)^{g−k} ∫_{E_{2k}}^{E_{2k+1}} \frac{Δ′(z) dz}{\sqrt{1 − Δ(z)^{2}}}

(9.57)  = \frac{−2i}{N} ∑_{k=0}^{j−1}(−1)^{g−k} arcsin(Δ(z))|_{E_{2k}}^{E_{2k+1}} = −\frac{2πi}{N}j.

Which proves the assertion, since we infer 2N\underline{Â}_{p_{0}}(∞_{+}) ∈ ℤ^{g}. □

Further we obtain in the periodic case for the Floquet multipliers

(9.58)  m(p) = ψ(p, N) = (−1)^{N}sgn(A) exp (N ∫_{p_{0}}^{p} ω̂_{∞_{+},∞_{−}})

and ã = \sqrt[N]{|A|}, b̃ = \frac{B}{N}.

Next we will show that the Baker-Akhiezer function is even fixed by its (finite) poles and its behavior at infinity

Lemma 9.7. Let D\underline{μ̂} ∈ M_{g} be nonspecial (i.e., i(D\underline{μ̂}) = 0). Then there is a meromorphic function ψ(n, .) on M fulfilling

(9.59)  (ψ(n)) ≥ −D\underline{μ̂} + n(D_{∞_{+}} − D_{∞_{−}}).

Moreover, ψ(n, .) is unique up to a multiple depending only on n and the divisor D_{zer}(n) ∈ M_{g} defined by

(9.60)  (ψ(n)) = D_{zer}(n) − D\underline{μ̂} + n(D_{∞_{+}} − D_{∞_{−}}),

is also nonspecial.

Proof. Abbreviate D_{n} = D\underline{μ̂} − n(D_{∞_{+}} − D_{∞_{−}}). The Riemann-Roch theorem (cf. Theorem (A.2)) implies existence of at least one Baker-Akhiezer function since r(−D_{n}) = 1 + i(D\underline{μ̂} − n(D_{∞_{+}} − D_{∞_{−}}) ≥ 1. Moreover, since there is a function satisfying (9.60), we obtain i(D_{zer}(n)) = i(D_{n}) = r(−D_{n}) − 1 by (A.39) and Riemann-Roch.

There is nothing to prove for n = 0, since L(−D\underline{μ̂}) = ℂ and D_{zer}(0) = D\underline{μ̂}. Next let us consider n = 1. From L(−D\underline{μ̂}) = ℂ we see L(−D\underline{μ̂} + D_{∞_{−}}) = \{0\} and hence r(−D_{1}) ≤ 1 (otherwise, if r(−D_{1}) > 1, a proper linear combination of two elements would lie in L(−D\underline{μ̂} + D_{∞_{−}}), a contradiction). Now suppose the claim holds for n > 1. Then it also holds for n+1 since r(−D_{n+1}) = r(−D_{n+1}−(ψ(n))) = r(−D_{zer}(n) + (D_{∞_{+}} − D_{∞_{−}})) = 1 again by (A.39). So by induction the claim holds for n ∈ ℕ. Similarly, one can show that it holds for n ∈ −ℕ. □

Now using (9.47) it is not hard to show that (near ∞_{±})

(9.61)  a(n)ϕ(p, n) + \frac{a(n − 1)}{ϕ(p, n − 1)} + b(n) − π(p) = O(\frac{1}{z})

or equivalently

(9.62) a(n)ψ(p, n + 1) + a(n − 1)ψ(p, n − 1) − (π(p) − b(n))ψ(p, n) = z^{∓n}O(\frac{1}{z}).

The function on the left fulfills the requirements for a Baker-Akhiezer function (9.59) and D_{∞_{+}} + D_{∞_{−}} is part of D_{zer}(n). Since D_{zer}(n) is nonspecial, this is impossible (by Lemma A.20) unless the left hand side is zero. This is clearly an independent proof that ψ(p, n) satisfies τψ(p, n) = π(p)ψ(p, n).

Remark 9.8. From (9.61) we also obtain a different representation of b(n). If we choose p = p_{0} it reads

(9.63) b(n) = E_{0} + ã\frac{θ(\underline{z}(n − 1))}{θ(\underline{z}(n))} \frac{θ(\underline{z}(p_{0}, n + 1))}{θ(\underline{z}(p_{0}, n))} + ã\frac{θ(\underline{z}(n))}{θ(\underline{z}(n − 1))} \frac{θ(\underline{z}(p_{0}, n − 1))}{θ(\underline{z}(p_{0}, n))}.

We note that (8.101)–(8.104) can be easily verified independently by considering poles, zeros, and the behavior at ∞_{±}. In particular, by expanding (8.104) around ∞_{±} and comparing the second term in the expansion we get our trace relation

(9.64)  b(n) = \frac{1}{2} ∑_{j=0}^{2g+1} E_{j} − ∑_{j=1}^{g} μ_{j}(n).

Upon evaluating the integral

(9.65)  I = \frac{1}{2πi} ∫_{∂M̂} π(.)d ln(θ(\underline{z}(., n))),

we get from the residue theorem

(9.66)  I = ∑_{j=1}^{g} μ_{j}(n) + ∑_{p∈\{∞_{±}\}}res_{p}(π(.)d ln(θ(\underline{z}(., n))),

and by direct calculation (using (A.83))

(9.67)  I = ∑_{j=1}^{g} ∫_{a_{j}} πζ_{j}.

Which yields the relation

(9.68) b(n) = \frac{1}{2} ∑_{j=0}^{2g+1} E_{j} − ∑_{j=1}^{g} ∫_{a_{j}} πζ_{j} + ∑_{j=1}^{g} c_{j}(g)\frac{∂}{∂w_{j}} ln (\frac{θ(\underline{w} + \underline{z}(n))}{θ(\underline{w} + \underline{z}(n − 1))})|_{\underline{w}=0},

and thus

(9.69)  ∑_{j=1}^{g} ∫_{a_{j}} πζ_{j} = ∑_{j=1}^{g} λ_{j}.

9.3. The elliptic case, genus one

In this section we will show how the formulas of the last section can be expressed in terms of Jacobi’s elliptic functions if g = 1. All integrals are evaluated using the tables in [38].

Suppose E_{0} < E_{1} < E_{2} < E_{3} are given and R_{4}^{1/2}(z) is defined as usual. We introduce

(9.70) k = \sqrt{\frac{E_{2} − E_{1}}{E_{3} − E_{1}} \frac{E_{3} − E_{0}}{E_{2} − E_{0}}} ∈ (0, 1), k′ = \sqrt{\frac{E_{3} − E_{2}}{E_{3} − E_{1}} \frac{E_{1} − E_{0}}{E_{2} − E_{0}}} ∈ (0, 1),

such that k^{2} + k′^{2} = 1, and

(9.71)  u(z) = \sqrt{\frac{E_{3} − E_{1}}{E_{3} − E_{0}} \frac{E_{0} − z}{E_{1} − z}} , C = \frac{2}{\sqrt{(E_{3} − E_{1})(E_{2} − E_{0})}}.

Furthermore, we will use the following notation for Jacobi’s elliptic integrals of the first

(9.72)  ∫_{0}^{z} \frac{dx}{\sqrt{(1 − x^{2})(1 − k^{2}x^{2})}} = F (z, k),

second

(9.73)  ∫_{0}^{z} \sqrt{\frac{1 − x^{2}}{1 − k^{2}x^{2}}} dx = E(z, k),

and third kind

(9.74)  ∫_{0}^{z} \frac{dx}{(1 − α^{2}x^{2})\sqrt{(1 − x^{2})(1 − k^{2}x^{2})}} = Π(z, α^{2}, k).

We set F (1, k) = K(k), E(1, k) = E(k), and Π(1, α^{2}, k) = Π(α^{2}, k), where k ∈ (0, 1), α^{2} ∈ ℝ and all roots are assumed to be positive for x ∈ (0, 1). We remark that Π(z, α^{2}, k) has simple poles at z^{2} = α^{−2} and that E(z, k) has a simple pole at ∞.

First we determine ζ, the abelian differential of the first kind (in the charts (Π_{±}, z)).

(9.75)  ζ = c(1)\frac{dπ}{R_{4}^{1/2}} = c(1)\frac{dz}{±R_{4}^{1/2}(z)}.

The normalization ([38], 254.00)

(9.76)  ∫_{a_{1}} ζ = 2c(1) ∫_{E_{1}}^{E_{2}} \frac{dz}{R_{4}^{1/2}(z)} = 2c(1)CK(k) = 1

yields

(9.77)  c(1) = \frac{1}{2CK(k)}.

In addition, we obtain ([38], 252.00)

(9.78)  τ_{1,1} = ∫_{b_{1}} ζ = −2c(1) ∫_{E_{0}}^{E_{1}} \frac{dx}{R_{4}^{1/2}(x)} = 2ic(1)CK(k′) = i\frac{K(k′)}{K(k)}.

Next we are interested in Abel’s map A_{p_{0}}, p_{0} = (E_{0}, 0). Let p = (z, ±R_{4}^{1/2}(z)), then we have ([38], 252.00)

(9.79)  A_{p_{0}}(p) = [∫_{p_{0}}^{p} ζ] = [±c(1) ∫_{E_{0}}^{z} \frac{dx}{R_{4}^{1/2}(x)}] = [±\frac{F (u(z), k)}{2K(k)}]

and especially for p = ∞_{+}

(9.80)  A_{p_{0}}(∞_{+}) = [\frac{F (\sqrt{\frac{E_{3}−E_{1}}{E_{3}−E_{0}}} , k)}{2K(k)}].

The vector of Riemann constants is independent of the base point p_{0} and clearly given by

(9.81)  Ξ = [\frac{1 − τ_{1,1}}{2} ].

Now we turn to the Dirichlet eigenvalue μ_{1}(n). We recall

(9.82)  A_{p_{0}}(μ̂_{1}(n)) = A_{p_{0}}(μ̂_{1}(0)) − 2nA_{p_{0}}(∞_{+}).

If we set A_{p_{0}}(μ̂_{1}(0)) = [δ + \frac{τ_{1,1}}{2} ], where δ ∈ [0, 1), we obtain

(9.83) A_{p_{0}}(μ̂_{1}(n)) = [\frac{τ_{1,1}}{2} +σ_{1}(n) ∫_{E_{1}}^{μ_{1}(n)} \frac{c(1)dx}{R_{4}^{1/2}(x)}] = [δ + \frac{τ_{1,1}}{2} −n\frac{F (\sqrt{\frac{E_{3}−E_{1}}{E_{3}−E_{0}}} , k)}{K(k)}],

with μ̂_{1}(n) = (μ_{1}(n), σ_{1}(n)R_{4}^{1/2}(μ_{1}(n))). Evaluation of the integral yields ([38], 254.00)

(9.84)  ∫_{E_{1}}^{μ_{1}(n)} \frac{c(1)dx}{R_{4}^{1/2}(x)} = \frac{F (\sqrt{\frac{E_{2}−E_{0}}{E_{2}−E_{1}} \frac{μ_{1}(n)−E_{1}}{μ_{1}(n)−E_{0}}} , k)}{2K(k)}

and hence we infer

(9.85)  μ_{1}(n) = E_{1} \frac{1 − \frac{E_{2}−E_{1}}{E_{2}−E_{0}} \frac{E_{0}}{E_{1}} sn^{2}(2K(k)δ − 2nF (\sqrt{\frac{E_{3}−E_{1}}{E_{3}−E_{0}}} , k))}{1 − \frac{E_{2}−E_{1}}{E_{2}−E_{0}} sn^{2}(2K(k)δ − 2nF (\sqrt{\frac{E_{3}−E_{1}}{E_{3}−E_{0}}} , k))},

where sn(z) is Jacobi’s elliptic function (i.e., the inverse of F (z, k)). The sheet can be read off from the sign of sn(2K(k)δ − 2nF (…)).

Now we search a more explicit expression for the differential ω_{∞_{−},∞_{+}}. We first determine the constant λ_{1} from ([38], 254.00, 254.10, 340.01)

∫_{a_{1}} ω_{∞_{−},∞_{+}} = 2 ∫_{E_{1}}^{E_{2}} \frac{x − λ_{1}}{R_{4}^{1/2}(x)}dx

= 2C((E_{1} − E_{0})Π(\frac{E_{2} − E_{1}}{E_{2} − E_{0}} , k) + (E_{0} − λ_{1})K(k)) = 0, (9.86)

which implies

(9.87)  λ_{1} = E_{0} + \frac{(E_{1} − E_{0})}{K(k)}Π(\frac{E_{2} − E_{1}}{E_{2} − E_{0}} , k).

The b-period is given by (cf. (9.35))

(9.88)  ∫_{b_{1}} ω_{∞_{−},∞_{+}} = −2 ∫_{E_{0}}^{E_{1}} \frac{x − λ_{1}}{R_{4}^{1/2}(x)}dx = −2πi(1 − \frac{F (\sqrt{\frac{E_{3}−E_{1}}{E_{3}−E_{0}}} , k)}{K(k)}).

And for p = (z, ±R_{4}^{1/2}(z)) near p_{0} we have ([38], 251.00, 251.10, 340.01)

∫_{p_{0}}^{p} ω_{∞_{−},∞_{+}} = ± ∫_{E_{0}}^{z} \frac{x − λ_{1}}{R_{4}^{1/2}(x)}dx

= ±C(E_{1} − E_{0})((1 − \frac{Π(\frac{E_{2}−E_{1}}{E_{2}−E_{0}} , k)}{K(k)})F (u(z), k)

− Π(u(z), \frac{E_{3} − E_{0}}{E_{3} − E_{1}} , k)). (9.89)

To complete this section we recall that Riemann’s theta function can be expressed as a Jacobi theta function, namely

(9.90)  θ(z) = ϑ_{3}(z) = ∑_{n∈ℤ} eπi(2nz + τ^{1,1}n^{2}).

9.4. Some illustrations of the Riemann-Roch theorem

In this section we give some illustrations of the Riemann-Roch theorem (Theorem A.2) in connection with hyperelliptic Riemann surfaces not branched at infinity and obtain a basis for the vector space L(−nD_{−∞} − D\underline{μ̂}), n ∈ ℕ_{0}.

Theorem 9.9. Assume D\underline{μ̂} ≥ 0 to be nonspecial (i.e., i(D\underline{μ̂}) = 0) and of degree g. For n ∈ ℕ_{0}, a basis for the vector space L(−nD_{−∞} − D\underline{μ̂}) is given by,

\begin{cases}j−1 \\ ∏ & ∗ \\ ℓ=0 \end{cases} ϕ(., ℓ)\}_{j=0}^{n}, (9.91)

or equivalently, by

L(−nD_{−∞} − D\underline{μ̂}) = span \{ψ(., j)\}_{j=0}^{n} . (9.92)

166 9. Quasi-periodic Jacobi operators Proof. Since the above functions are linearly independent elements of the vector space L(−nD_{−∞} − D\underline{μ̂}) it remains to show that they are maximal. From i(D\underline{μ̂}) = 0 = i(nD_{−∞} + D\underline{μ̂}) and the Riemann-Roch theorem we obtain r(−nD_{∞} − D\underline{μ̂}) = 1 + n, (9.93) which proves the theorem. □

Scattering theory

In Section 7.5 we have considered perturbations of periodic Jacobi operators, that is, we have started developing scattering theory with periodic background. In this chapter we will investigate sequences a(n) > 0, b(n) which look asymptotically like the free ones (compare Section 1.3).

10.1. Transformation operators

To begin with, we state the main hypothesis which will be assumed to hold throughout this chapter.

Hypothesis H. 10.1. Suppose a, b ∈ ℓ(ℤ, ℝ) satisfy a > 0 and

(10.1)  n(1 − 2a(n)) ∈ ℓ^{1}(ℤ),  n b(n) ∈ ℓ^{1}(ℤ).

In addition, it will turn out convenient to introduce

(10.2)  A_{−}(n) = ∏_{m=−∞}^{n−1} 2a(m), A_{+}(n) = ∏_{m=n}^{∞} 2a(m), A = A_{−}(n)A_{+}(n).

From Section 1.3 we know that it is better to consider the free problem in terms of k = z − \sqrt{z − 1}\sqrt{z + 1} rather than in terms of the spectral parameter z itself.

By virtue of Lemma 7.10 we infer (upon choosing a_{p}(n) = 1/2 and b_{p}(n) = 0) the existence of solutions f_{±}(k, n) of (τ − (k + k^{−1})/2)f = 0 which asymptotically look like the free ones, that is,

(10.3)  f_{±}(k, n) = k^{±n}(1 + o(1)) as n → ±∞,

respectively, lim_{n→±∞} f_{±}(k, n)k^{∓n} = 1.

They are called Jost solutions and their properties are summarized in the following theorem.

Theorem 10.2. Let a(n), b(n) satisfy (H.10.1), then there exist Jost solutions f_{±}(k, n) of

(10.4)  τf_{±}(k, n) = \frac{k + k^{−1}}{2} f_{±}(k, n), 0 < |k| ≤ 1,

fulfilling

(10.5)  lim_{n→±∞} f_{±}(k, n)k^{∓n} = 1.

Moreover f̃(k, n) = A_{±}(n)k^{∓n}f_{±}(k, n) is holomorphic in the domain |k| < 1, continuous for |k| ≤ 1, and satisfies f̃(0, n) = 1.

Proof. For the case k ≠ 0 everything follows from Lemma 7.10. To show that the singularity of f̃(k, n) at k = 0 is removable, one needs to investigate the limit k → 0 in (7.88). To do this, use a_{p}(n) = 1/2, b_{p}(n) = 0, m_{p}^{±}(z) = k^{±1}, s_{p}(z, n, m) = \frac{k}{k^{2}−1}(k^{n−m} − k^{m−n}) (see Section 1.3) and compute the limit lim_{k→0} k^{n−m}K(z, n, m) = 2a(m) − 1. □

The previous theorem shows that we may set

(10.6)  f_{+}(k, n) = \frac{k^{n}}{A_{+}(n)}(1 + ∑_{j=1}^{∞} K_{+,j}(n)k^{j}), |k| ≤ 1,

where K_{+,j}(n) are the Fourier coefficients of f̃_{+}(k, n) given by

(10.7)  K_{+,j}(n) = \frac{1}{2πi} ∫_{|k|=1} f̃_{+}(k, n)k^{−j} \frac{dk}{k} , j ∈ ℕ.

Taking the limit n → ∞ in this formula (using the estimate (7.80) and dominated convergence) shows lim_{n→∞} K_{+,j}(n) = 0, j ∈ ℕ.

Inserting (10.6) into (10.4) (set K_{0}^{+}(n) = 1, K_{j}^{+}(n) = 0, j < 0) yields

K_{+,j+1}(n) − K_{+,j+1}(n − 1) = 2b(n)K_{+,j}(n) − K_{+,j−1}(n)

+ 4a(n)^{2}K_{+,j−1}(n + 1). (10.8)

Summing up (with respect to n) using lim_{n→∞} K_{+,j}(n) = 0, j ∈ ℕ, we obtain

K_{+,1}(n) = − ∑_{m=n+1}^{∞} 2b(m),

K_{+,2}(n) = − ∑_{m=n+1}^{∞} (2b(m)K_{+,1}(m) + (4a(m)^{2} − 1)),

K_{+,j+1}(n) = K_{+,j−1}(n + 1) − ∑_{m=n+1}^{∞} (2b(m)K_{+,j}(m)

+ (4a(m)^{2} − 1)K_{+,j−1}(m + 1)). (10.9)

A straightforward induction argument shows

(10.10)  |K_{+,j}(n)| ≤ D_{+,j}(n)C_{+}(n + [[\frac{j}{2}]] + 1), j ∈ ℕ,

where

(10.11)  C_{+}(n) = ∑_{m=n}^{∞} c(m), D_{+,m}(n) = ∏_{j=1}^{m−1}(1 + C_{+}(n + j)),

and c(n) = 2|b(n)| + |4a(n)^{2} − 1|. It is important to note that C_{+} ∈ ℓ_{+}^{1}(ℤ). In fact, summation by parts shows

(10.12)  ∑_{m=n}^{∞} m c(m) = (n − 1)C_{+}(n) + ∑_{m=n}^{∞} C_{+}(m)

since

(10.13)  lim_{n→∞} n C_{+}(n + 1) ≤ lim_{n→∞} ∑_{m=n+1}^{∞} m c(m) = 0.

Moreover, 1 ≤ D_{+,m}(n) ≤ D_{+}(n) = lim_{j→∞} D_{+,j}(n) and lim_{n→∞} D_{+}(n) = 1. This gives also an estimate for

|K_{+,j+1}(n) − K_{+,j+1}(n − 1)| ≤ 2|b(n)||K_{+,j}(n)| + |4a(n)^{2} − 1||K_{+,j−1}(n)|

+ |K_{+,j−1}(n + 1) − K_{+,j−1}(n)|

≤ c(n)D_{+}(n)C_{+}(n + [[\frac{j}{2}]] + 1) + |K_{+,j−1}(n + 1) − K_{+,j−1}(n)|. (10.14)

Using induction we obtain

(10.15) |K_{+,j}(n) − K_{+,j}(n − 1)| ≤ D_{+}(n)C_{+}(n)C_{+}(n + [[\frac{j − 1}{2}]] + 1) + c(n + [[\frac{j}{2}]]).

In addition, the estimate (10.10) for K_{+,j}(n) gives us an estimate on the rate of convergence in (10.3)

(10.16)  |f_{+}(k, n)A(n)k^{−n} − 1| ≤ \{ \begin{matrix}\frac{|k|}{1 − |k|}D_{+}(n)C_{+}(n + 1), & |k| < 1 \\ 2|k|D_{+}(n) ∑_{m=n}^{∞} C_{+}(m), & |k| ≤ 1 \end{matrix} .

Moreover, since K_{+,j}(n) ≤ C_{∞}(1 + C_{∞})^{j−1}, C_{∞} = lim_{n→−∞} C_{+}(n), we can take the limits (K_{0} = 1)

K_{1} = lim_{n→−∞} K_{+,1}(n) = −2 ∑_{m∈ℤ} b(m),
K_{2} = lim_{n→−∞} K_{+,2}(n) = − ∑_{m∈ℤ} (2b(m)K_{+,1}(m) + (4a(m)^{2} − 1))
K_{j+1} = lim_{n→−∞} K_{+,j+1}(n) = K_{j−1} − ∑_{m∈ℤ} (2b(m)K_{+,j}(m)

+ (4a(m)^{2} − 1)K_{+,j−1}(m + 1)). (10.17)

(It will become clear later why we have dropped the + subscript.)

Associated with K_{+}(n, n + j) = K_{+,j}(n) is the operator

(10.18)  (K_{+}f)(n) = \frac{1}{A_{+}(n)}(f(n) + ∑_{m=n+1}^{∞} K_{+}(n, m)f(m)), f ∈ ℓ_{+}^{∞}(ℤ),

which acts as a transformation operator for the pair τ_{0}, τ.

Theorem 10.3. Let τ_{0} be the free Jacobi difference expression. Then we have

(10.19)  τK_{+}f = K_{+}τ_{0}f,  f ∈ ℓ_{+}^{∞}(ℤ).

Proof. A straightforward calculation shows that τK_{+} = K_{+}τ_{0} is equivalent to (10.8). □

By our estimate (10.10), K_{+} maps ℓ_{+}^{p}(ℤ), p ∈ [1, ∞) ∪ {∞}, into itself. Moreover, by Lemma 7.8, K_{+} is an automorphism of ℓ_{+}^{∞}(ℤ). Considering f ∈ ℓ_{0}(ℤ) we see that its inverse L_{+} is given by

(10.20) (L_{+}f)(n) = A_{+}(n)(f(n) + ∑_{m=n+1}^{∞} \frac{L_{+}(n, m)}{∏_{j=n}^{m−1} 2a(j)}f(m)), f ∈ ℓ_{+}^{∞}(ℤ),

with

L_{+}(n, m) = −K_{+}(n, m) − ∑_{ℓ=n+1}^{m−1} K_{+}(n, ℓ)L_{+}(ℓ, m)

= − K_{+}(n, m) − ∑_{ℓ=n+1}^{m−1} L_{+}(n, ℓ)K_{+}(ℓ, m), m > n.

From L_{+}τ = τ_{0}L_{+} we infer that the coefficients L_{+,j}(n) = L_{+}(n, n + j) satisfy

L_{+,j+1}(n) − L_{+,j+1}(n − 1) = −2b(n + j)L_{+,j}(n) − 4a(n + j − 1)^{2}L_{+,j−1}(n)

+ L_{+,j−1}(n + 1) (10.21)

as well as an estimate of type (10.10). In particular, K_{+} is an automorphism of ℓ_{+}^{p}(ℤ), p ∈ [1, ∞) ∪ {∞}.

Similarly, we set

(10.22)  f_{−}(k, n) = \frac{k^{−n}}{A_{−}(n)}(1 + ∑_{j=1}^{∞} K_{−,j}(n)k^{j}), |k| ≤ 1,

which yields after inserting it into (1.19) (set K_{−,0}(n) = 1, K_{−,j}(n) = 0, j < 0)

K_{−,j+1}(n + 1) − K_{−,j+1}(n) = −2b(n)K_{−,j}(n) + K_{−,j−1}(n)

− 4a(n − 1)^{2}K_{−,j−1}(n − 1). (10.23)

As before we obtain (lim_{n→−∞} K_{−,j}(n) = 0, j ∈ ℕ)

K_{−,1}(n) = − ∑_{m=−∞}^{n−1} 2b(m),

K_{−,2}(n) = − ∑_{m=−∞}^{n−1} 2b(m)K_{−,1}(m) − (4a(m − 1)^{2} − 1),

K_{−,j+1}(n) = K_{−,j−1}(n) − ∑_{m=−∞}^{n−1} (2b(m)K_{−,j}(m)

− (4a(m − 1)^{2} − 1)K_{−,j−1}(m − 1)). (10.24)

Furthermore, K_{−,j}(n) = K_{−}(n, n − j) satisfies an estimate of type (10.10) and we can define an associated operator

(10.25)  (K_{−}f)(n) = \frac{1}{A_{−}(n)}(f(n) + ∑_{m=−∞}^{n−1} K_{−}(n, m)f(m)), f ∈ ℓ_{−}^{∞}(ℤ),

with inverse L_{−}.

10.2. The scattering matrix

Our next objective is to introduce the scattering matrix. For |k| = 1, k^{2} ≠ 1, the Jost solutions f_{±}(k, n), f_{±}(k^{−1}, n) are linearly independent since we have

(10.26)  W (f_{±}(k), f_{±}(k^{−1})) = ±\frac{1 − k^{2}}{2k} .

In particular, we may set

(10.27)  f_{±}(k, n) = α(k)f_{∓}(k^{−1}, n) + β_{∓}(k)f_{∓}(k, n), |k| = 1, k^{2} ≠ 1,

where

α(k) = \frac{W (f_{∓}(k), f_{±}(k))}{W (f_{∓}(k), f_{∓}(k^{−1}))} = \frac{2k}{1 − k^{2}} W (f_{+}(k), f_{−}(k)),
(10.28) β_{±}(k) = \frac{W (f_{∓}(k), f_{±}(k^{−1}))}{W (f_{±}(k), f_{±}(k^{−1}))} = ±\frac{2k}{1 − k^{2}} W (f_{∓}(k), f_{±}(k^{−1})).

Note that we have

(10.29)  \overline{α(k)} = α(\overline{k}), \overline{β_{±}(k)} = β_{±}(\overline{k}), β_{+}(k^{−1}) = −β_{−}(k),

and \overline{k} = k^{−1} for |k| = 1. Moreover, the Plücker identity (2.169) with f_{1} = f_{∓}, f_{2} = f_{±}, f_{3} = \overline{f_{∓}}, f_{4} = \overline{f_{±}}, implies

(10.30)  |α(k)|^{2} = 1 + |β_{±}(k)|^{2}, |β_{+}(k)| = |β_{−}(k)|.

The function α(k) can be defined for all |k| ≤ 1. Moreover, the power series of α(k) follows formally from

α(k) = \frac{2k}{1 − k^{2}} lim_{n→±∞} a(n)(f_{+}(k, n)f_{−}(k, n + 1) − f_{+}(k, n + 1)f_{−}(k, n))
(10.31) = \frac{1}{A} lim_{n→±∞} f̃_{∓}(k, n) = \frac{1}{A} ∑_{j=0}^{∞} K_{j}k^{j},

where K_{j} = lim_{n→∓∞} K_{±,j}(n) from (10.17). To make this precise we need to show that we can interchange limit and summation in the last line. This follows from |K_{±,j}(n)| ≤ C_{∞}(1 + C_{∞})^{j−1} at least for |k| < (1 + C_{∞})^{−1}. Hence it follows for all |k| < 1 since α(k) is holomorphic for |k| < 1.

The function α(k) has even more interesting properties. To reveal its connection with Krein’s spectral shift theory, we compute its derivative. By virtue of (2.29) we obtain

\frac{d}{dk} W (f_{+}(k), f_{−}(k)) = W_{m}(f_{+}(k), \frac{d}{dk} f_{−}(k)) + W_{n}(\frac{d}{dk} f_{+}(k), f_{−}(k))

− \frac{1 − k^{2}}{2k^{2}} ∑_{j=m+1}^{n} f_{+}(k, j)f_{−}(k, j). (10.32)

From (10.6), (10.22) we get \frac{d}{dk} f_{±}(k, n) = ±\frac{n}{k}f_{±}(n, k) + k^{±n}o(1), n → ±∞, and f_{±}(k, n) = k^{±n}(α(k) + o(1)), n → ∓∞, respectively. Hence we see

W_{m}(f_{+}(k), \frac{d}{dk} f_{−}(k)) = −m\frac{1 − k^{2}}{2k^{2}} α(k) − \frac{1}{2k^{2}} α(k) + o(1), m → −∞,
W_{n}(\frac{d}{dk} f_{+}(k), f_{−}(k)) = n\frac{1 − k^{2}}{2k^{2}} α(k) − \frac{1}{2}α(k) + o(1), n → +∞ (10.33)

and using this to evaluate the limits m → −∞, n → ∞ shows

(10.34)  \frac{d}{dk} α(k) = \frac{−1}{k} ∑_{n∈ℤ}(f_{+}(k, n)f_{−}(k, n) − α(k)).

Next, observe that, due to (H.10.1), H−H_{0} is trace class. Indeed, the multiplication operators by b(n), 2a(n) − 1 are trace class and trace class operators form an ideal in the Banach space of bounded linear operators. So we can rewrite this equation in the form

−\frac{d}{dz} ln α(k(z)) = ∑_{j∈ℤ}(G(z, n, n) − G_{0}(z, n, n))
(10.35)  = tr((H − z)^{−1} − (H_{0} − z)^{−1}),

that is, A α(k) (since A α(0) = 1) is the perturbation determinant of the pair H, H_{0}. This implies ([155], Theorem 1)

(10.36)  α(k(z)) = \frac{1}{A} exp ( ∫_{ℝ} \frac{ξ_{α}(λ)dλ}{λ − z}),

where

(10.37)  ξ_{α}(λ) = \frac{1}{π} lim_{ε↓0} arg α(k(λ + iε)) ∈ [−1, 1]

is of compact support. However, the reader should be warned that α(k(z)) is not Herglotz since we do not have 0 ≤ ξ_{α}(λ) ≤ 1 in general.

In addition, Krein’s result implies

(10.38)  tr(H^{ℓ} − (H_{0})^{ℓ}) = ℓ ∫_{ℝ} λ^{ℓ−1}ξ_{α}(λ)dλ.

The spectrum of H can be easily characterized upon specializing to the case a_{p}(n) = 1/2, b_{p}(n) = 0 in Theorem 7.11.

Theorem 10.4. Suppose (H.10.1). Then we have

σ_{ess}(H) = σ_{ac}(H) = [−1, 1],
(10.39)  σ_{p}(H) = \{λ = \frac{k + k^{−1}}{2} | |λ| > 1, α(k) = 0\}.

Moreover, the essential spectrum of H is purely absolutely continuous and the point spectrum of H is finite.

Hence we can denote the number of eigenvalues by N ∈ ℕ and set

(10.40)  σ_{p}(H) = \{λ_{j} = \frac{k_{j} + k_{j}^{−1}}{2} \}_{1≤j≤N}.

Note that because of (10.30) all zeros of of α(k) lie inside the unit circle. Moreover, there can be no non-real zeros (otherwise H would have a non-real eigenvalue).

(Note also α(0) = A^{−1} > 0 by (10.31).) Using Theorem 3.20 we can even provide a bound on the number of eigenvalues.

Theorem 10.5. Suppose (H.10.1). Then

(10.41)  dim Ran P_{(±1,±∞)}(H) ≤ 1 + (∑_{n∈ℤ} c_{±}(n))^{2} ,

where

(10.42)  c_{±}(n) = \sqrt{n(n + 1)}|1 − 2a(n)| + |n| max(∓b(n), 0).

Proof. We only prove the first bound, the second being similar. Upon replacing b(n) by min(b(n), 0) we can assume b(n) ≤ 0 (cf. [195], Theorem XIII.10). Our goal is to apply Theorem 3.20. Since the limit of G_{0}(λ, n, m) as λ → −1 does not exist, we will consider H_{±} and use

dim Ran P_{(−∞,−1)}(H) ≤ 1 + dim Ran P_{(−∞,−1)}(H_{+})

+ dim Ran P_{(−∞,−1)}(H_{−}). (10.43)

Where the last claim follows from the fact that between two eigenvalues of H, there can be at most one Dirichlet eigenvalue (see Lemma 3.7). Hence we need to consider H_{0,+}. From (0 < m ≤ n, λ < −1)

(10.44)  G_{0,+}(λ, n, m) = −2k \frac{1 − k^{2m}}{1 − k^{2}} k^{n−m}

(k = λ − \sqrt{λ^{2} − 1} ∈ (−1, 0)) we infer

(10.45)  |G_{0,+}(λ, n, m)| ≤ 2 min(m, n) ≤ 2\sqrt{m n},  λ ≤ −1.

Evaluating Theorem 3.20 as in Corollary 3.21 gives

(10.46)  dim Ran P_{(−∞,−1)}(H_{+}) ≤ (∑_{n=1}^{∞} c_{+}(n))^{2} .

The bound for H_{−} follows from reflection. □

We now define the scattering matrix

(10.47)  S(k) = ( \begin{matrix}T (k) & R_{−}(k) \\ R_{+}(k) & T (k) \end{matrix} ) , |k| = 1,

where T (k) = α(k)^{−1} and R_{±}(k) = α(k)^{−1}β_{±}(k). Equations (10.29) and (10.30) imply

Lemma 10.6. The scattering matrix S(k) is unitary and its determinant reads

(10.48)  det S(k) = \frac{T (k)}{T (\overline{k})}.

The coefficients T (k), R_{±}(k) are continuous except at possibly k = ±1 and fulfill

(10.49)  |T (k)|^{2} + |R_{±}(k)|^{2} = 1, T (k)R_{+}(\overline{k}) + T (\overline{k})R_{−}(k) = 0,

and \overline{T (k)} = T (\overline{k}), \overline{R_{±}(k)} = R_{±}(\overline{k}).

The quantity T (k) is called transmission coefficient and R_{±}(k) are called reflection coefficients with respect to left respectively right incidence. We remark

(10.50)  |T (k)|^{2} = 1 − |R_{±}(k)|^{2} ≥ C^{2}|1 − k^{2}|^{2},

where we can choose C = 2 max_{|k|=1} |W (f_{+}(k), f_{−}(k))|. In particular, ln(1 − |R_{±}(k)|^{2}) is integrable over the unit circle.

We proceed with a closer investigation of T (k). Using the Laurent expansion for α(k) one computes

(10.51)  T (k) = A(1 + k ∑_{m∈ℤ} 2b(m) + O(k^{2})).

The only poles of T (k) are at k_{j}, where λ_{j} = (k_{j} + k_{j}^{−1})/2 ∈ σ_{p}(H). The residue can be computed from (10.34) which yields (cf. (2.33))

(10.52)  res_{k_{j}}T (k) = −\frac{k_{j}γ_{+,j}}{μ_{j}} = −k_{j}μ_{j}γ_{−,j},

where

(10.53)  \frac{1}{γ_{±,j}} = ∑_{m∈ℤ} |f_{±}(m, k_{j})|^{2}, f_{−}(n, k_{j}) = μ_{j}f_{+}(n, k_{j}).

The constants γ_{±,j} are usually referred to as norming constants. The sets

(10.54)  S_{±}(H) = {R_{±}(k), |k| = 1; (k_{j}, γ_{±,j}), 1 ≤ j ≤ N}

are called left/right scattering data for H, respectively. We will show that they determine the sequences a, b uniquely in the following sections. As a first step toward this result we note

Lemma 10.7. Either one of the sets S_{±}(H) determines the other and T (k) via (Poisson-Jensen formula)

(10.55) T (k) = ( ∏_{j=1}^{N} \frac{|k_{j}|(k − k_{j}^{−1})}{k − k_{j}}) exp (\frac{1}{4πi} ∫_{|κ|=1}ln(1 − |R_{±}(κ)|^{2})\frac{κ + k}{κ − k} \frac{dκ}{κ}) ,

(for |k| < 1) and

(10.56)  \frac{R_{−}(k)}{R_{+}(\overline{k})} = −\frac{T (k)}{T (\overline{k})}, γ_{+,j}γ_{−,j} = (\frac{res_{k_{j}}T (k)}{k_{j}})^{2}.

Proof. It suffices to prove the formula for T (k) since evaluating the residues provides μ_{j} and thus γ_{∓,j} = μ_{j}^{∓2}γ_{±,j}. For this purpose consider

(10.57)  t(k) = ∏_{j=1}^{N} \frac{k − k_{j}}{kk_{j} − 1}T (k)

which is holomorphic for |k| < 1 and has neither poles nor zeros inside the unit disk. In addition, t(k) is continuous for |k| ≤ 1 and has at most simple zeros at k = ±1. Applying the residue theorem in a circle with radius r < 1 we obtain

\frac{1}{2πi} ∫_{|κ|=r} ln(t(κ))\frac{κ + k}{κ − k} \frac{dκ}{κ} = 2 ln(t(k)) − ln(t(0)),
(10.58)  \frac{1}{2πi} ∫_{|κ|=r^{−1}} ln(t(κ^{−1}))\frac{κ + k}{κ − k} \frac{dκ}{κ} = ln(t(0)).

Letting r → 1 (dominated convergence), using \overline{t(k)} = t(k^{−1}), |k| = 1, and adding up the resulting equations yields

(10.59)  \frac{1}{2πi} ∫_{|κ|=1} ln(|t(κ)|^{2})\frac{κ + k}{κ − k} \frac{dκ}{κ} = 2 ln(t(k)) mod 2πi,

where ln(|t(κ)|^{2}) ∈ ℝ. Thus we conclude

(10.60)  t(k)^{2} = exp (\frac{1}{2πi} ∫_{|κ|=1} ln(|t(κ)|^{2})\frac{κ + k}{κ − k} \frac{dκ}{κ}) .

This proves the claim upon observing |t(k)| = |T (k)|, |k| = 1, and T (0) > 0. □

Comparing coefficients in the asymptotic expansions of (10.55) and (10.36) gives another trace formula

(10.61)  K̃_{m} = \frac{−1}{π} ∫_{0}^{π}ln(1 − |R_{±}(e^{iφ}, t)|^{2}) cos(mφ)dφ + ∑_{j=1}^{N} \frac{k_{j}^{m} − k_{j}^{−m}}{m} ,

where K̃_{m} = K_{m} − ∑_{j=1}^{m−1} \frac{j}{m} K̃_{m−j}K_{j} are the expansion coefficients of

(10.62)  ln α(k) = − ln A + ∑_{j=1}^{∞} K̃_{j}k^{j}.

Moreover, expanding ln α(k(z)) one can express the traces tr(H^{ℓ} − (H_{0})^{ℓ}) in terms of the coefficients K_{m}, for instance,

tr(H − (H_{0})) = −\frac{1}{2}K_{1},
tr(H^{2} − (H_{0})^{2}) = −\frac{1}{16}(2K_{2} + K_{1}^{2}),

etc. . (10.63)

10.3. The Gel’fand-Levitan-Marchenko equations

Now we want to set up a procedure which allows the reconstruction of H from its scattering data S_{±}(H). This will be done by deriving an equation for K_{±}(n, m). In addition, the decay condition (H.10.1) is reflected by the estimates (10.10) and (10.15). These turn out to be equivalent to the corresponding estimates for K_{±}(n, m) and will lead back to (H.10.1) when reconstructing H from S_{±}(H).

Since K_{+}(n, m) are essentially the Fourier coefficients of f_{+}(k, n) we compute the Fourier coefficients of (10.27). Therefore we multiply

(10.64)  T (k)f_{−}(k, n) = (f_{+}(k^{−1}, n) + R_{+}(k)f_{+}(k, n))

by (2πi)^{−1}A_{+}(n)k^{n+j−1} (j ∈ ℕ_{0}) and integrate around the unit circle. We first evaluate the right hand side

\frac{A_{+}(n)}{2πi} ∫_{|k|=1} f_{+}(k^{−1}, n)k^{n+j} \frac{dk}{k} = K_{+}(n, n + j),

\frac{A_{+}(n)}{2πi} ∫_{|k|=1}R_{+}(k)f_{+}(k, n)k^{n+j} \frac{dk}{k} = ∑_{m=0}^{∞} K_{+}(n, n + m)F̃^{+}(2n + m + j),

(10.65)

where

(10.66)  F̃^{+}(n) = \frac{1}{2πi} ∫_{|k|=1} R_{+}(k)k^{n} \frac{dk}{k} ∈ ℓ^{2}(ℤ, ℝ)

are the Fourier coefficients of R_{+}(k^{−1}). By Parseval’s identity we have

(10.67)  ∑_{n∈ℤ} F̃^{+}(n)^{2} = \frac{1}{2πi} ∫_{|k|=1} |R_{+}(k)|^{2} \frac{dk}{k} < 1.

Next we evaluate the left hand side using the residue theorem (take a contour inside the unit disk enclosing all poles and let this contour approach the unit circle)

\frac{A_{+}(n)}{2πi} ∫_{|k|=1} T (k)f_{−}(k, n)k^{n+j} \frac{dk}{k}

= −A_{+}(n) ∑_{ℓ=1}^{N} \frac{k_{ℓ}γ_{+,ℓ}}{μ_{ℓ}} f_{−}(k_{ℓ}, n)k_{ℓ}^{n+j−1} + A_{+}(n)\frac{T (0)}{A_{−}(n)}δ_{0}(j)

= −A_{+}(n) ∑_{ℓ=1}^{N} γ_{+,ℓ}f_{+}(k_{ℓ}, n)k_{ℓ}^{n+j} + A_{+}(n)^{2}δ_{0}(j)

= − ∑_{m=0}^{∞} K_{+}(n, n + m) ∑_{ℓ=1}^{N} γ_{+,ℓ}k_{ℓ}^{2n+m+j} + A_{+}(n)^{2}δ_{0}(j). (10.68)

Defining the Gel’fand-Levitan-Marchenko operator

(10.69)  F_{n}^{+}f(j) = ∑_{m=0}^{∞} F^{+}(2n + m + j)f(m), f ∈ ℓ^{2}(ℕ_{0}),

where

(10.70)  F^{+}(n) = F̃^{+}(n) + ∑_{ℓ=1}^{N} γ_{+,ℓ}k_{ℓ}^{n},

leads us to the Gel’fand-Levitan-Marchenko equation

(10.71)  (1l + F_{n}^{+})K_{+}(n, n + .) = A_{+}(n)^{2}δ_{0}.

Similarly, multiplying

(10.72)  T (k)f_{+}(k, n) = (f_{−}(k^{−1}, n) + R_{−}(k)f_{−}(k, n))

by (2πi)^{−1}A_{−}(n)k^{−n+j−1} (j ∈ ℕ_{0}) and integrating around the unit circle we get for K_{−}(n, m)

(10.73)  (1l + F_{n}^{−})K_{−}(n, n + .) = A_{−}(n)^{2}δ_{0},

where

(10.74)  F_{n}^{−}f(j) = ∑_{m=0}^{∞} F^{−}(2n − m − j)f(m), f ∈ ℓ^{2}(ℕ_{0}),

and

(10.75)  F^{−}(n) = \frac{1}{2πi} ∫_{|k|=1} R_{−}(k)k^{−n} \frac{dk}{k} + ∑_{ℓ=1}^{N} γ_{−,ℓ}k_{ℓ}^{−n}.

Our next objective is to consider the Gel’fand-Levitan-Marchenko operator F_{n}^{+} a little closer. The structure of the Gel’fand-Levitan-Marchenko equation

K_{+}(n, n + j) + F^{+}(2n + j) + ∑_{ℓ=1}^{∞} F^{+}(2n + ℓ + j)K_{+}(n, n + ℓ)

= A_{+}(n)δ_{0}(j) (10.76)

suggests that the estimates (10.10) and (10.15) for K_{+}(n, n+j) should imply similar estimates for F^{+}(2n + j). This will be shown first.

As a warmup we prove a discrete Gronwall-type inequality.

Lemma 10.8. Suppose C_{1}, C_{2}, f ∈ ℓ_{+}^{∞}(ℤ, [0, ∞)), C_{3} ∈ ℓ_{+}^{1}(ℤ, [0, ∞)), C_{1}C_{3} < 1, and

(10.77)  f(n) ≤ C_{1}(n)(C_{2}(n) + ∑_{j=n}^{∞} C_{3}(j)f(j)).

Then we have

(10.78)  f(n) ≤ C_{1}(n)(C_{2}(n) + ∑_{j=n}^{∞} \frac{C_{1}(j)C_{2}(j)C_{3}(j)}{∏_{ℓ=n}^{j}(1 − C_{1}(ℓ)C_{3}(ℓ))}).

Proof. Using g(n) = ∑_{j=n}^{∞} C_{3}(j)f(j) one computes

(10.79)  g(n) − g(n + 1) ≤ C_{3}(n)C_{1}(n)(C_{2}(n) + g(n))

and hence (solving for g(n))

(10.80)  g(n) ≤ ∑_{j=n}^{∞} \frac{C_{1}(j)C_{2}(j)C_{3}(j)}{∏_{ℓ=n}^{j}(1 − C_{1}(ℓ)C_{3}(ℓ))}.

The rest follows from f(n) ≤ C_{1}(n)(C_{2}(n) + g(n)). □

Solving the Gel’fand-Levitan-Marchenko equation (10.76) for F_{j}(n) = |F^{+}(2n+ j)|, j = 1, 2, and using the estimate (10.10) we obtain

F (n) ≤ D_{+}(n)(2C_{+}(n + 1) + ∑_{ℓ=1}^{∞}(|F^{+}(2n + ℓ + 1)| + |F^{+}(2n + ℓ + 2)|)

×C_{+}(n + [[\frac{ℓ}{2}]] + 1))

≤ D_{+}(n)(2C_{+}(n + 1) + ∑_{ℓ=0}^{∞} |F^{+}(2n + ℓ + 2)|C_{+}(n + k + 1)

+ 2 ∑_{ℓ=1}^{∞} |F^{+}(2n + ℓ + 1)|C_{+}(n + k) + ∑_{ℓ=1}^{∞} |F^{+}(2n + ℓ + 2)|C_{+}(n + k)

(10.81) ≤ 2D_{+}(n)(C_{+}(n + 1) + ∑_{ℓ=n}^{∞} C_{+}(ℓ)F (ℓ)),

where F (n) = F_{1}(n) + F_{2}(n). And hence by the lemma

|F^{+}(2n + 1)| + |F^{+}(2n + 2)| ≤ 2D_{+}(n)C_{+}(n + 1)

×(1 + ∑_{j=n}^{∞} \frac{2D_{+}(j)C_{+}(j)}{∏_{ℓ=n}^{j}(1 − 2D_{+}(ℓ)C_{+}(ℓ))}). (10.82)

for n large enough. In summary we have

(10.83)  |F^{+}(2n + j)| ≤ O(1)C_{+}(n + [[\frac{j}{2}]] + 1),

where O(1) are terms of order zero as n → ∞. Next we want to derive an estimate for ∂F^{+}(2 · +j), j = 1, 2. Set G_{j}(n) = |F^{+}(2n + j) − F^{+}(2n + j + 2)|, j = 1, 2, then

G_{1}(n) ≤ |K_{+,1}(n) − K_{1,+}(n + 1)|

+ ∑_{j=1}^{∞} |F^{+}(2n + j + 1)K_{+,j}(n) − F^{+}(2n + j + 3)K_{+,j}(n + 1)|

≤ c(n + 1) + ∑_{j=1}^{∞} |F^{+}(2n + j + 1)||K_{+,j}(n) − K_{+,j}(n + 1)|

+ ∑_{j=1}^{∞} |F^{+}(2n + j + 1) − F^{+}(2n + j + 3)||K_{+,j}(n + 1)|

≤ c(n + 1)

+ O(1) ∑_{j=1}^{∞} C_{+}(n + [[\frac{j}{2}]] + 1)(C_{+}(n + 1)C_{+}(n + [[\frac{j+1}{2}]]) + c(n + [[\frac{j}{2}]]))

+ O(1) ∑_{j=1}^{∞} |F^{+}(2n + j + 1) − F^{+}(2n + j + 3)|C_{+}(n + [[\frac{j}{2}]] + 2)

(10.84) ≤ O(1)(c̃(n) + ∑_{j=n}^{∞}(G_{1}(j) + G_{2}(j))C_{+}(j + 1)),

where c̃(n) = c(n + 1) + C_{+}(n)C_{+}(n + 1). Similarly, we obtain

(10.85)  G_{2}(n) ≤ O(1)(c̃(n) + ∑_{j=n}^{∞}(G_{1}(j) + G_{2}(j))C_{+}(j + 1)).

Adding both equations gives an estimate for G(n) = G_{1}(n) + G_{2}(n) and again by the lemma

(10.86)  G(n) ≤ O(1)(c̃(n) + ∑_{j=n}^{∞} c̃(j)C_{+}(j + 1)).

As a consequence we infer

(10.87)  nG(n) ≤ O(1)(nc̃(n) + C_{+}(n) ∑_{j=n}^{∞} jc̃(j)) ∈ ℓ^{1}(ℕ).

Similar results hold for F^{−}. Let us summarize the conditions satisfied by S_{±}(H).

Hypothesis H. 10.9. The scattering data

(10.88)  S_{±}(H) = {R_{±}(k), |k| = 1; k_{j}, γ_{±,j}, 1 ≤ j ≤ N}

satisfy the following conditions.

(i). The consistency condition

(10.89)  \frac{R_{−}(k)}{R_{+}(\overline{k})} = −\frac{T (k)}{T (\overline{k})}, γ_{+,j}γ_{−,j} = (\frac{res_{k_{j}}T (k)}{k_{j}})^{2},

where T (k) is defined using (10.55).

(ii). The reflection coefficients R_{±}(k) are continuous except possibly at k = ±1 and fulfill

(10.90)  \overline{R_{±}(k)} = R_{±}(\overline{k}),  C^{2}|1 − k^{2}|^{2} ≤ 1 − |R_{±}(k)|^{2}

for some C > 0. The Fourier coefficients F̃^{±} of R_{±}(k^{∓1}) satisfy

(10.91)  ∑_{j=1}^{∞} j|F̃^{±}(±j) − F̃^{±}(±j ± 2)| < ∞.

(iii). The values k_{j} ∈ (−1, 0) ∪ (0, 1), 1 ≤ j ≤ N, are distinct and the norming constants γ_{±,j}, 1 ≤ j ≤ N, are positive.

Finally, we collect some properties of the operator F_{n}^{+}.

Theorem 10.10. Fix n ∈ ℤ and consider F_{n}^{+} : ℓ^{p}(ℕ_{0}) → ℓ^{p}(ℕ_{0}), p ∈ [1, ∞)∪{∞}, as above. Then F_{n}^{+} is a compact operator with norm

(10.92)  ∥F_{n}^{+}∥_{p} ≤ ∑_{m=2n}^{∞} |F^{+}(m)|.

For p = 2, F_{n}^{+} is a self-adjoint trace class operator satisfying

(10.93)  1l + F_{n}^{+} ≥ ε_{n} > 0, lim_{n→∞} ε_{n} = 1.

The trace of F_{n}^{+} is given by

(10.94)  tr(F_{n}^{+}) = ∑_{j=0}^{∞} F^{+}(2n + 2j) + ∑_{ℓ=1}^{N} γ_{+,ℓ} \frac{k_{ℓ}^{2n}}{1 − k_{ℓ}} .

Proof. The inequality for the norm of F_{n}^{+} is a special case of Young’s inequality. Moreover, cutting off F^{+} after finitely many terms gives a sequences of finite rank operators which converge to F_{n}^{+} in norm; hence F_{n}^{+} is compact.

Let f ∈ ℓ^{2}(ℕ_{0}) and abbreviate f̂(k) = ∑_{j=0}^{∞} f(j)k^{j}. Setting f(j) = 0 for j < 0 we obtain

(10.95) ∑_{j=0}^{∞} \overline{f(j)}F_{n}^{+}f(j) = \frac{1}{2πi} ∫_{|k|=1}R_{+}(k)k^{2n}|f̂(k)|^{2} \frac{dk}{k} + ∑_{ℓ=1}^{N} γ_{+,ℓ}k_{ℓ}^{2n}|f̂(k_{ℓ})|^{2}

from the convolution formula. Since \overline{R_{+}(k)} = R_{+}(\overline{k}) the integral over the imaginary part vanishes and the real part can be replaced by

Re(R_{+}(k)k^{2n}) = \frac{1}{2} (|1 + R_{+}(k)k^{2n}|^{2} − 1 − |R_{+}(k)k^{2n}|^{2})
(10.96)  = \frac{1}{2} (|1 + R_{+}(k)k^{2n}|^{2} + |T (k)|^{2}) − 1

(remember |T (k)|^{2} + |R_{+}(k)k^{2n}|^{2} = 1). This eventually yields the identity

∑_{j=0}^{∞} \overline{f(j)}(1l + F_{n}^{+})f(j) = ∑_{ℓ=1}^{N} γ_{+,ℓ}k_{ℓ}^{2n}|f̂(k_{ℓ})|^{2}

+ \frac{1}{4πi} ∫_{|k|=1} (|1 + R_{+}(k)k^{2n}|^{2} + |T (k)|^{2})|f̂(k)|^{2} \frac{dk}{k} , (10.97)

which establishes 1l+F_{n}^{+} ≥ 0. In addition, by virtue of |1+R_{+}(k)k^{2n}|^{2}+|T (k)|^{2} > 0 (a.e.), −1 is no eigenvalue and thus 1l + F_{n}^{+} ≥ ε_{n} for some ε_{n} > 0. That ε_{n} → 1 follows from ∥F_{n}^{+}∥ → 0.

To see that F_{n}^{+} is trace class we use the splitting F_{n}^{+} = F̃_{n}^{+} + ∑_{ℓ=1}^{N} F̃_{n}^{+,ℓ} according to (10.70). The operators F̃_{n}^{+,ℓ} are positive and trace class. The operator F̃_{n}^{+} is given by multiplication with k^{2n}R_{+}(k) in Fourier space and hence is trace class since |R_{+}(k)| is integrable. □

A similar result holds for F_{n}^{−}.

10.4. Inverse scattering theory

In this section we want to invert the process of scattering theory. Clearly, if S_{+}(H) (and thus F_{n}^{+}) is given, we can use the Gel’fand-Levitan-Marchenko equation (10.71) to reconstruct a(n), b(n) from F_{n}^{+}

a(n)^{2} = \frac{1}{4} \frac{⟨δ_{0}, (1l + F_{n}^{+})^{−1}δ_{0}⟩}{⟨δ_{0}, (1l + F_{n+1}^{+})^{−1}δ_{0}⟩},

(10.98)  b(n) = \frac{1}{2}(\frac{⟨δ_{1}, (1l + F_{n}^{+})^{−1}δ_{0}⟩}{⟨δ_{0}, (1l + F_{n}^{+})^{−1}δ_{0}⟩} − \frac{⟨δ_{1}, (1l + F_{n−1}^{+})^{−1}δ_{0}⟩}{⟨δ_{0}, (1l + F_{n−1}^{+})^{−1}δ_{0}⟩}).

In other words, the scattering data S_{+}(H) uniquely determine a, b. Since F_{n}^{+} is trace class we can use Kramer’s rule to express the above scalar products. If we delete the first row and first column in the matrix representation of 1l+F_{n}^{+} we obtain 1l+F_{n+1}^{+}. If we delete the first row and second column in the matrix representation of 1l + F_{n}^{+} we obtain an operator 1l + G_{n}^{+}. By Kramer’s rule we have

⟨δ_{0}, (1l + F_{n}^{+})^{−1}δ_{0}⟩ = \frac{det(1l + F_{n+1}^{+})}{det(1l + F_{n}^{+})},
(10.99)  ⟨δ_{1}, (1l + F_{n}^{+})^{−1}δ_{0}⟩ = \frac{det(1l + G_{n}^{+})}{det(1l + F_{n}^{+})},

where the determinants have to be interpreted as Fredholm determinants.

These formulas can even be used for practical computations. Let F_{n}^{N}, G_{n}^{N} be the finite rank operators obtained from F_{n}^{+}, G_{n}^{+}, respectively, by setting F^{+}(2n + j) = 0 for j > N. Then we have F_{n}^{N} → F_{n}^{+}, G_{n}^{N} → G_{n}^{+} in norm and hence also (1l + F_{n}^{N})^{−1} → (1l + F_{n}^{+})^{−1} as N → ∞. Furthermore,

⟨δ_{0}, (1l + F_{n}^{+})^{−1}δ_{0}⟩ = lim_{N→∞}⟨δ_{0}, (1l + F_{n}^{N})^{−1}δ_{0}⟩
= lim_{N→∞} \frac{det(1l + F_{n+1}^{N})}{det(1l + F_{n}^{N})},
⟨δ_{1}, (1l + F_{n}^{+})^{−1}δ_{0}⟩ = lim_{N→∞}⟨δ_{1}, (1l + F_{n}^{N})^{−1}δ_{0}⟩
(10.100)  = lim_{N→∞} \frac{det(1l + G_{n}^{N})}{det(1l + F_{n}^{N})},

where

F_{n}^{N} = ( \begin{matrix}F^{+}(2n + r + s), & r + s ≤ N \\ 0, & r + s > N \end{matrix} )_{0≤r,s} ,

(10.101)  G_{n}^{N} = ( \begin{matrix}F^{+}(2n + s + 1) − δ_{0,s}, & r = 0 \\ F^{+}(2n + r + s + 2), & r > 0, r + s ≤ N \\ 0, & r + s > N \end{matrix} )_{0≤r,s} .

For this procedure it is clearly interesting to know when given sets S_{±} are the scattering data of some operator H. In the previous section we have seen that Hypothesis 10.9 is necessary. In this section we will show that it is also sufficient for S_{±} to be the scattering data of some Jacobi operator H. We prove a preliminary lemma first.

Lemma 10.11. Suppose a given set S_{+} satisfies (H.10.9) (ii) and (iii). Then the sequences a_{+}(n), b_{+}(n) defined as in (10.98) satisfy n|2a_{+}(n) − 1|, n|b_{+}(n)| ∈ ℓ^{1}(ℕ). In addition, f_{+}(k, n) = A_{+}(n)^{−1} ∑_{m=n}^{∞} K_{+}(n, m)k^{m}, where K_{+}(n, m) = A_{+}(n)^{−2}⟨δ_{m}, (1l + F_{n}^{+})^{−1}δ_{0}⟩ and A_{+}(n)^{2} = ⟨δ_{0}, (1l + F_{n}^{+})^{−1}δ_{0}⟩, satisfies τ_{+}f = \frac{k+k^{−1}}{2} f.

Proof. We abbreviate

(10.102)  C(n) = ∑_{j=0}^{∞} |F^{+}(n + j) − F^{+}(n + j + 2)|, C_{1}(n) = ∑_{j=n}^{∞} C(j)

182 10. Scattering theory and observe C(n + 1) ≤ C(n) and

|F^{+}(n)| + |F^{+}(n + 1)| ≤ ∑_{j=0}^{∞} |F^{+}(n + 2j) − F^{+}(n + 2j + 2)|

+ ∑_{j=0}^{∞} |F^{+}(n + 2j + 1) − F^{+}(n + 2j + 3)| = C(n). (10.103)

Moreover, let K̃_{+,j}(n) = ⟨δ_{j}, (1l + F_{n}^{+})^{−1}δ_{0}⟩ − δ_{0,j} = A_{+}(n)^{2}K_{+}(n, n + j) − δ_{0,j}. Then

(10.104) K̃_{+,j}(n) + F^{+}(2n + j) + ∑_{ℓ=0}^{∞} K̃_{+,ℓ}(n)F^{+}(2n + ℓ + j) = 0

or equivalently K̃_{+,j}(n) = (1l + F_{n}^{+})^{−1}F^{+}(2n + j). Using

(10.105) ∑_{ℓ=0}^{∞} |K̃_{+,ℓ}(n)| ≤ N(n)C_{1}(2n), N(n) = ∥(1l + F_{n}^{+})^{−1}∥_{1},

we obtain

|K̃_{+,j}(n)| ≤ C(2n + j) + C(2n + j) ∑_{ℓ=0}^{∞} |K̃_{+,ℓ}(n)|
(10.106)  ≤ C(2n + j)(1 + N(n)C_{1}(2n)).

Taking differences in (10.104) we see

|K̃_{+,j}(n) − K̃_{+,j}(n + 1)| ≤ |F^{+}(2n + j) − F^{+}(2n + j + 2)|

+ ∑_{ℓ=0}^{∞} (|F^{+}(2n + j + ℓ) − F^{+}(2n + j + ℓ + 2)||K̃_{+,ℓ}(n)|

+ |F^{+}(2n + j + ℓ + 2)||K̃_{+,ℓ}(n) − K̃_{+,ℓ}(n + 1)|) (10.107) Using

(1l + F_{n}^{+})(K̃_{+,j}(n) − K̃_{+,j}(n + 1)) = F^{+}(2n + j + 2) − F^{+}(2n + j)

+ ∑_{ℓ=0}^{∞}(F^{+}(2n + j + ℓ + 2) − F^{+}(2n + j + ℓ))K̃_{+,ℓ}(n + 1) (10.108)

and hence

∑_{j=0}^{∞} |K̃_{+,j}(n) − K̃_{+,j}(n + 1)| ≤ N(n) ∑_{j=1}^{∞} (|F^{+}(2n + j + 2) − F^{+}(2n + j)|

+ ∑_{ℓ=0}^{∞} |F^{+}(2n + j + ℓ + 2) − F^{+}(2n + j + ℓ)||K̃_{+,ℓ}(n + 1)|)

≤ O(1)C(2n) (10.109) we finally infer

|K̃_{+,j}(n) − K̃_{+,j}(n + 1)| ≤ |F^{+}(2n + j) − F^{+}(2n + j + 2)|

+ O(1)C(2n)C(2n + j). (10.110) Thus n|K̃_{+,j}(n)−K̃_{+,j}(n+1)| ∈ ℓ^{1}(ℕ) and the same is true for 4a_{+}(n)^{2} −1, b_{+}(n).

Next we turn to f_{+}(k, n). Abbreviate

(ΔK)(n, m) = 4a_{+}(n)^{2}K(n + 1, m) + K(n − 1, m) + 2b_{+}(n)K(n, m)

− K(n, m + 1) − K(n, m − 1), (10.111)

then f_{+}(k, n) satisfies τ_{+}f = \frac{k+k^{−1}}{2} f if ΔK_{+} = 0 (cf. (10.8)). We have K_{+}(n, n) = 1 and K_{+}(n, n + 1) = − ∑_{m=n+1}^{∞} 2b_{+}(m) by construction. Moreover, using the Gel’fand-Levitan-Marchenko equation

(10.112)  K_{+}(n, m) = −F^{+}(n + m) − ∑_{ℓ=n+1}^{∞} K_{+}(n, ℓ)F^{+}(ℓ + m)

one checks

(10.113)  (ΔK_{+})(n, m) = − ∑_{ℓ=n+1}^{∞}(ΔK_{+})(n, ℓ)F^{+}(ℓ + m), m > n + 1.

By Lemma 7.8 this equation has only the trivial solution ΔK_{+} = 0 and hence the proof is complete. □

A corresponding result holds for given data S_{−}. Now we can prove the main result of this section.

Theorem 10.12. Hypothesis (H.10.9) is necessary and sufficient for two sets S_{±} to be the left/right scattering data of a unique Jacobi operator H associated with sequences a, b satisfying (H.10.1).

Proof. Necessity has been established in the previous section. Moreover, by the previous lemma we know existence of sequences a_{±}, b_{±} and corresponding solutions f_{±}(k, n) associated with S_{±}, respectively. Hence it remains to establish the relation between these two parts. The key will be the equation T (k)f_{∓}(k, n) = f_{±}(k^{−1}, n)+ R_{±}(k)f_{±}(k, n) and the consistency condition.

We start by defining

(10.114)  ϕ_{+,j}(n) = \frac{1}{A_{+}(n)} ∑_{m=0}^{∞} K_{+,m}(n)F̃^{+}(2n + m + j)

implying

(10.115)  ∑_{j=−∞}^{∞} ϕ_{+,j}(n)k^{−n−j} = R_{+}(k)f_{+}(k, n).

Using the Gel’fand-Levitan-Marchenko equation we further infer

ϕ_{+,j}(n) = A_{+}(n)δ_{0}(j) − \frac{1}{A_{+}(n)}(K_{+,j}(n) + ∑_{m=0}^{∞} K_{+,m}(n) ∑_{ℓ=0}^{N} γ_{+,ℓ}k_{ℓ}^{2n+m+j})

(10.116) = A_{+}(n)δ_{0}(j) − \frac{K_{+,j}(n)}{A_{+}(n)} − ∑_{ℓ=1}^{N} γ_{+,ℓ}f_{+}(k_{ℓ}, n)k_{ℓ}^{n+j}.

Next, computing the Fourier series we see

R_{+}(k)f_{+}(k, n) = ∑_{j=1}^{∞} ϕ_{+,−j}(n)k^{−n+j} + A_{+}(n)k^{−n} − f_{+}(k^{−1}, n)

− ∑_{ℓ=1}^{N} γ_{+,ℓ}f_{+}(k_{ℓ}, n)\frac{k_{ℓ}^{n}k^{−n}}{1 − k_{ℓ}k^{−1}} (10.117)

and hence

(10.118)  T (k)h_{−}(k, n) = f_{+}(k^{−1}, n) + R_{+}(k)f_{+}(k, n),

where

(10.119) h_{−}(k, n) = \frac{k^{−n}}{T (k)}(A_{+}(n) + ∑_{j=1}^{∞} ϕ_{+,−j}(n)k^{j} − ∑_{ℓ=1}^{N} γ_{+,ℓ}f_{+}(k_{ℓ}, n)\frac{k_{ℓ}^{n}k}{k − k_{ℓ}} ).

Similarly, one obtains

(10.120) h_{+}(k, n) = \frac{k^{n}}{T (k)}(A_{−}(n) + ∑_{j=1}^{∞} ϕ_{−,j}(n)k^{j} − ∑_{ℓ=1}^{N} γ_{−,ℓ}f_{−}(k_{ℓ}, n)\frac{k_{ℓ}^{−n}k}{k − k_{ℓ}} ),

where

(10.121)  ϕ_{−,j}(n) = \frac{1}{A_{−}(n)} ∑_{m=0}^{∞} K_{−,m}(n)F̃^{+}(2n − m − j).

In particular, we see that k^{∓n}h_{±}(k, n) are holomorphic inside the unit circle. In addition, by virtue of the consistency condition (H.10.9)(i) we obtain

(10.122)  h_{±}(k_{ℓ}, n) = (res_{k_{j}}T (k))^{−1}γ_{∓,ℓ}k_{ℓ}f_{∓}(k_{ℓ}, n)

and (use also |T (k)|^{2} + |R_{±}(k)|^{2} = 1)

(10.123)  T (k)f_{∓}(k, n) = h_{±}(k^{−1}, n) + R_{±}(k)h_{±}(k, n).

Eliminating (e.g.) R_{−}(k) from the last equation and T (k)h_{+}(k, n) = f_{−}(k^{−1}, n) + R_{−}(k)f_{−}(k, n) we see

f_{−}(\overline{k}, n)h_{−}(k, n) − f_{−}(k, n)h_{−}(\overline{k}, n) =
(10.124) T (k)(h_{−}(k, n)h_{+}(k, n) − f_{−}(k, n)f_{+}(k, n)) = G(k, n), |k| = 1.

Equation (10.122) shows that all poles of G(k, n) = ∑_{j∈ℤ} G_{j}k^{j} inside the unit circle are removable and hence G_{j} = 0, j < 0. Moreover, by G(\overline{k}, n) = −G(k, n) we see G_{−j} = −G_{j} and thus G(k, n) = 0. If we compute 0 = lim_{k→0} h_{−}(k, n)h_{+}(k, n) − f_{−}(k, n)f_{+}(k, n) we infer A_{−}(n)A_{+}(n) = T (0) and thus a_{−}(n) = a_{+}(n) ≡ a(n).

Our next aim is to show h_{±}(k, n) = f_{±}(k, n). To do this we consider the function

(10.125)  H(k, n) = ∑_{j∈ℤ} H_{j}k^{j} = \frac{h_{−}(k, n)}{f_{−}(k, n)} = \frac{f_{+}(k, n)}{h_{+}(k, n)}, |k| ≤ 1,

which is meromorphic inside the unit disk with H_{0} = 1. However, if we choose −n large enough, then k^{n}f_{−}(k, n) cannot vanish and hence H(k, n) is holomorphic inside the unit disc implying H_{j} = 0, j < 0. And since H(\overline{k}, n) = H(k, n), |k| = 1, we even have H_{j} = 0, j > 0, that is, H(k, n) = 1. Similarly, considering H(k, n)^{−1}, we see H(k, n) = 1 for +n large enough.

Thus we know h_{±}(k, n) = f_{±}(k, n), |k| ≤ 1, at least for |n| large enough. In particular, h_{±}(k, n) ∈ ℓ_{±}^{2}(ℤ), |k| < 1, and hence h_{+}(k, n), f_{−}(k, n) and f_{+}(k, n), h_{−}(k, n) can be used to compute the Weyl m-functions of H_{−} and H_{+}, respectively. Since the Weyl m-functions of H_{−} and H_{+} are equal for some n, we conclude H_{−} = H_{+} ≡ H (i.e., b_{−}(n) = b_{+}(n) ≡ b(n)).

Up to this point we have constructed an operator H which has the correct reflection coefficients and the correct eigenvalues. To finish the proof we need to show that our operator also has the correct norming constants. This follows from (cf. (10.34))

(10.126)  ∑_{n∈ℤ} f_{−}(k_{ℓ}, n)f_{+}(k_{ℓ}, n) = k_{ℓ}(res_{k_{j}}T (k))^{−1}

and (10.122). □

Spectral deformations – Commutation methods

11.1. Commuting first order difference expressions

The idea of this section is to factor a given Jacobi operator into the product of two simpler operators, that is, two operators associated with first order difference expressions. Since it is not clear how to do this, we will first investigate the other direction, that is, we take a first order difference expression and multiply it with its adjoint. The main reason why we take the adjoint as second operator is, since this choice ensures self-adjointness of the resulting product (i.e., we get a Jacobi operator). In addition, changing the order of multiplication produces a second operator whose spectral properties are closely related to those of the first. In fact, we have the well-known result by von Neumann (see, e.g., [241], Theorem 5.39 and [54], Theorem 1).

Theorem 11.1. Suppose A is a bounded operator with adjoint A^{∗}. Then A A^{∗} and A^{∗}A are non-negative self-adjoint operators which are unitarily equivalent when restricted to the orthogonal complements of their corresponding null-spaces. Moreover, the resolvents of A A^{∗} and A^{∗}A for z ∈ ℂ\(σ(A^{∗}A) ∪ {0}) are related by

(A A^{∗} − z)^{−1} = \frac{1}{z}(A(A^{∗}A − z)^{−1}A^{∗} − 1),
(11.1)  (A^{∗}A − z)^{−1} = \frac{1}{z}(A^{∗}(A A^{∗} − z)^{−1}A − 1).

Let ρ_{o}, ρ_{e} be two real-valued sequences fulfilling

(11.2)  ρ_{o}, ρ_{e} ∈ ℓ^{∞}(ℤ),  −ρ_{o}(n), ρ_{e}(n) > 0.

The second requirement is for convenience only (it will imply a_{j}(n) < 0 below). Now consider the (bounded) operators

(11.3)  \begin{aligned}A & ℓ^{2}(ℤ) & → & ℓ^{2}(ℤ) \\ & f(n) & ↦ & ρ_{o}(n)f(n + 1) + ρ_{e}(n)f(n), \\ A^{∗} & ℓ^{2}(ℤ) & → & ℓ^{2}(ℤ) \\ & f(n) & ↦ & ρ_{o}(n − 1)f(n − 1) + ρ_{e}(n)f(n), \end{aligned}

where A^{∗} is the adjoint operator of A. Using them we can construct two nonnegative operators H_{1} = A^{∗}A and H_{2} = AA^{∗}. A direct calculation shows

(11.4)  H_{j}f(n) = τ_{j}f(n) = a_{j}(n)f(n + 1) + a_{j}(n − 1)f(n − 1) + b_{j}(n)f(n),

with

(11.5)  \begin{matrix}a_{1}(n) = ρ_{o}(n)ρ_{e}(n) & b_{1}(n) = ρ_{o}(n − 1)^{2} + ρ_{e}(n)^{2} \\ a_{2}(n) = ρ_{o}(n)ρ_{e}(n + 1) & b_{2}(n) = ρ_{o}(n)^{2} + ρ_{e}(n)^{2} \end{matrix} .

Due to (11.2) we have a_{j}, b_{j} ∈ ℓ^{∞}(ℤ) and a_{j}(n) < 0, b_{j}(n) > 0. Next, observe that the quantities

(11.6)  ϕ_{1}(n) = −\frac{ρ_{e}(n)}{ρ_{o}(n)},  ϕ_{2}(n) = −\frac{ρ_{o}(n)}{ρ_{e}(n + 1)}

satisfy

(11.7)  a_{j}(n)ϕ_{j}(n) + \frac{a_{j}(n − 1)}{ϕ_{j}(n − 1)} = −b_{j}(n).

Hence our approach gives us positive solutions

(11.8)  τ_{j}u_{j} = 0,  u_{j}(n) = \begin{matrix}n−1 \\ ∏ & ∗ \\ m=n_{0} \end{matrix} ϕ_{j}(m) > 0

for free. Moreover, it shows that a given operator H_{1} can only be factorized in this manner if a positive solution u_{1} of τ_{1}u_{1} = 0 exists. On the other hand, since we have

ρ_{o}(n) = −\sqrt{−\frac{a_{1}(n)}{ϕ_{1}(n)}} = −\sqrt{−a_{2}(n)ϕ_{2}(n)},

(11.9)  ρ_{e}(n) = \sqrt{−a_{1}(n)ϕ_{1}(n)} = \sqrt{−\frac{a_{2}(n − 1)}{ϕ_{2}(n − 1)}},

a positive solution is the only necessary ingredient for such a factorization. Together with Section 2.3, where we have seen that positive solutions exist if H_{j} ≥ 0 (see also Remark 2.10) we obtain

Corollary 11.2. Suppose a(n) < 0. Then a Jacobi operator is non-negative, H ≥ 0, if and only if there exists a positive solution of the corresponding Jacobi equation.

11.2. The single commutation method

In this section we invert the process of the last section. This will result in a method for inserting eigenvalues below the spectrum of a given Jacobi operator H. It is usually referred to as single commutation method and a detailed study is the topic of this and the next section.

To assure existence of the aforementioned necessary positive solutions, we will need a < 0 throughout Sections 11.2 and 11.3.

Hypothesis H. 11.3. Suppose a, b ∈ ℓ^{∞}(ℤ, ℝ) and a(n) < 0.

Let H be a given Jacobi operator. We pick σ_{1} ∈ [−1, 1] and λ_{1} < inf σ(H). Furthermore, recall the minimal positive solutions u_{±}(λ_{1}, n) constructed in Section 2.3 and set

(11.10)  u_{σ_{1}}(λ_{1}, n) = \frac{1 + σ_{1}}{2} u_{+}(λ_{1}, n) + \frac{1 − σ_{1}}{2} u_{−}(λ_{1}, n).

By the analysis of Section 2.3 (cf. (2.68)), any positive solution can be written in this form up to normalization.

Motivated by (11.9) we consider the sequences

(11.11) ρ_{o,σ_{1}}(n) = −\sqrt{−\frac{a(n)u_{σ_{1}}(λ_{1}, n)}{u_{σ_{1}}(λ_{1}, n + 1)}}, ρ_{e,σ_{1}}(n) = \sqrt{−\frac{a(n)u_{σ_{1}}(λ_{1}, n + 1)}{u_{σ_{1}}(λ_{1}, n)}}

which are bounded because of

(11.12)  |\frac{a(n)u_{σ_{1}}(n + 1)}{u_{σ_{1}}(n)}| + |\frac{a(n − 1)u_{σ_{1}}(n − 1)}{u_{σ_{1}}(n)}| = |λ_{1} − b(n)|.

Thus we can define a corresponding operator A_{σ_{1}} on ℓ^{2}(ℤ) together with its adjoint A_{σ_{1}}^{∗},

A_{σ_{1}}f(n) = ρ_{o,σ_{1}}(n)f(n + 1) + ρ_{e,σ_{1}}(n)f(n),
(11.13)  A_{σ_{1}}^{∗}f(n) = ρ_{o,σ_{1}}(n − 1)f(n − 1) + ρ_{e,σ_{1}}(n)f(n).

For simplicity of notation we will not distinguish between the operator A_{σ_{1}} and its difference expression.

By Theorem 11.1, A_{σ_{1}}^{∗}A_{σ_{1}} is a positive self-adjoint operator. A quick calculation shows

(11.14)  H = A_{σ_{1}}^{∗}A_{σ_{1}} + λ_{1}.

Commuting A_{σ_{1}}^{∗} and A_{σ_{1}} yields a second positive self-adjoint operator A_{σ_{1}}A_{σ_{1}}^{∗} and further the commuted operator

(11.15)  H_{σ_{1}} = A_{σ_{1}}A_{σ_{1}}^{∗} + λ_{1}.

The next theorem characterizes H_{σ_{1}}.

Theorem 11.4. Assume (H.11.3) and let λ_{1} < inf σ(H), σ_{1} ∈ [−1, 1]. Then the self-adjoint operator H_{σ_{1}} is associated with

(11.16)  (τ_{σ_{1}}f)(n) = a_{σ_{1}}(n)f(n + 1) + a_{σ_{1}}(n − 1)f(n − 1) + b_{σ_{1}}(n)f(n),

where

a_{σ_{1}}(n) = −\frac{\sqrt{a(n)a(n + 1)u_{σ_{1}}(λ_{1}, n)u_{σ_{1}}(λ_{1}, n + 2)}}{u_{σ_{1}}(λ_{1}, n + 1)},
b_{σ_{1}}(n) = λ_{1} − a(n)(\frac{u_{σ_{1}}(λ_{1}, n)}{u_{σ_{1}}(λ_{1}, n + 1)} + \frac{u_{σ_{1}}(λ_{1}, n + 1)}{u_{σ_{1}}(λ_{1}, n)})
(11.17)  = b(n) + ∂^{∗} \frac{a(n)u_{σ_{1}}(λ_{1}, n)}{u_{σ_{1}}(λ_{1}, n + 1)}

satisfy (H.11.3). The equation τ_{σ_{1}}v = λ_{1}v has the positive solution

(11.18)  v_{σ_{1}}(λ_{1}, n) = \frac{1}{\sqrt{−a(n)u_{σ_{1}}(λ_{1}, n)u_{σ_{1}}(λ_{1}, n + 1)}},

which is an eigenfunction of H_{σ_{1}} if and only if σ_{1} ∈ (−1, 1). H − λ_{1} and H_{σ_{1}} − λ_{1} restricted to the orthogonal complements of their corresponding one-dimensional null-spaces are unitarily equivalent and hence (11.19)

\begin{aligned}σ(H_{σ_{1}}) & = & \{ \begin{matrix}σ(H) ∪ \{λ_{1}\}, & σ_{1} ∈ (−1, 1) \\ σ(H), & σ_{1} ∈ \{−1, 1\} \end{matrix} , & σ_{ac}(H_{σ_{1}}) & = & σ_{ac}(H), \\ σ_{pp}(H_{σ_{1}}) & = & \{ \begin{matrix}σ_{pp}(H) ∪ \{λ_{1}\}, & σ_{1} ∈ (−1, 1) \\ σ_{pp}(H), & σ_{1} ∈ \{−1, 1\} \end{matrix} , & σ_{sc}(H_{σ_{1}}) & = & σ_{sc}(H). \end{aligned}

In addition, the operator

(11.20)  A_{σ_{1}}f(n) = \frac{W_{n}(u_{σ_{1}}(λ_{1}), f)}{\sqrt{−a(n)u_{σ_{1}}(λ_{1}, n)u_{σ_{1}}(λ_{1}, n + 1)}}

acts as transformation operator

(11.21)  τ_{σ_{1}}A_{σ_{1}}f = A_{σ_{1}}τf.

Moreover, one obtains

(11.22)  W_{σ_{1}}(A_{σ_{1}}u(z), A_{σ_{1}}v(z)) = (z − λ_{1})W (u(z), v(z))

for solutions u(z), v(z) of τu = zu, where W_{σ_{1},n}(u, v) = a_{σ_{1}}(n)(u(n)v(n + 1) − u(n + 1)v(n)). The resolvent of H_{σ_{1}} for z ∈ ℂ\(σ(H) ∪ {λ_{1}}) is given by

(11.23)  (H_{σ_{1}} − z)^{−1} = \frac{1}{z − λ_{1}} (A_{σ_{1}}(H − z)^{−1}A_{σ_{1}}^{∗} − 1),

or, in terms of Green’s functions for n ≥ m, z ∈ ℂ\(σ(H) ∪ {λ_{1}}),

(11.24)  G_{σ_{1}}(z, n, m) = \frac{A_{σ_{1}}u_{−}(z, m)A_{σ_{1}}u_{+}(z, n)}{(z − λ_{1})W (u_{−}(z), u_{+}(z))}.

Hence, u_{σ_{1},±}(z, n) = ±A_{σ_{1}}u_{±}(z, n) satisfy u_{σ_{1},±}(z, n) ∈ ℓ_{±}(ℤ) and are the minimal solutions of (H_{σ_{1}} − z)u = 0 for z < λ_{1}. In addition, we have

(11.25)  ∑_{n∈ℤ} v_{σ_{1}}(λ_{1}, n)^{2} = \frac{4}{1 − σ_{1}^{2}} W (u_{−}(λ_{1}), u_{+}(λ_{1}))^{−1}, σ_{1} ∈ (−1, 1),

and, if τu(λ) = λu(λ), u(λ, .) ∈ ℓ^{2}(ℤ),

(11.26)  ∑_{n∈ℤ}(A_{σ_{1}}u(λ, n))^{2} = (λ − λ_{1}) ∑_{n∈ℤ} u(λ, n)^{2}.

Proof. The unitary equivalence together with equation (11.23) follow from Theorem 11.1. Equations (11.20) – (11.22) are straightforward calculations. Equation (11.24) follows from the obvious fact that A_{σ_{1}}u_{±}(z, n) ∈ ℓ_{±}^{2}(ℤ) and (11.22). Moreover,

(11.27)  A_{σ_{1}}u_{±}(λ_{1}, n) = ∓\frac{1 ∓ σ_{1}}{2} W (u_{−}(λ_{1}), u_{+}(λ_{1}))v_{σ_{1}}(λ_{1}, n)

shows that v_{σ_{1}}(λ_{1}, n) is in ℓ^{2}(ℤ) if and only if σ_{1} ∈ (−1, 1). In addition, we infer ±A_{σ_{1}}u_{±}(λ_{1}, n) ≥ 0 since W (u_{−}(λ_{1}), u_{+}(λ_{1})) < 0 (by G(λ_{1}, n, n) < 0) and hence a simple continuity argument implies ±A_{σ_{1}}u_{±}(z, n) > 0 for z < λ_{1}. Next, if σ_{1} ∈ (−1, 1), we can use (11.27) to compute the residue of (11.24) at z = λ_{1} and compare this with (2.34) to obtain (11.25). To see (11.26) use ⟨A_{σ_{1}}u, A_{σ_{1}}u⟩ = ⟨u, A_{σ_{1}}^{∗}A_{σ_{1}}u⟩. The rest is simple. □

Remark 11.5. (i). Multiplying u_{σ_{1}} with a positive constant leaves all formulas and, in particular, H_{σ_{1}} invariant. (ii). We can also insert eigenvalues into the highest spectral gap, that is, above the spectrum of H, upon considering −H. Then λ > sup(σ(H)) implies that we don’t have positive but rather alternating solutions and all our previous calculations carry over with minor changes. (iii). We can weaken (H.11.3) by requiring a(n) ≠ 0 instead of a(n) < 0. Everything stays the same with the only difference that u_{±} are not positive but change sign in such a way that (2.52) stays positive. Moreover, the signs of a_{σ_{1}}(n) can also be prescribed arbitrarily by altering the signs of ρ_{o,σ_{1}} and ρ_{e,σ_{1}}. (iv). The fact that v_{σ_{1}} ∈ ℓ^{2}(ℤ) if and only if σ_{1} ∈ (−1, 1) gives an alternate proof for

(11.28) ∑_{n=0}^{±∞} \frac{1}{−a(n)u_{σ_{1}}(λ_{1}, n)u_{σ_{1}}(λ_{1}, n + 1)} < ∞ if and only if σ_{1} ∈ \begin{matrix}[−1, 1) \\ (−1, 1] \end{matrix}

(cf. Lemma 2.9 (iv)). (v). In the case σ_{1} = ±1 the second minimal solution for z = λ_{1} is given by (cf. (1.51))

(11.29)  u_{σ_{1},∓}(λ_{1}, n) = v_{σ_{1}}(n) ∑_{j=±∞}^{{n+1 \atop n}} u_{σ_{1}}(λ_{1}, n), σ_{1} = ±1.

(vi). The formula (11.22) can be generalized to

W_{σ_{1}}(A_{σ_{1}}u(z), A_{σ_{1}}v(ẑ))

= (z − λ_{1})W (u(z), v(ẑ)) + \frac{(z − ẑ)v(ẑ, n + 1)}{u(λ_{1}, n + 1)}W_{n}(u(λ_{1}), u(z))

= −\frac{a(n)a(n + 1)}{u(λ_{1}, n + 1)}C_{n}(u(λ_{1}), u(z), v(ẑ)), (11.30)

where C_{n}(…) denotes the N-dimensional Casoratian

(11.31)  C_{n}(u_{1}, …, u_{N}) = det(u_{i}(n + j − 1))_{1≤i,j≤N}.

In the remainder of this section we will show some connections between the single commutation method and some other theories. We start with the Weyl mfunctions.

Lemma 11.6. Assume (H.11.3). The Weyl m̃-functions m̃_{σ_{1},±}(z) of H_{σ_{1}}, σ_{1} ∈ [−1, 1] in terms of m̃_{±}(z), the ones of H, read

(11.32) m̃_{σ_{1},±}(z) = \frac{∓u_{σ_{1}}(λ_{1}, 1)}{a(1)u_{σ_{1}}(λ_{1}, 2)}(1 ± \frac{(z − λ_{1})m̃_{±}(z)}{1 ∓ \frac{a(0)}{β} m̃_{±}(z)}),  β = −\frac{u_{σ_{1}}(λ_{1}, 1)}{u_{σ_{1}}(λ_{1}, 0)}.

Proof. The above formulas follows after a straightforward calculation using (11.24) and (2.2). □

From Lemma 8.1 (iii) we obtain the useful corollary

Corollary 11.7. The operator H_{σ_{1}} is reflectionless if and only if H is.

Moreover, we even have the following result.

Lemma 11.8. The operators H_{n}^{∞} − λ_{1} and (H_{σ_{1}})_{n−1}^{β_{σ_{1}}} − λ_{1} (resp. H_{n}^{β} − λ_{1} and (H_{σ_{1}})_{n}^{∞}−λ_{1}) are unitarily equivalent when restricted to the orthogonal complements of their corresponding one-dimensional null-spaces, where

(11.33)  β = −\frac{u_{σ_{1}}(λ_{1}, 1)}{u_{σ_{1}}(λ_{1}, 0)},  β_{σ_{1}} = −\frac{v_{σ_{1}}(λ_{1}, 1)}{v_{σ_{1}}(λ_{1}, 0)}.

Proof. We recall Remark 1.11 and we set λ_{1} = 0 without loss of generality. Choose Ã_{σ_{1}} = P_{n}A_{σ_{1}}P_{n}+ρ_{e,σ_{1}}(n)⟨δ_{n+1}, .⟩δ_{n}, then one computes Ã_{σ_{1}}^{∗} Ã_{σ_{1}} = P_{n}HP_{n} and Ã_{σ_{1}} Ã_{σ_{1}}^{∗} = P_{n−1}^{β_{σ_{1}}}H_{σ_{1}}P_{n−1}^{β_{σ_{1}}}. Similarly, choosing Ã_{σ_{1}} = P_{n}A_{σ_{1}}P_{n} + ρ_{e,σ_{1}}(n − 1)⟨δ_{n}, .⟩δ_{n−1}, one obtains Ã_{σ_{1}}^{∗} Ã_{σ_{1}} = P_{n}^{β}HP_{n}^{β} and Ã_{σ_{1}} Ã_{σ_{1}}^{∗} = P_{n}H_{σ_{1}}P_{n}. □

Finally, we turn to scattering theory and assume (H.10.1) (cf. Remark 11.5). We denote the set of eigenvalues by σ_{p}(H) = {λ_{j}}_{j∈J}, where J ⊆ ℕ is a finite index set, and use the notation introduced in Chapter 10.

Lemma 11.9. Suppose H satisfies (H.10.1) and let H_{σ_{1}} be constructed as in Theorem 11.4 with

(11.34)  u_{σ_{1}}(λ_{1}, n) = \frac{1 + σ_{1}}{2} f_{+}(k_{1}, n) + \frac{1 − σ_{1}}{2} f_{−}(k_{1}, n).

Then the transmission T_{σ_{1}}(k) and reflection coefficients R_{±,σ_{1}}(k) of H_{σ_{1}} in terms of the corresponding scattering data T (k), R_{±}(k) of H are given by

T_{σ_{1}}(k) = \frac{1 − k k_{1}}{k − k_{1}} T (k), R_{±,σ_{1}}(k) = k^{±1} \frac{k − k_{1}}{1 − k k_{1}} R_{±}(k), σ_{1} ∈ (−1, 1),
(11.35)  T_{σ_{1}}(k) = T (k), R_{±,σ_{1}}(k) = \frac{k_{1}^{σ_{1}} − k^{∓1}}{k_{1}^{σ_{1}} − k^{±1}} R_{±}(k), σ_{1} ∈ \{−1, 1\},

where k_{1} = λ_{1} + \sqrt{λ_{1}^{2} − 1} ∈ (−1, 0). Moreover, the norming constants γ_{±,σ_{1},j} associated with λ_{j} ∈ σ_{pp}(H_{σ_{1}}) in terms of γ_{±,j}, corresponding to H, read

γ_{±,σ_{1},j} = |k_{j}|^{±1} \frac{1 − k_{j}k_{1}}{(k_{j} − k_{1})}γ_{±,j}, j ∈ J, σ_{1} ∈ (−1, 1),

γ_{±,σ_{1},1} = (\frac{1 − σ_{1}}{1 + σ_{1}} )^{±1} |1 − k_{1}^{∓2}| T (k_{1}), σ_{1} ∈ (−1, 1),
(11.36)  γ_{±,σ_{1},j} = |k_{1}^{σ_{1}} − k_{j}^{∓1}|γ_{±,j}, j ∈ J, σ_{1} ∈ {−1, 1}.

Proof. The claims follow easily after observing that up to normalization the Jost solutions of H_{σ_{1}} are given by A_{σ_{1}}f_{±}(k, n) (compare (11.24)). □

11.3. Iteration of the single commutation method

From the previous section it is clear that the single commutation method can be iterated. In fact, choosing λ_{2} < λ_{1} and σ_{2} ∈ [−1, 1] we can define

(11.37)  u_{σ_{1},σ_{2}}(λ_{2}, n) = \frac{1 + σ_{2}}{2} u_{+,σ_{1}}(λ_{2}, n) + \frac{1 − σ_{2}}{2} u_{−,σ_{1}}(λ_{2}, n)

and corresponding sequences ρ_{o,σ_{1},σ_{2}}, ρ_{e,σ_{1},σ_{2}}. As before, the associated operators A_{σ_{1},σ_{2}}, A_{σ_{1},σ_{2}}^{∗} satisfy

(11.38)  H_{σ_{1}} = A_{σ_{1},σ_{2}}^{∗}A_{σ_{1},σ_{2}} − λ_{2}

and a further commutation then yields the operator

(11.39)  H_{σ_{1},σ_{2}} = A_{σ_{1},σ_{2}}A_{σ_{1},σ_{2}}^{∗} − λ_{2}.

Clearly, if one works out the expression for a_{σ_{1},σ_{2}} and b_{σ_{1},σ_{2}} (by simply plugging in) we get rather clumsy formulas. Moreover, these formulas will get even worse in each step. The purpose of this section is to show that the resulting quantities after N steps can be expressed in terms of determinants.

Theorem 11.10. Assume (H.11.3). Let H be as in Section 11.2 and choose

(11.40)  λ_{N} < ⋯ < λ_{2} < λ_{1} < inf σ(H), σ_{ℓ} ∈ [−1, 1], 1 ≤ ℓ ≤ N ∈ ℕ.

Then we have

a_{σ_{1},...,σ_{N}} (n) = −\sqrt{a(n)a(n + N)}\frac{\sqrt{C_{n}(U_{σ_{1},...,σ_{N}})C_{n+2}(U_{σ_{1},...,σ_{N}})}}{C_{n+1}(U_{σ_{1},...,σ_{N}})},
b_{σ_{1},...,σ_{N}} (n) = λ_{N} − a(n)\frac{C_{n+2}(U_{σ_{1},...,σ_{N−1}})C_{n}(U_{σ_{1},...,σ_{N}})}{C_{n+1}(U_{σ_{1},...,σ_{N−1}})C_{n+1}(U_{σ_{1},...,σ_{N}})}

− a(n + N − 1)\frac{C_{n}(U_{σ_{1},...,σ_{N−1}})C_{n+1}(U_{σ_{1},...,σ_{N}})}{C_{n+1}(U_{σ_{1},...,σ_{N−1}})C_{n}(U_{σ_{1},...,σ_{N}})}

= b(n) + ∂^{∗}a(n)\frac{D_{n}(U_{σ_{1},...,σ_{N}})}{C_{n+1}(U_{σ_{1},...,σ_{N}})}
(11.41)  = b(n + N) − ∂^{∗}a(n + N)\frac{D̃_{n+1}(U_{σ_{1},...,σ_{N}})}{C_{n+1}(U_{σ_{1},...,σ_{N}})},

where (U_{σ_{1},...,σ_{N}}) = (u_{σ_{1}}^{1}, …, u_{σ_{N}}^{N}) and

(11.42)  u_{σ_{ℓ}}^{ℓ}(n) = \frac{1 + σ_{ℓ}}{2} u_{+}(λ_{ℓ}, n) + (−1)^{ℓ+1} \frac{1 − σ_{ℓ}}{2} u_{−}(λ_{ℓ}, n).

Here C_{n}(…) denotes the Casoratian (11.31) and

D_{n}(u_{1}, …, u_{N}) = det ( \begin{matrix}u_{i}(n), & j = 1 \\ u_{i}(n + j), & j > 1 \end{matrix} )_{1≤i,j≤N} ,
(11.43)  D̃_{n}(u_{1}, …, u_{N}) = det ( \begin{matrix}u_{i}(n + j − 1), & j < N \\ u_{i}(n + N), & j = N \end{matrix} )_{1≤i,j≤N} .

The corresponding Jacobi operator H_{σ_{1},...,σ_{N}} is self-adjoint with spectrum

(11.44)  σ(H_{σ_{1},...,σ_{N}}) = σ(H) ∪ {λ_{ℓ} | σ_{ℓ} ∈ (−1, 1), 1 ≤ ℓ ≤ N}.

In addition, we have

ρ_{o,σ_{1},...,σ_{N}} (n) =

−\sqrt{−a(n)\frac{C_{n+2}(U_{σ_{1},...,σ_{N−1}})C_{n}(U_{σ_{1},...,σ_{N}})}{C_{n+1}(U_{σ_{1},...,σ_{N−1}})C_{n+1}(U_{σ_{1},...,σ_{N}})}},

ρ_{e,σ_{1},...,σ_{N}} (n) =

\sqrt{−a(n + N − 1)\frac{C_{n}(U_{σ_{1},...,σ_{N−1}})C_{n+1}(U_{σ_{1},...,σ_{N}})}{C_{n+1}(U_{σ_{1},...,σ_{N−1}})C_{n}(U_{σ_{1},...,σ_{N}})}}. (11.45)

and

(11.46)  A_{σ_{1},...,σ_{N}} ⋯ A_{σ_{1}}f(n) = \frac{( ∏_{j=0}^{ℓ−1} \sqrt{−a(n + j)})C_{n}(U_{σ_{1},...,σ_{N}} , f)}{\sqrt{C_{n}(U_{σ_{1},...,σ_{N}})C_{n+1}(U_{σ_{1},...,σ_{N}})}}.

The principal solutions of τ_{σ_{1},...,σ_{ℓ}}u = λu for 1 ≤ ℓ ≤ N, λ < λ_{ℓ} are given by

(11.47)  u_{σ_{1},...,σ_{ℓ},±}(λ, n) = (±1)^{ℓ}A_{σ_{1},...,σ_{ℓ}} ⋯ A_{σ_{1}}u_{±}(λ, n)

and

(11.48) u_{σ_{1},...,σ_{ℓ}}(λ_{ℓ}, n) = \frac{1 + σ_{ℓ}}{2} u_{σ_{1},...,σ_{ℓ−1},+}(λ_{ℓ}, n) + \frac{1 − σ_{ℓ}}{2} u_{σ_{1},...,σ_{ℓ−1},−}(λ_{ℓ}, n)

is used to define H_{σ_{1},...,σ_{ℓ}}.

To prove this theorem we need a special case of Sylvester’s determinant identity (cf. [91], Sect. II.3, [128]).

Lemma 11.11. Let \underline{M} be an arbitrary n × n matrix (n ≥ 2). Denote by \underline{M}^{(i,j)} the matrix with the i-th column and the j-th row deleted and by \underline{M}^{(i,j,l,m)} the matrix with the i-th and l-th column and the j-th and m-th row deleted. Then we have

det \underline{M}^{(n,n)} det \underline{M}^{(n−1,n−1)} − det \underline{M}^{(n−1,n)} det \underline{M}^{(n,n−1)}

= det \underline{M}^{(n,n,n−1,n−1)} det \underline{M}. (11.49)

Proof. Obviously we can assume \underline{M} to be nonsingular. First observe that interchanging the i-th row and the j-th row for 1 ≤ i, j ≤ n − 2 leaves the formula invariant. Furthermore, adding an arbitrary multiple of the j-th row to the i-th row with 1 ≤ j ≤ n − 2, 1 ≤ i ≤ n, also leaves the formula unchanged. Thus we may assume \underline{M} to be of the form

(11.50)  ( \begin{matrix}M_{11} & M_{12} & … & … & M_{1n} \\ & M_{22} & & & ⋮ \\ & & ... & & ⋮ \\ & & & M_{n−1,n−1} & M_{n−1,n} \\ & & & M_{n,n−1} & M_{n,n} \end{matrix} ) ,

if \underline{M}^{(n,n,n−1,n−1)} is nonsingular, and of the form

(11.51)  ( \begin{matrix}0 & M_{12} & … & … & M_{1n} \\ ⋮ & ⋮ & & & ⋮ \\ 0 & M_{n−2,2} & & & ⋮ \\ M_{n−1,1} & … & … & … & M_{n−1,n} \\ M_{n,1} & … & … & … & M_{n,n} \end{matrix} ) ,

if \underline{M}^{(n,n,n−1,n−1)} is singular. But for such matrices the claim can be verified directly. □

Proof. (of Theorem 11.10) It is enough to prove (11.41) and (11.46), the remaining assertions then follow from these two equations. We will use a proof by induction on N. All formulas are valid for N = 1 and we have

f_{σ_{1},...,σ_{N+1}}(n) = \frac{−\sqrt{−a_{σ_{1},...,σ_{N}} (n)}C_{n}(u_{σ_{1},...,σ_{N}} (λ_{N}), f)}{\sqrt{u_{σ_{1},...,σ_{N}} (λ_{N}, n)u_{σ_{1},...,σ_{N}} (λ_{N}, n + 1)}},
a_{σ_{1},...,σ_{N+1}}(n) = −\sqrt{a_{σ_{1},...,σ_{N}} (n)a_{σ_{1},...,σ_{N}} (n + 1)} ×

\frac{\sqrt{u_{σ_{1},...,σ_{N}} (λ_{N}, n)u_{σ_{1},...,σ_{N}} (λ_{N}, n + 2)}}{u_{σ_{1},...,σ_{N}} (λ_{N}, n + 1)}, (11.52)

where f_{σ_{1},...,σ_{ℓ}} = A_{σ_{1},...,σ_{N}} ⋯ A_{σ_{1}}f. Equation (11.46) follows after a straightforward calculation using (by Lemma 11.11)

C_{n}(U_{σ_{1},...,σ_{N}} , f)C_{n+1}(U_{σ_{1},...,σ_{N+1}}) − C_{n+1}(U_{σ_{1},...,σ_{N}} , f)C_{n}(U_{σ_{1},...,σ_{N+1}})

= C_{n+1}(U_{σ_{1},...,σ_{N}})C_{n}(U_{σ_{1},...,σ_{N+1}}, f). (11.53)

The formula for a_{σ_{1},...,σ_{N}} is then a simple calculation.

The first formula for b_{σ_{1},...,σ_{N}} follows after solving the corresponding Jacobi equation (τ_{σ_{1},...,σ_{N}} −λ_{N})f_{σ_{1},...,σ_{N}} = 0 for it. The remaining two follow upon letting λ → −∞ in (11.47) and comparing with (6.27). The rest is straightforward. □

For the sake of completeness we also extend Lemma 11.9. For brevity we assume σ_{ℓ} ∈ (−1, 1).

Lemma 11.12. Suppose H satisfies (H.10.1) and let H_{σ_{1},...,σ_{N}}, σ_{ℓ} ∈ (−1, 1,), 1 ≤ ℓ ≤ N, be constructed as in Theorem 11.10 with

u_{σ_{ℓ}}^{ℓ}(n) = \frac{1 + σ_{ℓ}}{2} f_{+}(k_{ℓ}, n) + (−1)^{ℓ+1} \frac{1 − σ_{ℓ}}{2} f_{−}(k_{ℓ}, n). (11.54)

Then the transmission T_{σ_{1},...,σ_{N}} (k) and reflection coefficients R_{σ_{1},...,σ_{N} ,±}(k) of the operator H_{σ_{1},...,σ_{N}} in terms of the corresponding scattering data T (k), R_{±}(k) of H are given by

T_{σ_{1},...,σ_{N}} (k) = ( ∏_{ℓ=1}^{N} \frac{1 − k k_{ℓ}}{k − k_{ℓ}} ) T (k),

R_{σ_{1},...,σ_{N} ,±}(k) = k^{±N} ( ∏_{ℓ=1}^{N} \frac{k − k_{ℓ}}{1 − k k_{ℓ}} ) R_{±}(k), (11.55)

where k_{ℓ} = λ_{ℓ} + \sqrt{λ_{ℓ}^{2} − 1} ∈ (−1, 0), 1 ≤ ℓ ≤ N. Moreover, the norming constants γ_{±,σ_{1},...,σ_{N} ,j} associated with λ_{j} ∈ σ_{pp}(H_{σ_{1},...,σ_{N}}) in terms of γ_{±,j} corresponding to H read

γ_{±,σ_{1},...,σ_{N} ,j} = (\frac{1−σ_{j}}{1+σ_{j}} )^{±1} |k_{j}|^{−2∓(N−1)} \frac{∏_{ℓ=1}^{N} |1 − k_{j}k_{ℓ}|}{∏N_{{ℓ=1 \atop ℓ≠j}} |k_{j} − k_{l}|}T (k_{j}), 1 ≤ j ≤ N,

γ_{±,σ_{1},...,σ_{N} ,j} = |k_{j}|^{±N} ∏_{ℓ=1}^{N} \frac{1 − k_{j}k_{ℓ}}{|k_{j} − k_{ℓ}|}γ_{±,j}, j ∈ J. (11.56)

Proof. Observe that

u_{σ_{1},σ_{2}}(λ_{2}, n) = \frac{1 + σ_{2}}{2} A_{σ_{1}}f_{+}(k_{2}, n) + \frac{1 − σ_{2}}{2} A_{σ_{1}}f_{−}(k_{2}, n)
(11.57)  = c(\frac{1 + σ̂_{2}}{2} f_{+,σ_{1}}(k_{2}, n) + \frac{1 − σ̂_{2}}{2} f_{−,σ_{1}}(k_{2}, n)),

where c > 0 and σ_{2}, σ̂_{2} are related via

(11.58)  \frac{1 + σ̂_{2}}{1 − σ̂_{2}} = \frac{1}{k_{2}} \frac{1 + σ_{1}}{1 − σ_{1}} .

The claims now follow from Lemma 11.9 after extending this result by induction.

□

11.4. Application of the single commutation method

In this section we want to apply the single commutation method to Jacobi operators associated with quasi-periodic sequences (a, b) as in Section 9.2 (cf. (9.48)). We choose p_{1} = (λ_{1}, σ_{1}R_{2g+2}^{1/2}(λ_{1})) and ϕ(n) = ϕ(p_{1}, n). To ensure ϕ(n) > 0 we require π(p_{1}) ≤ E_{0}, σ_{1} ∈ {±1}. Using the representations of a, b, ϕ in terms of theta functions

a(n) = −ã\sqrt{\frac{θ(\underline{z}(n + 1))θ(\underline{z}(n − 1))}{θ(\underline{z}(n))^{2}}},

b(n) = b̃ + ∑_{j=1}^{g} c_{j}(g)\frac{∂}{∂w_{j}} ln (\frac{θ(\underline{w} + \underline{z}(n))}{θ(\underline{w} + \underline{z}(n − 1))})|_{\underline{w}=0},

(11.59)  ϕ(n) = \sqrt{\frac{θ(\underline{z}_{0}(n − 1))}{θ(\underline{z}_{0}(n + 1))}} \frac{θ(\underline{z}(p_{1}, n + 1))}{θ(\underline{z}(p_{1}, n))} exp ( ∫_{p_{0}}^{p_{1}}ω_{∞_{+},∞_{−}}),

(where all roots are chosen with positive sign) yields

ρ_{o,σ_{1}}(n) = −\sqrt{−\frac{a(n)}{ϕ(n)}}

= −\sqrt{ã\frac{θ(\underline{z}(n + 1))}{θ(\underline{z}(n))} \frac{θ(\underline{z}(p_{1}, n))}{θ(\underline{z}(p_{1}, n + 1))}} exp ( − ∫_{p_{0}}^{p_{1}} \frac{ω_{∞_{+},∞_{−}}}{2} ),
ρ_{e,σ_{1}}(n) = \sqrt{−a(n)ϕ(n)}
(11.60)  = \sqrt{ã\frac{θ(\underline{z}_{0}(n − 1))}{θ(\underline{z}_{0}(n))} \frac{θ(\underline{z}(p_{1}, n + 1))}{θ(\underline{z}(p_{1}, n))}} exp ( ∫_{p_{0}}^{p_{1}} \frac{ω_{∞_{+},∞_{−}}}{2} ).

Next we calculate

ϕ_{σ_{1}}(n) = \frac{ρ_{o,σ_{1}}(n)}{ρ_{e,σ_{1}}(n + 1)}

(11.61)  = \sqrt{\frac{θ(\underline{z}(p_{1}, n))}{θ(\underline{z}(p_{1}, n + 2))}} \frac{θ(\underline{z}(n + 1))}{θ(\underline{z}(n))} exp ( − ∫_{p_{0}}^{p_{1}}ω_{∞_{+},∞_{−}}).

This suggests to define

(11.62)  \underline{z}_{σ_{1}}(p, n) = \underline{Â}_{p_{0}}(p) − \underline{α̂}_{p_{0}}(D_{\underline{μ̂}_{σ_{1}}} ) + 2n\underline{Â}_{p_{0}}(∞_{+}) − \underline{Ξ̂}_{p_{0}},

associated with a new Dirichlet divisor D_{\underline{μ̂}_{σ_{1}}} given by

(11.63)  \underline{α}_{p_{0}}(D_{\underline{μ̂}_{σ_{1}}} ) = \underline{α}_{p_{0}}(D\underline{μ̂}) − \underline{A}_{p_{0}}(p_{1}) − \underline{A}_{p_{0}}(∞_{+}).

Lemma 11.8 says that we have p_{1} = λ̂_{g+1}^{β}(0) and

(11.64)  μ̂_{σ_{1},j}(0) = λ̂_{j}^{β}(0), 1 ≤ j ≤ g,

where β = ϕ(0). This can also be obtained directly by comparing (11.63) and (9.11).

Upon using

(11.65)  \underline{z}(p_{1}, n) = \underline{z}_{σ_{1}}(∞_{+}, n − 1),  \underline{z}(∞_{+}, n) = \underline{z}_{σ_{1}}(p_{1}^{∗}, n),

we obtain (again all roots are chosen with positive sign)

a_{σ_{1}}(n) = −ã\sqrt{\frac{θ(\underline{z}_{σ_{1}}(n + 1))θ(\underline{z}_{σ_{1}}(n − 1))}{θ(\underline{z}_{σ_{1}}(n))^{2}}},

b_{σ_{1}}(n) = b̃ + ∑_{j=1}^{g} c_{j}(g)\frac{∂}{∂w_{j}} ln (\frac{θ(\underline{w} + \underline{z}_{σ_{1}}(n))}{θ(\underline{w} + \underline{z}_{σ_{1}}(n − 1))})|_{\underline{w}=0},

(11.66) ϕ_{σ_{1}}(n) = \sqrt{\frac{θ(\underline{z}_{σ_{1}}(n − 1))}{θ(\underline{z}_{σ_{1}}(n + 1))}} \frac{θ(\underline{z}_{σ_{1}}(p_{1}^{∗}, n + 1))}{θ(\underline{z}_{σ_{1}}(p_{1}^{∗}, n))} exp ( ∫_{p_{0}}^{p_{1}^{∗}}ω_{∞_{+},∞_{−}}).

Remark 11.13. If we choose π(p_{1}) ≥ E_{2g+2} instead of π(p_{1}) ≤ E_{1} the operator H_{σ_{1}} is well-defined but the sequences ρ_{o,σ_{1}}, ρ_{e,σ_{1}} become purely imaginary.

Now let us choose π(p_{1}) ∈ [E_{2j−1}, E_{2j}]. Then neither ρ_{o,σ_{1}}, ρ_{e,σ_{1}} nor H_{σ_{1}} are well-defined. But if we ignore this and make (formally) a second commutation at a point p_{2} = (λ_{2}, σ_{2}R_{2g+2}^{1/2}(λ_{2})) with π(p_{2}) ∈ [E_{2j−1}, E_{2j}], we get an operator H_{2} associated with a Dirichlet divisor D_{\underline{μ̂}_{σ_{1},σ_{2}}} given by

\underline{α}_{p_{0}}(D_{\underline{μ̂}_{σ_{1},σ_{2}}} ) = \underline{α}_{p_{0}}(D_{\underline{μ̂}_{σ_{1}}} ) − \underline{A}_{p_{0}}(p_{1}) − \underline{A}_{p_{0}}(∞_{+})
(11.67)  = \underline{α}_{p_{0}}(D\underline{μ̂}) − \underline{A}_{p_{0}}(p_{1}) − \underline{A}_{p_{0}}(p_{2}) − 2\underline{A}_{p_{0}}(∞_{+}).

Since 2[i Im(\underline{A}_{p_{0}}(p_{1}))] = [0], H_{2} is a well-defined operator. Moreover, choosing p_{1} = μ̂_{j} in the first step implies that the Dirichlet eigenvalue μ̂_{j} is formally replaced by one at ∞_{−}. The second step corrects this by moving the Dirichlet eigenvalue to the point p_{2}^{∗} = (μ, σR_{2g+2}^{1/2}(μ))) and the factor 2\underline{A}_{p_{0}}(∞_{+}) shifts the sequences by one. We will undo the shift and denote the corresponding sequences by a_{(μ,σ)}, b_{(μ,σ)}. From Theorem 11.10 we know that they are equivalently given by

a_{(μ,σ)}(n + 1) = −\sqrt{a(n)a(n + 2)}

×\sqrt{\frac{C_{n}(ψ_{σ_{j}}(μ_{j}), ψ_{−σ}(μ))C_{n+2}(ψ_{σ_{j}}(μ_{j}), ψ_{−σ}(μ))}{C_{n+1}(ψ_{σ_{j}}(μ_{j}), ψ_{−σ}(μ))^{2}}},

(11.68) b_{(μ,σ)}(n + 1) = b(n) − ∂^{∗} \frac{ψ_{σ_{j}}(μ_{j}, n)ψ_{−σ}(μ, n + 1)}{C_{n}(ψ_{σ_{j}}(μ_{j}), ψ_{−σ}(μ))},

where the n + 1 on the left-hand-side takes the aforementioned shift of reference point into account. Thus, applying this procedure g times we can replace all Dirichlet eigenvalues (μ_{j}, σ_{j}) by (μ̃_{j}, σ̃_{j}) proving that the sequences associated with μ̃_{j}, 1 ≤ j ≤ g, are given by

a_{(μ̃_{1},σ̃_{1}),...,(μ̃_{g},σ̃_{g})}(n) = \sqrt{a(n − g)a(n + g)}

×\frac{\sqrt{C_{n−g}(Ψ_{(μ̃_{1},σ̃_{1}),...,(μ̃_{g},σ̃_{g})})C_{n−g+2}(Ψ_{(μ̃_{1},σ̃_{1}),...,(μ̃_{g},σ̃_{g})})}}{C_{n−g+1}(Ψ_{(μ̃_{1},σ̃_{1}),...,(μ̃_{g},σ̃_{g})})},

b_{(μ̃_{1},σ̃_{1}),...,(μ̃_{g},σ̃_{g})}(n) = b(n) − ∂^{∗} a(n)\frac{D̃_{n−g+1}(Ψ_{(μ̃_{1},σ̃_{1}),...,(μ̃_{g},σ̃_{g})})}{C_{n−g+1}(Ψ_{(μ̃_{1},σ̃_{1}),...,(μ̃_{g},σ̃_{g})})},

(11.69)

where (Ψ_{(μ̃_{1},σ̃_{1}),...,(μ̃_{g},σ̃_{g})}) = (ψ_{σ_{1}}(μ_{1}), ψ_{−σ̃_{1}}(μ̃_{1}), …, ψ_{σ_{g}}(μ_{g}), ψ_{−σ̃_{g}}(μ̃_{g})). We will generalize this method to arbitrary operators in Section 11.8.

11.5. A formal second commutation

Our next aim is to remove the condition that H is bounded from below. To do this we apply the single commutation method twice which will produce a method for inserting eigenvalues into arbitrary spectral gaps of a given Jacobi operator (not only into the lowest). This method is hence called the double commutation method.

Let H be a given operator (a < 0 for the moment) and let u_{±}(λ_{1}, .), λ_{1} < σ(H), be a positive solution. Define

Af(n) = ρ_{o}(n)f(n + 1) + ρ_{e}(n)f(n),
(11.70)  A^{∗}f(n) = ρ_{o}(n − 1)f(n − 1) + ρ_{e}(n)f(n),

where

(11.71)  ρ_{o}(n) = \sqrt{−\frac{a(n)u_{±}(λ_{1}, n)}{u_{±}(λ_{1}, n + 1)}},  ρ_{e}(n) = −\sqrt{−\frac{a(n)u_{±}(λ_{1}, n + 1)}{u_{±}(λ_{1}, n)}}.

Then we have H = A^{∗}A + λ_{1} and we can define a second operator

(11.72)  H̃ = AA^{∗} + λ_{1}.

Fix γ_{±} > 0. Then by (11.18) and (11.29)

(11.73)  v(λ_{1}, n) = \frac{c_{γ_{±}}(λ_{1}, n)}{\sqrt{−a(n)u_{±}(λ_{1}, n)u_{±}(λ_{1}, n + 1)}}

11.5. A formal second commutation 199 solves H̃v = λ_{1}v, where

(11.74)  c_{γ_{±}}(λ_{1}, n) = \frac{1}{γ_{±}} ∓\begin{matrix}n \\ ∑ & ∗ \\ j=±∞ \end{matrix} u_{±}(λ_{1}, j)^{2}.

Now define

ρ_{o,γ_{±}}(n) = ρ_{e}(n + 1)\sqrt{\frac{c_{γ_{±}}(λ_{1}, n)}{c_{γ_{±}}(λ_{1}, n + 1)}},
(11.75)  ρ_{e,γ_{±}}(n) = ρ_{o}(n)\sqrt{\frac{c_{γ_{±}}(λ_{1}, n + 1)}{c_{γ_{±}}(λ_{1}, n)}}

and introduce corresponding operators A_{γ_{±}}, A_{γ_{±}}^{∗} on ℓ^{2}(ℤ) by

(A_{γ_{±}}f)(n) = ρ_{o,γ_{±}}(n)f(n + 1) + ρ_{e,γ_{±}}(n)f(n),
(11.76)  (A_{γ_{±}}^{∗}f)(n) = ρ_{o,γ_{±}}(n − 1)f(n − 1) + ρ_{e,γ_{±}}(n)f(n).

A simple calculation shows that A_{γ_{±}}^{∗}A_{γ_{±}} = A_{σ_{1}}A_{σ_{1}}^{∗} and hence

(11.77)  H_{σ_{1}} = A_{γ_{±}}^{∗}A_{γ_{±}} + λ_{1}.

Performing a second commutation yields the doubly commuted operator

(11.78)  H_{γ_{±}} = A_{γ_{±}}A_{γ_{±}}^{∗} + λ_{1}.

Explicitly, one verifies

(11.79)  (H_{γ_{±}}f)(n) = a_{γ_{±}}(n)f(n + 1) + a_{γ_{±}}(n − 1)f(n − 1) − b_{γ_{±}}(n)f(n),

with

a_{γ_{±}}(n) = a(n + 1)\frac{\sqrt{c_{γ_{±}}(λ_{1}, n)c_{γ_{±}}(λ_{1}, n + 2)}}{c_{γ_{±}}(λ_{1}, n + 1)},
b_{γ_{±}}(n) = b(n + 1) ∓ (\frac{a(n)u_{±}(λ_{1}, n)u_{±}(λ_{1}, n + 1)}{c_{γ_{±}}(λ_{1}, n)}

− \frac{a(n + 1)u_{±}(λ_{1}, n + 1)u_{±}(λ_{1}, n + 2)}{c_{γ_{±}}(λ_{1}, n + 1)}). (11.80)

Note that

(11.81)  u_{γ_{±}}(λ_{1}, n) = \frac{u_{±}(λ_{1}, n + 1)}{\sqrt{c_{γ_{±}}(λ_{1}, n)c_{γ_{±}}(λ_{1}, n + 1)}}

is a solution of H_{γ_{±}}u = λ_{1}u fulfilling u_{γ_{±}}(λ_{1}) ∈ ℓ^{2}(±∞, n_{0}) and because of

(11.82) ∑_{n∈ℤ} |u_{γ_{±}}(λ_{1}, n)|^{2} = γ_{±} ± lim_{n→∓∞} \frac{1}{c_{γ_{±}}(λ_{1}, n)},

we have u_{γ_{±}}(λ_{1}) ∈ ℓ^{2}(ℤ). Furthermore, if u is a solution of Hu = zu (for arbitrary z ∈ ℂ), then

u_{γ_{±}}(z, n) = \frac{1}{z − λ_{1}} AA_{γ_{±}}u(z, n)
(11.83)  = \frac{c_{γ_{±}}(λ_{1}, n)u(z, n + 1) ± \frac{1}{z−λ_{1}} u_{±}(z, n + 1)W_{n}(u_{±}(λ_{1}), u(z))}{\sqrt{c_{γ_{±}}(λ_{1}, n)c_{γ_{±}}(λ_{1}, n + 1)}}

is a solution of H_{γ_{±}}u = zu.

Now observe that H_{γ_{±}} remains well-defined even if u_{±} is no longer positive. This applies, in particular, in the case where u_{±}(λ_{1}) has zeros and hence all intermediate operators A_{σ_{1}}, A_{γ_{±}}, H̃, etc., become ill-defined. Thus, to define H_{γ_{±}}, it suffices to assume the existence of a solution u_{±}(λ_{1}) which is square summable near ±∞. This condition is much less restrictive than the existence of a positive solution (e.g., the existence of a spectral gap for H around λ_{1} is sufficient in this context).

One expects that formulas analogous to (11.19) will carry over to this more general setup. That this is actually the case will be shown in Theorem 11.16. Hence the double commutation method (contrary to the single commutation method) enables one to insert eigenvalues not only below the spectrum of H but into arbitrary spectral gaps of H.

11.6. The double commutation method

In this section we provide a complete characterization of the double commutation method for Jacobi operators. Since Theorem 11.1 only applies when λ_{1} < σ(H), we need a different approach. We start with a linear transformation which turns out to be unitary when restricted to proper subspaces of ℓ^{2}(ℤ). We use this transformation to construct an operator H_{γ_{1}} from a given background operator H. This operator H_{γ_{1}} will be the doubly commuted operator of H as discussed in the previous section. The following abbreviation will simplify our notation considerably

(11.84)  ⟨f, g⟩_{m}^{n} = ∑_{j=m}^{n} \overline{f(j)}g(j).

Let u ∈ ℓ_{−}^{2}(ℤ) be a given real-valued sequence. Choose a constant γ ∈ [−∥u∥^{−2}, ∞) or γ = ∞ and define

(11.85)  c_{γ}(n) = \frac{1}{γ} + ⟨u, u⟩_{−∞}^{n}, γ ≠ 0

(setting ∞^{−1} = 0). Consider the following linear transformation

(11.86)  \begin{aligned}U_{γ} : & ℓ^{2}(ℤ) & → & ℓ(ℤ) \\ & f(n) & ↦ & \sqrt{\frac{c_{γ}(n)}{c_{γ}(n−1)}}f(n) − u_{γ}(n)⟨u, f⟩_{−∞}^{n} \end{aligned} ,

(U_{0} = 1l), where

(11.87)  u_{γ}(n) = \frac{u(n)}{\sqrt{c_{γ}(n − 1)c_{γ}(n)}},

(u_{0} = 0). We note that U_{γ} can be defined on ℓ_{−}^{2}(ℤ) and U_{γ}u = γ^{−1}u_{γ}, γ ≠ 0. Furthermore,

(11.88)  u_{γ}(n)^{2} = ∂^{∗} \frac{1}{c_{γ}(n)},

and hence

(11.89)  ∥u_{γ}∥^{2} = \{ \begin{matrix}γ, & u ∉ ℓ^{2}(ℤ) \\ \frac{γ^{2}∥u∥^{2}}{1+γ∥u∥^{2}} , & u ∈ ℓ^{2}(ℤ) \end{matrix}

implying u_{γ} ∈ ℓ^{2}(ℤ) if −∥u∥^{−2} < γ < ∞. If γ = −∥u∥^{−2}, γ = ∞ we only have u_{γ} ∈ ℓ_{−}^{2}(ℤ), u_{γ} ∈ ℓ_{+}^{2}(ℤ), respectively. In addition, we remark that for f_{γ} = U_{γ}f we have

u_{γ}(n)f_{γ}(n) = −∂^{∗} \frac{⟨u, f⟩_{−∞}^{n}}{c_{γ}(n)} ,
(11.90)  |f_{γ}(n)|^{2} = |f(n)|^{2} + ∂^{∗} \frac{|⟨u, f⟩_{−∞}^{n}|^{2}}{c_{γ}(n)} .

Summing over n and taking limits (if γ = ∞ use Cauchy-Schwarz) shows

⟨u_{γ}, f_{γ}⟩_{−∞}^{n} = \{ \begin{matrix}c_{γ}(n)^{−1}⟨u, f⟩_{−∞}^{n}, & γ ∈ ℝ \\ \frac{⟨u,f⟩}{∥u∥^{2}} − c_{∞}(n)^{−1}⟨u, f⟩_{n+1}^{∞}, & γ = ∞ \end{matrix} ,

(11.91)  ⟨f_{γ}, f_{γ}⟩_{−∞}^{n} = ⟨f, f⟩_{−∞}^{n} − \frac{|⟨u, f⟩_{−∞}^{n}|^{2}}{c_{γ}(n)} .

Clearly, the last equation implies U_{γ} : ℓ^{2}(ℤ) → ℓ^{2}(ℤ). In addition, we remark that this also shows U_{γ} : ℓ_{−}^{2}(ℤ) → ℓ_{−}^{2}(ℤ).

Denote by P, P_{γ} the orthogonal projections onto the one-dimensional subspaces of ℓ^{2}(ℤ) spanned by u, u_{γ} (set P, P_{γ} = 0 if u, u_{γ} ∉ ℓ^{2}(ℤ)), respectively. Define

(11.92) \begin{aligned}U_{γ}^{−1} : & ℓ^{2}(ℤ) & → & ℓ(ℤ) \\ & g(n) & ↦ & \{ \begin{matrix}\sqrt{\frac{c_{γ}(n−1)}{c_{γ}(n)}}g(n) + u(n)⟨u_{γ}, g⟩_{−∞}^{n}, & γ ∈ ℝ \\ \sqrt{\frac{c_{∞}(n−1)}{c_{∞}(n)}}g(n) − u(n)⟨u_{∞}, g⟩_{n+1}^{∞}, & γ = ∞ \end{matrix} \end{aligned}

and note

(11.93)  c_{γ}^{−1}(n) = \{ \begin{matrix}γ − ⟨u_{γ}, u_{γ}⟩_{−∞}^{n}, & γ ∈ ℝ \\ ∥u∥^{−2} + ⟨u_{∞}, u_{∞}⟩_{n+1}^{∞}, & γ = ∞ \end{matrix} .

As before one can show U_{γ}^{−1} : (1l − P_{γ})ℓ^{2}(ℤ) → ℓ^{2}(ℤ) and one verifies

(11.94)  \begin{matrix}U_{γ}U_{γ}^{−1} = 1l, U_{γ}^{−1}U_{γ} = 1l, & γ ∈ ℝ, \\ U_{∞}U_{∞}^{−1} = 1l, U_{∞}^{−1}U_{∞} = 1l − P, & γ = ∞. \end{matrix}

If P = 0, γ ∈ (−∥u∥^{−2}, ∞), then U_{γ}U_{γ}^{−1} = 1l should be understood on (1l−P_{γ})ℓ^{2}(ℤ) since U_{γ}^{−1}u_{γ} ∉ ℓ^{2}(ℤ) by

(11.95)  U_{γ}^{−1}u_{γ} = \{ \begin{matrix}γu, & γ ∈ ℝ \\ ∥u∥^{−2}u, & γ = ∞ \end{matrix} .

Summarizing,

Lemma 11.14. The operator U_{γ} is unitary from (1l − P)ℓ^{2}(ℤ) onto (1l − P_{γ})ℓ^{2}(ℤ) with inverse U_{γ}^{−1}. If P, P_{γ} ≠ 0, then U_{γ} can be extended to a unitary transformation Ũ_{γ} on ℓ^{2}(ℤ) by

(11.96)  Ũ_{γ} = U_{γ}(1l − P) + \sqrt{1 + γ∥u∥^{2}} U_{γ}P.

Proof. The first identity in (11.91) shows that U_{γ} maps (1l − P)ℓ^{2}(ℤ) onto (1l − P_{γ})ℓ^{2}(ℤ). Unitarity follows from (11.91) and

(11.97)  lim_{n→∞} \frac{|⟨u, f⟩_{−∞}^{n}|^{2}}{⟨u, u⟩_{−∞}^{n}} = 0

for any f ∈ ℓ^{2}(ℤ) if u ∉ ℓ^{2}(ℤ). In fact, suppose ∥f∥ = 1, pick m and n > m so large that ⟨f, f⟩_{m}^{∞} ≤ ε/2 and ⟨u, u⟩_{−∞}^{m−1}/⟨u, u⟩_{−∞}^{n} ≤ ε/2. Splitting up the sum in the numerator and applying Cauchy’s inequality then shows that the limit of (11.97) is smaller than ε. The rest follows from (11.89). □

We remark that (11.91) plus the polarization identity implies

(11.98)  ⟨f_{γ}, g_{γ}⟩_{−∞}^{n} = ⟨f, g⟩_{−∞}^{n} − \frac{⟨f, u⟩_{−∞}^{n}⟨u, g⟩_{−∞}^{n}}{c_{γ}(n)} ,

where f_{γ} = U_{γ}f, g_{γ} = U_{γ}g.

Next, we take a Jacobi operator associated with bounded sequences a, b as usual. We assume (λ_{1}, γ_{1}) to be of the following kind.

Hypothesis H. 11.15. Suppose (λ, γ) satisfies the following conditions. (i). u_{−}(λ, n) exists. (ii). γ ∈ [−∥u_{−}(λ)∥^{−2}, ∞) ∪ {∞}.

Note that by Lemma 2.2 a sufficient condition for u_{−}(λ_{1}, n) to exist is λ ∈ ℝ\σ_{ess}(H_{−}).

We now use Lemma 11.14 with u(n) = u_{−}(λ_{1}, n), γ = γ_{1}, c_{γ_{1}}(λ_{1}, n) = c_{γ}(n) to prove

Theorem 11.16. Suppose (H.11.15) and let H_{γ_{1}} be the operator associated with

(11.99)  (τ_{γ_{1}}f)(n) = a_{γ_{1}}(n)f(n + 1) + a_{γ_{1}}(n − 1)f(n − 1) + b_{γ_{1}}(n)f(n),

where

a_{γ_{1}}(n) = a(n)\frac{\sqrt{c_{γ_{1}}(λ_{1}, n − 1)c_{γ_{1}}(λ_{1}, n + 1)}}{c_{γ_{1}}(λ_{1}, n)},
(11.100)  b_{γ_{1}}(n) = b(n) − ∂^{∗} \frac{a(n)u_{−}(λ_{1}, n)u_{−}(λ_{1}, n + 1)}{c_{γ_{1}}(λ_{1}, n)}.

Then

(11.101)  H_{γ_{1}}(1l − P_{γ_{1}}(λ_{1})) = U_{γ_{1}}HU_{γ_{1}}^{−1}(1l − P_{γ_{1}}(λ_{1}))

and τ_{γ_{1}}u_{γ_{1},−}(λ_{1}) = λ_{1}u_{γ_{1},−}(λ_{1}), where

(11.102)  u_{γ_{1},−}(λ_{1}, n) = \frac{u_{−}(λ_{1}, n)}{\sqrt{c_{γ_{1}}(λ_{1}, n − 1)c_{γ_{1}}(λ_{1}, n)}}.

Proof. Let f be a sequence which is square summable near −∞ implying that τf is also square summable near −∞. Then a straightforward calculation shows

(11.103)  τ_{γ_{1}}(U_{γ_{1}}f) = U_{γ_{1}}(τf).

The rest follows easily. □

Corollary 11.17. Suppose u_{−}(λ_{1}) ∉ ℓ^{2}(ℤ). (i). If γ_{1} > 0, then H and (1l − P_{γ_{1}}(λ_{1}))H_{γ_{1}} are unitarily equivalent. Moreover, H_{γ_{1}} has the additional eigenvalue λ_{1} with eigenfunction u_{γ_{1},−}(λ_{1}). (ii). If γ_{1} = ∞, then H and H_{γ_{1}} are unitarily equivalent.

Suppose u_{−}(λ_{1}) ∈ ℓ^{2}(ℤ) (i.e., λ_{1} is an eigenvalue of H). (i). If γ_{1} ∈ (−∥u_{−}(λ_{1})∥^{−2}, ∞), then H and H_{γ_{1}} are unitarily equivalent (using Ũ_{γ_{1}} from Lemma 11.14).

(ii). If γ_{1} = −∥u_{−}(λ_{1})∥^{−2}, ∞, then (1l − P (λ_{1}))H, H_{γ_{1}} are unitarily equivalent, that is, the eigenvalue λ_{1} is removed.

Remark 11.18. (i). By choosing λ_{1} ∈ σ_{ac}(H) ∪ σ_{sc}(H) (provided the continuous spectrum is not empty and a solution satisfying (H.11.15) exists) we can use the double commutation method to construct operators with eigenvalues embedded in the continuous spectrum. (ii). Multiplying γ_{1} by some constant c^{2} > 0 amounts to the same as multiplying u_{−}(λ_{1}, .) by c^{−1}.

The previous theorem tells us only how to transfer solutions of τu = zu into solutions of τ_{γ_{1}}v = zv if u is square summable near −∞. The following lemma treats the general case.

Lemma 11.19. The sequence

(11.104)  u_{γ_{1}}(z, n) = \frac{c_{γ_{1}}(λ_{1}, n)u(z, n) − \frac{1}{z−λ_{1}} u_{−}(λ_{1}, n)W_{n}(u_{−}(λ_{1}), u(z))}{\sqrt{c_{γ_{1}}(λ_{1}, n − 1)c_{γ_{1}}(λ_{1}, n)}},

z ∈ ℂ\{λ_{1}} solves τ_{γ_{1}}u = zu if u(z) solves τu = zu. If u(z) is square summable near −∞, we have u_{γ_{1}}(z, n) = (U_{γ_{1}}u)(z, n), justifying our notation. Furthermore, we note

(11.105)  |u_{γ_{1}}(z, n)|^{2} = |u(z, n)|^{2} + \frac{1}{|z − λ_{1}|^{2}} ∂^{∗} \frac{|W_{n}(u_{−}(λ_{1}), u(z))|^{2}}{c_{γ_{1}}(λ_{1}, n)} ,

and

(11.106)  W_{γ_{1},n}(u_{−,γ_{1}}(λ_{1}), u_{γ_{1}}(z)) = \frac{W_{n}(u_{−}(λ_{1}), u(z))}{c_{γ_{1}}(λ_{1}, n)},

where W_{γ_{1},n}(u, v) = a_{γ_{1}}(n)(u(n)v(n + 1) − u(n + 1)v(n)). Hence u_{γ_{1}} is square summable near ∞ if u is. If û_{γ_{1}}(ẑ) is constructed analogously, then

W_{γ_{1},n}(u_{γ_{1}}(z), û_{γ_{1}}(ẑ)) = W_{n}(u(z), û(ẑ)) + \frac{1}{c_{γ_{1}}(λ_{1}, n)} \frac{z − ẑ}{(z − λ_{1})(ẑ − λ_{1})} ×

W_{n}(u_{−}(λ_{1}), u(z))W_{n}(u_{−}(λ_{1}), û(ẑ)). (11.107)

Proof. All facts are tedious but straightforward calculations. □

As a consequence we get.

Corollary 11.20. The Green function G_{γ_{1}}(z, n, m) of H_{γ_{1}} is given by (n > m)

(11.108)  G_{γ_{1}}(z, n, m) = \frac{u_{+,γ_{1}}(z, n)u_{−,γ_{1}}(z, m)}{W (u_{−}(z), u_{+}(z))},

where

(11.109) u_{±,γ_{1}}(z, n) = \frac{c_{γ_{1}}(λ_{1}, n)u_{±}(z, n) − \frac{1}{z−λ_{1}} u_{−}(λ_{1}, n)W_{n}(u_{−}(λ_{1}), u_{±}(z))}{\sqrt{c_{γ_{1}}(λ_{1}, n − 1)c_{γ_{1}}(λ_{1}, n)}}.

Next, we turn to Weyl m-functions. Without loss of generality we assume u_{−}(λ_{1}, n) = s_{β}(λ_{1}, n, 0), that is,

(11.110)  u_{−}(λ_{1}, 0) = − sin(α), u_{−}(λ_{1}, 0) = cos(α), α ∈ [0, π)

(change γ_{1} if necessary — see Remark 11.18 (ii)).

Theorem 11.21. Let m̃_{±}^{β}(z), m̃β̃ ; γ_{1},±(z) denote the Weyl m̃-functions of H, H_{γ_{1}} respectively. Then we have

(11.111) m̃β̃ ; γ_{1},±(z) = \frac{c_{γ_{1}}(λ_{1}, 0)}{c_{γ_{1}}(λ_{1}, −1)} \frac{1 + β̃^{2}}{1 + β^{2}} (m̃_{±}^{β}(z) ∓ \frac{c_{γ_{1}}(λ_{1}, 0)^{−1}}{z − λ_{1}} ± \frac{βc_{γ_{1}}(λ_{1}, 1)^{−1}}{a(0)(1 + β̃^{2})} ),

where β = cot(α) and

(11.112)  β̃ = \sqrt{\frac{c_{γ_{1}}(λ_{1}, −1)}{c_{γ_{1}}(λ_{1}, 1)}}β.

Proof. Consider the sequences

\frac{\sqrt{c_{γ_{1}}(λ_{1}, −1)c_{γ_{1}}(λ_{1}, 1)}}{c_{γ_{1}}(λ_{1}, 0)}s_{β,γ_{1}}(z, n),

\frac{c_{γ_{1}}(λ_{1}, 1)}{c_{γ_{1}}(λ_{1}, 0)} \frac{1 + β̃^{2}}{1 + β^{2}} (c_{β,γ_{1}}(z, n) − δs_{β,γ_{1}}(z, n)), (11.113)

(where δ is defined in (11.116) below) constructed from the fundamental system c_{β}(z, n), s_{β}(z, n) for τ (cf. (2.71)) as in Lemma 11.19. They form a fundamental system for τ_{γ_{1}} corresponding to the initial conditions associated with β̃. Now use (11.106) to evaluate (2.73). □

As a straightforward consequence we obtain the following lemma.

Corollary 11.22. (i). The operator H_{γ_{1}} is a reflectionless if and only if H is (cf. Lemma 8.1). (ii). Near z = λ_{1} we have

m̃β̃ ; γ_{1},+(z) = −\frac{γ_{+}^{β}(λ_{1}) + c_{γ_{1}}(λ_{1}, 0)^{−1}}{z − λ_{1}} + O(z − λ_{1})^{0},
(11.114) m̃β̃ ; γ_{1},−(z) = −\frac{(c_{γ_{1}}(λ_{1}, 0)(γ_{1}c_{γ_{1}}(λ_{1}, 0) − 1))^{−1}}{z − λ_{1}} + O(z − λ_{1})^{0},

where γ_{+}^{β}(λ_{1}) = (⟨u_{−}(λ_{1}), u_{−}(λ_{1})⟩_{1}^{∞})^{−1} ≥ 0 is the norming constant of H_{+}^{β} at λ_{1}.

As a direct consequence of Theorem 11.21 we infer

Theorem 11.23. Given H and H_{γ_{1}} the respective Weyl M-matrices M^{β}(z) and M β̃ ; γ_{1}(z) are related by

M β̃ ; γ_{1},0,0(z) = \frac{c_{γ_{1}}(λ_{1}, 0)}{c_{γ_{1}}(λ_{1}, 1)} \frac{1 + β^{2}}{1 + β̃^{2}} M_{0,0}^{β}(z),
M β̃ ; γ_{1},0,1(z) = \frac{c_{γ_{1}}(λ_{1}, 0)}{\sqrt{c_{γ_{1}}(λ_{1}, −1)c_{γ_{1}}(λ_{1}, 1)}}(M_{0,1}^{β}(z) + δM_{0,0}^{β}(z)),

M β̃ ; γ_{1},1,1(z) = \frac{c_{γ_{1}}(λ_{1}, 0)}{c_{γ_{1}}(λ_{1}, −1)} \frac{1 + β̃^{2}}{1 + β^{2}} (M_{1,1}^{β}(z) + 2δM_{0,1}^{β}(z)

+ δ^{2}M_{0,0}^{β}(z)), (11.115)

where

(11.116)  δ = \frac{a(0)c_{γ_{1}}(λ_{1}, 0)^{−1}}{z − λ_{1}} − \frac{βc_{γ_{1}}(λ_{1}, 1)^{−1}}{1 + β̃^{2}} .

The quantities M β̃ ; γ_{1},0,0(z) and M β̃ ; γ_{1},0,1(z) are both holomorphic near z = λ_{1}. More-

over, M β̃ ; γ_{1},1,1(z) has a simple pole with residue −(γ_{1}c_{γ_{1}}(λ_{1}, 0))^{−1} at z = λ_{1}.

Using the previous corollary plus weak convergence of π^{−1}Im(F (λ + iε))dλ to the corresponding spectral measure dρ(λ) as ε ↓ 0 (cf. Lemma B.3) implies

Corollary 11.24. The matrix measures dρ^{β}, dρβ̃ ; γ_{1} corresponding to M^{β}(z), M β̃ ; γ_{1}(z) are related by

dρβ̃ ; γ_{1},0,0(λ) = \frac{c_{γ_{1}}(λ_{1}, 0)}{c_{γ_{1}}(λ_{1}, −1)} \frac{1 + β^{2}}{1 + β̃^{2}} dρ_{0,0}^{β}(λ),
dρβ̃ ; γ_{1},0,1(λ) = \frac{c_{γ_{1}}(λ_{1}, 0)}{\sqrt{c_{γ_{1}}(λ_{1}, −1)c_{γ_{1}}(λ_{1}, 1)}}(dρ_{0,1}^{β}(λ) + δdρ_{0,0}^{β}(λ)),

dρβ̃ ; γ_{1},1,1(λ) = \frac{c_{γ_{1}}(λ_{1}, 0)}{c_{γ_{1}}(λ_{1}, −1)} \frac{1 + β^{2}}{1 + β̃^{2}} (dρ_{1,1}^{β}(λ) + 2δdρ_{0,1}^{β}(λ)

+ δ^{2}dρ_{0,0}^{β}(λ)) + (γ_{1}c_{γ_{1}}(λ_{1}, 0))^{−1}dΘ(λ − λ_{1}). (11.117)

Equivalently

(11.118)  dρβ̃ ; γ_{1}(λ) = Udρ^{β}(λ)U^{T} + \frac{1}{γ_{1}c_{γ_{1}}(λ_{1}, 0)} ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ,

where

(11.119)  U = \sqrt{\frac{c_{γ_{1}}(λ_{1}, 0)}{c_{γ_{1}}(λ_{1}, −1)} \frac{1 + β̃^{2}}{1 + β^{2}}} ( \begin{matrix}\frac{1+β̃^{2}}{1+β^{2}} \sqrt{\frac{c_{γ_{1}} (λ_{1},−1)}{c_{γ_{1}} (λ_{1},1)}} & 0 \\ δ & 1 \end{matrix} ) .

Remark 11.25. We can interchange the role of −∞ and ∞ in this section by substituting −∞ → ∞, ⟨., ..⟩_{−∞}^{n} → ⟨., ..⟩_{n+1}^{∞}, u_{−} → u_{+}.

Our next aim is to show how the scattering data of the operators H, H_{γ_{1}} are related, where H_{γ_{1}} is defined as in Theorem 11.16 using the Jost solution f_{−}(k, n) (see Theorem 10.2).

Lemma 11.26. Let H be a given Jacobi operator satisfying (H.10.1). Then the doubly commuted operator H_{γ_{1}}, γ_{1} > 0, defined via u_{−}(λ_{1}, n) = f_{−}(k_{1}, n), λ_{1} = (k_{1} + k_{1}^{−1})/2 as in Theorem 11.16, has the transmission and reflection coefficients

T_{γ_{1}}(k) = sgn(k_{1})\frac{k k_{1}−1}{k−k_{1}}T (k),

R_{−,γ_{1}}(k) = R_{−}(k), R_{+,γ_{1}}(k) = (\frac{k−k_{1}}{k k_{1}−1})^{2} R_{+}(k), (11.120)

where z and k are related via z = (k +k^{−1})/2. Furthermore, the norming constants γ_{−,j} corresponding to λ_{j} ∈ σ_{pp}(H), j ∈ J, (cf. (10.53)) remain unchanged except for an additional eigenvalue λ_{1} with norming constant γ_{−,1} = γ_{1} if u_{−}(λ_{1}) ∉ ℓ^{2}(ℤ) respectively with norming constant γ̃_{−,1} = γ_{−,1} + γ_{1} if u_{−}(λ_{1}) ∈ ℓ^{2}(ℤ) and γ_{−,1} denotes the original norming constant of λ_{1} ∈ σ_{p}(H).

Proof. By Lemma 11.19 the Jost solutions f_{±,γ_{1}}(k, n) are up to a constant given by

(11.121)  \frac{c_{γ_{1}}(λ_{1}, n − 1)f_{±}(k, n) − \frac{1}{z−λ_{1}} u_{−}(λ_{1}, n)W_{n−1}(u_{−}(λ_{1}), f_{±}(k))}{\sqrt{c_{γ_{1}}(λ_{1}, n − 1)c_{γ_{1}}(λ_{1}, n)}}.

This constant is easily seen to be 1 for f_{−,γ_{1}}(k, n). Thus we can compute R_{−}(λ) using (11.107) (the second unknown constant cancels). The rest follows by a straightforward calculation. □

11.7. Iteration of the double commutation method

Finally, we demonstrate how to iterate the double commutation method. We choose a given background operator H and further (λ_{1}, γ_{1}) according to (H.11.15). Next use u_{−}(λ_{1}) to define the transformation U_{γ_{1}} and the operator H_{γ_{1}}. In the second step, we choose (λ_{2}, γ_{2}) and another function u_{−}(λ_{2}) to define u_{−,γ_{1}}(λ_{2}) = U_{γ_{1}}u_{−}(λ_{2}), a corresponding transformation U_{γ_{1},γ_{2}}, and an operator H_{γ_{1},γ_{2}}. Applying this procedure N-times results in

Theorem 11.27. Let H be a given background operator and let (λ_{ℓ}, γ_{ℓ}), 1 ≤ ℓ ≤ N, satisfy (H.11.15). Define the following matrices (1 ≤ ℓ ≤ N)

C^{ℓ}(n) = (\frac{δ_{r,s}}{γ_{r}} + ⟨u_{−}(λ_{r}), u_{−}(λ_{s})⟩_{−∞}^{n})_{1≤r,s≤ℓ} ,

C^{ℓ}(f, g)(n) = ( \begin{matrix}C^{ℓ−1}(n)_{r,s}, & r,s≤ℓ−1 \\ ⟨f, u_{−}(λ_{s})⟩_{−∞}^{n}, & s≤ℓ−1,r=ℓ \\ ⟨u_{−}(λ_{r}), g⟩_{−∞}^{n}, & r≤ℓ−1,s=ℓ \\ ⟨f, g⟩_{−∞}^{n}, & r=s=ℓ \end{matrix} )_{1≤r,s≤ℓ} ,

D_{j}^{ℓ}(n) = ( \begin{matrix}C^{ℓ}(n)_{r,s}, & r,s≤ℓ \\ u_{−}(λ_{s}, n − j), & s≤ℓ,r=ℓ+1 \\ u_{−}(λ_{r}, n), & r≤ℓ,s=ℓ+1 \\ δ_{0,j}, & r=s=ℓ+1 \end{matrix} )_{1≤r,s≤ℓ+1} ,

(11.122)  U^{ℓ}(f(n)) = ( \begin{matrix}C^{ℓ}(n)_{r,s}, & r,s≤ℓ \\ ⟨u_{−}(λ_{s}), f⟩_{−∞}^{n}, & s≤ℓ,r=ℓ+1 \\ u_{−}(λ_{r}, n), & r≤ℓ,s=ℓ+1 \\ f, & r=s=ℓ+1 \end{matrix} )_{1≤r,s≤ℓ+1} .

Then we have (set C^{0}(n) = 1, U^{0}(f) = f)

(11.123)  ⟨U_{γ_{1},...,γ_{ℓ−1}} ⋯ U_{γ_{1}}f, U_{γ_{1},...,γ_{ℓ−1}} ⋯ U_{γ_{1}}g⟩_{−∞}^{n} = \frac{det C^{ℓ}(f, g)(n)}{det C^{ℓ−1}(n)}

and

(11.124)  U_{γ_{1},...,γ_{ℓ}} ⋯ U_{γ_{1}}f(n) = \frac{det U^{ℓ}(f)(n)}{\sqrt{det C^{ℓ}(n − 1) det C^{ℓ}(n)}}.

In particular, we obtain

c_{γ_{1},...,γ_{ℓ}}(λ_{ℓ}, n) = \frac{1}{γ_{ℓ}} + ⟨U_{γ_{1},...,γ_{ℓ−1}} ⋯ U_{γ_{1}}u_{−}(λ_{ℓ}), U_{γ_{1},...,γ_{ℓ−1}} ⋯ U_{γ_{1}}u_{−}(λ_{ℓ})⟩_{−∞}^{n}
(11.125)  = \frac{det C^{ℓ}(n)}{det C^{ℓ−1}(n)}

and hence

(11.126)  ∏_{ℓ=1}^{N} c_{γ_{1},...,γ_{ℓ}}(λ_{ℓ}, n) = det C^{N}(n).

The corresponding operator H_{γ_{1},...,γ_{N}} is associated with

a_{γ_{1},...,γ_{N}} (n) = a(n)\frac{\sqrt{det C^{N}(n − 1) det C^{N}(n + 1)}}{det C^{N}(n)},
(11.127)  b_{γ_{1},...,γ_{N}} (n) = b(n) − ∂^{∗}a(n)\frac{det D_{1}^{N}(n + 1)}{det C^{N}(n)},

and we have

H_{γ_{1},...,γ_{N}} (1l − ∑_{j=1}^{N} P_{γ_{1},...,γ_{N}} (λ_{j}))

= (U_{γ_{1},...,γ_{N}} ⋯ U_{γ_{1}})H(U_{γ_{1}}^{−1} ⋯ U_{γ_{1},...,γ_{N}}^{−1} )(1l − ∑_{j=1}^{N} P_{γ_{1},...,γ_{N}} (λ_{j})). (11.128)

Here P_{γ_{1},...,γ_{N}} (λ_{j}) denotes the projection onto the one-dimensional subspace spanned by

(11.129)  u_{γ_{1},...,γ_{N} ,−}(λ_{ℓ}, n) = \frac{det U_{ℓ}^{N}(n)}{\sqrt{det C^{N}(n − 1) det C^{N}(n)}},

where U_{ℓ}^{N}(n) denotes the matrix U^{N}(u_{−}(λ_{ℓ}))(n) with ℓ’th column and row dropped.

Proof. We start with (11.123). Using Lemma 11.11 we obtain

det C^{ℓ} det C^{ℓ}(f, g) − det C^{ℓ}(u_{−}(λ_{ℓ}), g) det C^{ℓ}(f, u_{−}(λ_{ℓ}))

= det C^{ℓ−1} det C^{ℓ+1}(f, g), (11.130)

which proves (11.123) together with a look at (11.98) by induction on N. Next, (11.125) easily follows from (11.123). Similarly,

det C^{ℓ} det U^{ℓ−1}(f) − det U^{ℓ−1}(u_{−}(λ_{ℓ})) det C^{ℓ}(u_{−}(λ_{ℓ}), f)

= det C^{ℓ−1} det U^{ℓ}(f), (11.131)

and (11.92) prove (11.124). To show the formula for b_{γ_{1},...,γ_{N}} we use that (11.124) with f = u_{−}(z) has an expansion of the type (6.27). Hence we need to expand U^{N}(u_{−}(z)) in powers of z and compare coefficients with (6.27). We can assume u_{−}(z, 0) = 1 implying ⟨u_{−}(λ), u_{−}(z)⟩_{−∞}^{n} = (z − λ)^{−1}W_{n}(u_{−}(λ), u_{−}(z)) = O(z^{n}) as |z| → ∞. Now we see that ⟨u_{−}(λ), u_{−}(z)⟩_{−∞}^{n} = u_{−}(λ, n)u_{−}(z, n) + u_{−}(λ, n − 1)u_{−}(z, n − 1) + O(z^{n−2}) and thus

(11.132) det U^{N}(u_{−}(z))(n) = u(z, n) det D_{0}^{N}(n)+u(z, n−1) det D_{1}^{N}(n)+O(z^{n−2}).

Using (6.27) to expand u(z, n) and comparing coefficients as indicated above shows det D_{0}^{N}(n) = det C^{N}(n − 1) plus the formula for b_{γ_{1},...,γ_{N}}. The rest follows in a straightforward manner. □

Remark 11.28. (i). Using (11.107) instead of (11.98) one infers that

(11.133)  u_{γ_{1},...,γ_{N}} (z, n) = \frac{det U^{N}(u(z, n))}{\sqrt{det C^{N}(n − 1) det C^{N}(n)}},

where

(11.134) U^{N}(u(z, n)) = ( \begin{matrix}C^{N}(n)_{r,s} & r,s≤N \\ \frac{1}{z−λ_{s}} W_{n}(u_{−}(λ_{s}), u(z)) & s≤N,r=N+1 \\ u_{−}(λ_{r}, n) & r≤N,s=N+1 \\ u(z, n) & r=s=N+1 \end{matrix} )_{1≤r,s≤N+1}

satisfies τ_{γ_{1},...,γ_{N}} u = zu if u(z, .) satisfies τu = zu. (ii). Equation (11.129) can be rephrased as

(u_{γ_{1},...,γ_{N} ,−}(λ_{1}, n), …, u_{γ_{1},...,γ_{N} ,−}(λ_{N}, n)) =

\sqrt{\frac{det C^{N}(n)}{det C^{N}(n − 1)}}(C^{N}(n))^{−1}(u_{−}(λ_{1}, n), …, u_{−}(λ_{N}, n)), (11.135)

where (C^{ℓ}(n))^{−1} is the inverse matrix of C^{ℓ}(n). (iii). The ordering of the pairs (λ_{j}, γ_{j}), 1 ≤ j ≤ N, is clearly irrelevant (interchanging row i, j and column i, j leaves all determinants unchanged). Moreover, if λ_{i} = λ_{j}, then (λ_{i}, γ_{i}), (λ_{j}, γ_{j}) can be replaced by (λ_{i}, γ_{i} + γ_{j}) (by the first assertion it suffices to verify this for N = 2).

Finally, we also extend Lemma 11.26. For simplicity we assume u_{−}(λ_{j}, n) ∉ ℓ^{2}(ℤ), γ_{j} > 0, 1 ≤ j ≤ N.

Lemma 11.29. Let H be a given Jacobi operator satisfying (H.10.1). Then the operator H_{γ_{1},...,γ_{N}}, defined via u_{−}(λ_{ℓ}, n) = f_{−}(k_{ℓ}, n) with λ_{ℓ} = (k_{ℓ} + k_{ℓ}^{−1})/2 in ℝ\σ(H_{γ_{1},...,γ_{ℓ−1}}), 1 ≤ ℓ ≤ N, has the transmission and reflection coefficients

T_{γ_{1},...,γ_{N}} (k) = ( ∏_{ℓ=1}^{N} sgn(k_{ℓ})\frac{k k_{ℓ} − 1}{k − k_{ℓ}})T (k),
R_{−,γ_{1},...,γ_{N}} (k) = R_{−}(k),
(11.136)  R_{+,γ_{1},...,γ_{N}} (k) = (∏_{ℓ=1}^{N} (\frac{k − k_{ℓ}}{k k_{ℓ} − 1})^{2}) R_{+}(k),

where z = (k + k^{−1})/2. Furthermore, the norming constants γ_{−,j} corresponding to λ_{j} ∈ σ_{p}(H), j ∈ J, (cf. (10.53)) remain unchanged and the additional eigenvalues λ_{ℓ} have norming constants γ_{−,ℓ} = γ_{ℓ}.

Remark 11.30. Of special importance is the case a(n) = 1/2, b(n) = 0. Here we have f_{±}(k, n) = k^{±n}, T (k) = 1, and R_{±}(k) = 0. We know from Section 10.4 that R_{±}(k), |k| = 1 together with the point spectrum and corresponding norming constants uniquely determine a(n), b(n). Hence we infer from Lemma 11.12 that H_{γ_{1},...,γ_{N}} constructed from u_{−}(λ_{ℓ}, n) = k_{ℓ}^{−n} as in Theorem 11.27 and H_{σ_{1},...,σ_{N}} constructed from u_{σ_{ℓ}}^{ℓ} = \frac{1+σ_{ℓ}}{2} k_{ℓ}^{n} + (−1)^{ℓ+1} \frac{1−σ_{ℓ}}{2} k_{ℓ}^{−n} as in Theorem 11.10 coincide if

(11.137)  γ_{j} = (\frac{1 − σ_{j}}{1 + σ_{j}} )^{−1} |k_{j}|^{−1−N} \frac{∏_{ℓ=1}^{N} |1 − k_{j}k_{ℓ}|}{∏N_{{ℓ=1 \atop ℓ≠j}} |k_{j} − k_{ℓ}|}, 1 ≤ j ≤ N.

11.8. The Dirichlet deformation method

In this section we will see that the method of changing individual Dirichlet eigenvalues found in Section 11.4 for quasi-periodic finite-gap sequences works in a much more general setting.

Let us formulate our basic hypothesis.

Hypothesis H.11.31. (i). Let (E_{0}, E_{1}) be a spectral gap of H, that is, (E_{0}, E_{1})∩ σ(H) = {E_{0}, E_{1}}. (ii). Suppose μ_{0} ∈ σ_{d}(H_{σ_{0}}) ∩ [E_{0}, E_{1}]. (iii). Let (μ, σ) ∈ [E_{0}, E_{1}] × {±} and μ ∈ (E_{0}, E_{1}) or μ ∈ σ_{d}(H).

Remark 11.32. If μ_{0} is an eigenvalue of two of the operators H, H_{−}, H_{+}, then it is also one of the third. Hence, if μ_{0} ∈ σ_{d}(H_{σ_{0}})\σ_{d}(H_{−σ_{0}}), then μ_{0} ∈ (E_{0}, E_{1}) and if μ_{0} ∈ σ_{d}(H_{σ_{0}}) ∩ σ_{d}(H_{−σ_{0}}), then μ_{0} ∈ {E_{0}, E_{1}}. (The choice of σ_{0} in the latter case is irrelevant). Condition (ii) thus says that μ_{0} = E_{0,1} is only allowed if E_{0,1} is a discrete eigenvalue of H. Similarly in (iii) for μ.

Our next objective is to define the operator H_{(μ,σ)} as in Section 11.4 by performing two single commutations using u_{σ_{0}}(μ_{0}, .) and u_{−σ}(μ, .). Since H_{(μ_{0},σ_{0})} = H, we will assume (μ, σ) ≠ (μ_{0}, σ_{0}) without restriction.

Due to our assumption (H.11.31), we can find solutions u_{σ_{0}}(μ_{0}, .), u_{−σ}(μ, .) (cf. Lemma 2.2) and define

(11.138)  W_{(μ,σ)}(n) = \{ \begin{matrix}\frac{W_{n}(u_{σ_{0}} (μ_{0}),u_{−σ}(μ))}{μ−μ_{0}} , & μ ≠ μ_{0} \\ ∑_{m=σ_{0}∞}^{n} u_{σ_{0}}(μ_{0}, m)^{2}, & (μ, σ) = (μ_{0}, −σ_{0}) \end{matrix} ,

where ∑_{m=+∞}^{n} = − ∑_{m=n+1}^{∞}. The motivation for the case (μ, σ) = (μ_{0}, −σ_{0}) follows from (assuming u_{−σ}(μ, m) holomorphic w.r.t. μ – compare (2.32))

lim_{μ→μ_{0}} \frac{W_{n}(u_{σ_{0}}(μ_{0}), u_{σ_{0}}(μ))}{μ − μ_{0}} = lim_{μ→μ_{0}} ∑_{m=σ_{0}∞}^{n} u_{σ_{0}}(μ_{0}, m)u_{σ_{0}}(μ, m)

(11.139)  = ∑_{m=σ_{0}∞}^{n} u_{σ_{0}}(μ_{0}, m)^{2}.

From the proof of Theorem 4.19 we infer

Lemma 11.33. Suppose (H.11.31), then

(11.140)  W_{(μ,σ)}(n + 1)W_{(μ,σ)}(n) > 0, n ∈ ℤ.

Thus the sequences

a_{(μ,σ)}(n) = a(n)\sqrt{\frac{W_{(μ,σ)}(n − 1)W_{(μ,σ)}(n + 1)}{W_{(μ,σ)}(n)^{2}}},

(11.141)  b_{(μ,σ)}(n) = b(n) − ∂^{∗} \frac{a(n)u_{σ_{0}}(μ_{0}, n)u_{−σ}(μ, n + 1)}{W_{(μ,σ)}(n)},

are both well-defined and we can consider the associated difference expression

(11.142) (τ_{(μ,σ)}u)(n) = a_{(μ,σ)}(n)u(n + 1) + a_{(μ,σ)}(n − 1)u(n − 1) + b_{(μ,σ)}(n)u(n).

However, we do not know at this point whether a_{(μ,σ)}(n), b_{(μ,σ)}(n) are bounded or not. This will be ignored for the moment. The next lemma collects some basic properties which follow either from Section 11.3 (choosing N = 2 in Theorem 11.10) or can be verified directly.

Lemma 11.34. Let

(11.143) (A_{(μ,σ)}u)(z, n) = \frac{W_{(μ,σ)}(n)u(z, n) − \frac{1}{z−μ_{0}} u_{−σ}(μ, n)W_{n}(u_{σ_{0}}(μ_{0}), u(z))}{\sqrt{W_{(μ,σ)}(n − 1)W_{(μ,σ)}(n)}},

where u(z) solves τu = zu for z ∈ ℂ\{μ_{0}}. Then we have

(11.144)  τ_{(μ,σ)}(A_{(μ,σ)}u)(z, n) = z(A_{(μ,σ)}u)(z, n)

and (11.145)

|(A_{(μ,σ)}u)(z, n)|^{2} = |u(z, n)|^{2} + \frac{1}{|z − μ_{0}|^{2}} \frac{u_{−σ}(μ, n)}{u_{σ_{0}}(μ_{0}, n)}∂^{∗} \frac{|W_{n}(u_{σ_{0}}(μ_{0}), u(z))|^{2}}{W_{(μ,σ)}(n)} .

Moreover, the sequences

u_{μ_{0}}(n) = \frac{u_{−σ}(μ, n)}{\sqrt{W_{(μ,σ)}(n − 1)W_{(μ,σ)}(n)}},
(11.146)  u_{μ}(n) = \frac{u_{σ_{0}}(μ_{0}, n)}{\sqrt{W_{(μ,σ)}(n − 1)W_{(μ,σ)}(n)}},

satisfy τ_{(μ,σ)}u = μ_{0}u, τ_{(μ,σ)}u = μu respectively. Note also u_{σ_{0}}(μ_{0}, 0) = u_{μ}(0) = 0 and

(11.147)  u_{μ_{0}}(n)u_{μ}(n) = ∂^{∗} \frac{1}{W_{(μ,σ)}(n)}.

In addition, let u(z), û(z) satisfy τu = zu, then

W_{(μ,σ),n}(u_{μ_{0}}, A_{(μ,σ)}u(z)) = \frac{W_{n}(u_{−σ}(μ), u(z))}{W_{(μ,σ)}(n)},

W_{(μ,σ),n}(u_{μ}, A_{(μ,σ)}u(z)) = \frac{z − μ}{z − μ_{0}} \frac{W_{n}(u_{σ_{0}}(μ_{0}), u(z))}{W_{(μ,σ)}(n)},

W_{(μ,σ),n}(A_{(μ,σ)}u(z), A_{(μ,σ)}û(ẑ)) = \frac{z − μ}{z − μ_{0}} W_{n}(u(z), û(ẑ))

+ \frac{z − ẑ}{(z − μ_{0})(ẑ − μ_{0})} \frac{W_{n}(u_{σ}(μ), u(z))W_{n}(u_{−σ_{0}}(μ_{0}), û(ẑ))}{W_{(μ,σ)}(n)}, (11.148)

where W_{(μ,σ),n}(u, v) = a_{(μ,σ)}(n)(u(n)v(n + 1) − u(n + 1)v(n)).

Since we do not know whether a_{(μ,σ)}(n), b_{(μ,σ)}(n) are bounded or not, we need to (temporarily) introduce suitable boundary conditions for τ_{(μ,σ)} (compare Section 2.6). They will soon turn out to be superfluous.

Let ω ∈ {±} and

(11.149)  BC_{ω}(f) = \{ \begin{matrix}lim_{n→ω∞} W_{n}(u_{ω}, f) = 0 if & τ_{(μ,σ)} is l.c. at ω∞ \\ 0  if & τ_{(μ,σ)} is l.p. at ω∞ \end{matrix} ,

where u_{ω} is u_{ω} = u_{μ} or u_{ω} = u_{μ_{0}}

Using this boundary conditions we define

(11.150)  \begin{aligned}H_{(μ,σ)} : & 𝔇(H_{(μ,σ)}) & → & ℓ^{2}(ℤ) \\ & f & ↦ & τ_{(μ,σ)}f \end{aligned} ,

where the domain of H_{(μ,σ)} is explicitly given by

(11.151) 𝔇(H_{(μ,σ)}) = {f ∈ ℓ^{2}(ℤ)|τ_{(μ,σ)}f ∈ ℓ^{2}(ℤ), BC_{−}(f) = BC_{+}(f) = 0}.

Furthermore, H_{(μ,σ),±} denote the corresponding Dirichlet half-line operators with respect to the base point n_{0} = 0.

We first give a complete spectral characterization of the half-line operators H_{(μ,σ),±}. In addition, this will provide all necessary results for the investigation of H_{(μ,σ)}.

To begin with we compute the Weyl m̃-functions of H_{(μ,σ)}.

Theorem 11.35. Let m̃_{±}(z) and m̃_{(μ,σ),±}(z) denote the Weyl m̃-functions of H and H_{(μ,σ)} respectively. Then we have

(11.152)  m̃_{(μ,σ),±}(z) = \frac{1}{1 + γ_{(μ,σ)}} (\frac{z − μ_{0}}{z − μ} m̃_{±}(z) ∓ \frac{γ_{(μ,σ)}}{z − μ} ),

where

(11.153)  γ_{(μ,σ)} = \{ \begin{matrix}−σ(μ − μ_{0})m̃_{−σ}(μ), & μ ≠ μ_{0} \\ −\frac{σ_{0}u_{σ_{0}} (μ_{0},σ_{0}1)^{2}}{∑_{n∈σ_{0}ℕ} u_{σ_{0}} (μ_{0},n)^{2}} , & (μ, σ) = (μ_{0}, −σ_{0}) \end{matrix} .

In particular, H_{(μ,σ),±} and thus a_{(μ,σ)}, b_{(μ,σ)} are bounded.

Proof. We first note that

c_{(μ,σ)}(z, n) = \frac{z − μ_{0}}{z − μ} (A_{(μ,σ)}c)(z, n) − \frac{γ_{(μ,σ)}}{z − μ} a(0)(A_{(μ,σ)}s)(z, n),
(11.154) s_{(μ,σ)}(z, n) = \sqrt{1 + γ_{(μ,σ)}}(A_{(μ,σ)}s)(z, n)

constructed from the fundamental system c(z, n), s(z, n) for τ form a fundamental system for τ_{(μ,σ)} corresponding to the same initial conditions. Furthermore, note

(11.155)  \frac{W_{(μ,σ)}(1)}{W_{(μ,σ)}(0)} = 1 + γ_{(μ,σ)}, \frac{W_{(μ,σ)}(0)}{W_{(μ,σ)}(−1)} = 1.

Now the result follows upon evaluating (cf. Section 2.6)

(11.156)  m̃_{(μ,σ),±}(z) = \frac{±1}{a_{(μ,σ)}(0)} lim_{n→±∞} \frac{W_{(μ,σ),n}(c_{(μ,σ)}(z), u_{ω})}{W_{(μ,σ),n}(s_{(μ,σ)}(z), u_{ω})}.

Using (11.148) one obtains for u_{ω} = u_{μ}(n), u_{μ_{0}}(n) and v_{ω} = u_{σ_{0}}(μ_{0}, n), u_{σ_{0}}(μ_{0}, n), respectively,

(11.157) m̃_{(μ,σ),±}(z) = \frac{1}{1 + γ_{(μ,σ)}} (\frac{z − μ_{0}}{z − μ} \frac{±1}{a(0)} lim_{n→±∞} \frac{W_{n}(c(z), v_{ω})}{W_{n}(s(z), v_{ω})} ∓ \frac{γ_{(μ,σ)}}{z − μ} ).

Hence the claim is evident. □

In particular, since H_{(μ,σ)} is bounded (away from μ, μ_{0} the spectrum of H_{(μ,σ),±} is unchanged), we can forget about the boundary conditions (11.149) and H_{(μ,σ)} is defined on ℓ^{2}(ℤ). As a consequence we note Corollary 11.36. The sequences

(11.158)  u_{(μ,σ),±}(z, n) = (A_{(μ,σ)}u_{±})(z, n), z ∈ ℂ\{μ, μ_{0}},

are square summable near ±∞. Moreover, the same is true for

u_{(μ,σ),−σ}(μ, n) = W_{(μ,σ)}(n)u_{μ_{0}}(n) − u_{μ}(n) ∑_{m=−σ∞}^{n} u_{−σ}(μ, m)^{2},

(11.159) u_{(μ,σ),σ_{0}}(μ_{0}, n) = W_{(μ,σ)}(n)u_{μ}(n) − u_{μ_{0}}(n) ∑_{m=σ_{0}∞}^{n} u_{σ_{0}}(μ_{0}, m)^{2}

and

u_{(μ,σ),σ}(μ, n) = u_{μ}(n), μ ∉ σ_{d}(H),
(11.160)  u_{(μ,σ),−σ_{0}}(μ_{0}, n) = u_{μ_{0}}(n), μ_{0} ∉ σ_{d}(H).

If μ or μ_{0} ∈ σ_{d}(H) one has to replace the last formulas by

u_{(μ,σ),σ}(μ, n) = W_{(μ,σ)}(n)u_{μ_{0}}(n) − u_{μ}(n) ∑_{m=σ∞}^{n} u_{−σ}(μ, m)^{2},

(11.161) u_{(μ,σ),−σ_{0}}(μ_{0}, n) = W_{(μ,σ)}(n)u_{μ}(n) − u_{μ_{0}}(n) ∑_{m=−σ_{0}∞}^{n} u_{σ_{0}}(μ_{0}, m)^{2},

respectively.

Proof. Follows immediately from

(11.162) u_{(μ,σ),±}(z, n) = \frac{c_{±}(z)}{1 + γ_{(μ,σ)}} \frac{z − μ_{0}}{z − μ} (\frac{c_{(μ,σ)}(z, n)}{a_{(μ,σ)}(0)} ∓ m̃_{(μ,σ),±}(z)s_{(μ,σ)}(z, n))

if

(11.163)  u_{±}(z, n) = c_{±}(z)(\frac{c(z, n)}{a(0)} ∓ m̃_{±}(z)s(z, n)).

If z = μ, μ_{0} one can assume u_{±}(z) holomorphic with respect to z near μ, μ_{0} and consider limits (compare Section 2.2). □

Next we are interested in the pole structure of m̃_{(μ,σ),±}(z) near z = μ, μ_{0}. A straightforward investigation of (11.152) using the Herglotz property of m̃_{(μ,σ),±}(z) shows

Corollary 11.37. We have

(11.164)  m̃_{(μ,σ),ω}(z) = \{ \begin{matrix}−\frac{γ_{μ}}{z−μ} + O(z − μ)^{0}, & ω = σ \\ O(z − μ)^{0}, & ω = −σ \end{matrix} , ω ∈ \{±\},

where

(11.165) γ_{μ} = \{ \begin{matrix}(μ − μ_{0})(m̃_{+}(μ) + m̃_{−}(μ)), & μ ≠ μ_{0} \\ \frac{u_{−}(μ,−1)^{2}}{∑_{n=−∞}^{−1} u_{−}(μ,n)^{2}} + \frac{u_{+}(μ,1)^{2}}{∑_{n=1}^{+∞} u_{+}(μ,n)^{2}} , & (μ, σ) = (μ_{0}, −σ_{0}) \end{matrix} ≥ 0.

In addition, γ_{μ} = 0 if μ ∈ σ_{d}(H)\{μ_{0}} and γ_{μ} > 0 otherwise. If μ ≠ μ_{0}, then

(11.166)  m̃_{(μ,σ),±}(z) = O(z − μ_{0})^{0},

and note γ_{μ} = a(0)^{−2}(μ − μ_{0})g(μ, 0)^{−1}.

Using the previous corollary plus weak convergence of π^{−1}Im(m̃_{±}(λ + iε))dλ to the corresponding spectral measure dρ̃_{±}(λ) as ε ↓ 0 implies

Lemma 11.38. Let dρ̃_{±}(λ) and dρ̃_{(μ,σ),±}(λ) be the respective spectral measures of m̃_{±}(z) and m̃_{(μ,σ),±}(z). Then we have

(11.167) dρ̃_{(μ,σ),±}(λ) = \frac{1}{1 + γ_{(μ,σ)}} (\frac{λ − μ_{0}}{λ − μ} dρ̃_{±}(λ)+\{ \begin{matrix}γ_{μ} dΘ(λ − μ) & σ = ± \\ 0 & σ = ∓ \end{matrix} \} ),

where γ_{(μ,σ)}, γ_{μ} are defined in (11.153), (11.165) and dΘ is the unit point measure concentrated at 0.

Let P_{±}(μ_{0}), P_{(μ,σ),±}(μ) denote the orthogonal projections onto the subspaces spanned by u_{σ_{0}}(μ_{0}, .), u_{μ}(.) in ℓ^{2}(±ℕ), respectively. Then the above results clearly imply

Theorem 11.39. The operators (1l−P_{±}(μ_{0}))H_{±} and (1l−P_{(μ,σ),±}(μ))H_{(μ,σ),±} are unitarily equivalent. Moreover, μ ∉ σ(H_{(μ,σ),−σ}) and μ_{0} ∉ σ(H_{(μ,σ),±})\{μ}.

If μ ∉ σ_{d}(H) or (μ, σ) = (μ_{0}, −σ_{0}), then μ ∈ σ(H_{(μ,σ),σ}) and thus

(11.168)  σ(H_{(μ,σ),±}) = (σ(H_{±})\\{μ_{0}\}) ∪ \{ \begin{matrix}\{μ\}, & σ = ± \\ ∅, & σ = ∓ \end{matrix} .

Otherwise, that is, if μ ∈ σ_{d}(H)\{μ_{0}}, then μ ∉ σ(H_{(μ,σ),σ}) and thus

(11.169)  σ(H_{(μ,σ),±}) = σ(H_{±})\{μ_{0}}.

In essence, Theorem 11.39 says that, as long as μ ∉ σ_{d}(H)\{μ_{0}}, the Dirichlet datum (μ_{0}, σ_{0}) is rendered into (μ, σ), whereas everything else remains unchanged. If μ ∈ σ_{d}(H)\{μ_{0}}, that is, if we are trying to move μ_{0} to an eigenvalue, then μ_{0} is removed. This latter case reflects the fact that we cannot move μ_{0} to an eigenvalue E without moving the Dirichlet eigenvalue on the other side of E to E at the same time (compare Section 8.2).

Remark 11.40. (i). For f ∈ ℓ(ℕ) set

(A_{(μ,σ),+}f)(n) = \sqrt{\frac{W_{(μ,σ)}(n)}{W_{(μ,σ)}(n − 1)}}f(n)

− u_{−σ_{0},(μ,σ)}(μ_{0}, n) ∑_{j=1}^{n} u_{σ_{0}}(μ_{0}, j)f(j),

(A_{(μ,σ),+}^{−1}f)(n) = \sqrt{\frac{W_{(μ,σ)}(n − 1)}{W_{(μ,σ)}(n)}}f(n)

− u_{−σ}(μ, n) ∑_{j=1}^{n} u_{σ,(μ,σ)}(μ, j)f(j). (11.170)

Then we have A_{(μ,σ),+}A_{(μ,σ),+}^{−1} = A_{(μ,σ),+}^{−1}A_{(μ,σ),+} = 1l_{ℓ(ℕ)} and A_{(μ,σ),+} acts as transformation operator

(11.171)  τ_{(μ,σ),+} = A_{(μ,σ),+}τ_{+}A_{(μ,σ),+}^{−1}.

Similarly, for f ∈ ℓ(−ℕ) set

(A_{(μ,σ),−}f)(n) = \sqrt{\frac{W_{(μ,σ)}(n)}{W_{(μ,σ)}(n − 1)}}f(n)

− u_{−σ_{0},(μ,σ)}(μ_{0}, n) ∑_{j=n+1}^{1} u_{σ_{0}}(μ_{0}, j)f(j),

(A_{(μ,σ),−}^{−1}f)(n) = \sqrt{\frac{W_{(μ,σ)}(n − 1)}{W_{(μ,σ)}(n)}}f(n)

− u_{−σ}(μ, n) ∑_{j=n+1}^{1} u_{σ,(μ,σ)}(μ, j)f(j). (11.172)

Then we have A_{(μ,σ),−}A_{(μ,σ),−}^{−1} = A_{(μ,σ),−}^{−1}A_{(μ,σ),−} = 1l_{ℓ(−ℕ)} and

(11.173)  τ_{(μ,σ),−} = A_{(μ,σ),−}τ_{−}A_{(μ,σ),−}^{−1}.

(ii). Note that the case (μ, σ) = (μ_{0}, −σ_{0}) corresponds to the double commutation method with γ_{1} = ∞ (cf. Section 11.6). Furthermore, the operators A_{(μ,σ),±} are unitary when restricted to proper subspaces of ℓ^{2}(±ℕ) in this case. (iii). Due to the factor \frac{z−μ_{0}}{z−μ} in front of m̃_{(μ,σ),±}(z), all norming constants (i.e., the negative residues at each pole of m̃_{(μ,σ),±}(z)) are altered. (iv). Clearly, by Theorem 11.35, the Dirichlet deformation method preserves reflectionless properties (cf. Lemma 8.1).

Having these results at our disposal, we can now easily deduce all spectral properties of the operator H_{(μ,σ)}. First of all, Theorem 11.35 yields for the Weyl matrix of H_{(μ,σ)}.

Theorem 11.41. The respective Weyl M-matrices M(z), M_{(μ,σ)}(z) of H, H_{(μ,σ)} are related by

M_{(μ,σ),0,0}(z) = \frac{1}{(1 + γ_{(μ,σ)})^{2}} \frac{z − μ}{z − μ_{0}} M_{0,0}(z),
M_{(μ,σ),0,1}(z) = \frac{1}{1 + γ_{(μ,σ)}} (M_{0,1}(z) + \frac{γ_{(μ,σ)}}{z − μ_{0}} a(0)M_{0,0}(z)),
M_{(μ,σ),1,1}(z) = \frac{z − μ_{0}}{z − μ} M_{1,1}(z) − 2\frac{γ_{(μ,σ)}}{z − μ_{0}} a(0)M_{0,1}(z)

+ \frac{γ_{(μ,σ)}^{2}}{(z − μ_{0})(z − μ)}a(0)^{2}M_{0,0}(z). (11.174)

Moreover, M_{(μ,σ),j,k}(z, m, n), j, k ∈ {0, 1}, are holomorphic near z = μ, μ_{0}.

Given the connection between M(z) and M_{(μ,σ)}(z), we can compute the corresponding Herglotz matrix measure of M_{(μ,σ)}(z) as in Lemma 11.38.

Lemma 11.42. The matrix measures dρ and dρ_{(μ,σ)} corresponding to M(z) and M_{(μ,σ)}(z) are related by

dρ_{(μ,σ),0,0}(λ) = \frac{1}{(1 + γ_{(μ,σ)})^{2}} \frac{λ − μ}{λ − μ_{0}} dρ_{0,0}(λ),
dρ_{(μ,σ),0,1}(λ) = \frac{1}{1 + γ_{(μ,σ)}} (dρ_{0,1}(λ) + \frac{γ_{(μ,σ)}}{λ − μ_{0}} a(0)dρ_{0,0}(λ)),
dρ_{(μ,σ),1,1}(λ) = \frac{λ − μ_{0}}{λ − μ} dρ_{1,1}(λ) − 2\frac{γ_{(μ,σ)}}{λ − μ_{0}} a(0)dρ_{0,1}(λ)

+ \frac{γ_{(μ,σ)}^{2}}{(λ − μ_{0})(λ − μ)}a(0)^{2}dρ_{0,0}(λ). (11.175)

Equivalently

dρ_{(μ,σ)}(λ) = \frac{1}{(z − μ)(z − μ_{0})} ×

( \begin{matrix}\frac{z−μ}{1+γ_{(μ,σ)}} & 0 \\ a(0)γ_{(μ,σ)} & z − μ_{0} \end{matrix} ) dρ(λ) ( \begin{matrix}\frac{z−μ}{1+γ_{(μ,σ)}} & a(0)γ_{(μ,σ)} \\ 0 & z − μ_{0} \end{matrix} ) . (11.176)

This finally leads to our main theorem in this section

Theorem 11.43. Let H, H_{(μ,σ)} be defined as before. Denote by P (μ_{0}) and P (μ) the orthogonal projections corresponding to the spaces spanned by u_{σ_{0}}(μ_{0}, .) and u_{−σ}(μ, 0) in ℓ^{2}(ℤ) respectively. Then (1l−P (μ_{0})−P (μ))H and H_{(μ,σ)} are unitarily equivalent. In particular, H and H_{(μ,σ)} are unitarily equivalent if μ, μ_{0} ∉ σ_{d}(H).

Remark 11.44. By inspection, Dirichlet deformations produce the commuting diagram

[figure: (μ_{1}, σ_{1}); (μ_{0}, σ_{0}); (μ_{2}, σ_{2})]

for (μ_{j}, σ_{j}), 0 ≤ j ≤ 2, according to (H.11.31). In fact, this can be easily verified using (the first two) formulas for the spectral measures (11.175) which uniquely determine the corresponding operator.

Remark 11.45. We have seen in Theorem 11.39 that the Dirichlet deformation method cannot create a situation where a discrete eigenvalue E of H is rendered into a Dirichlet eigenvalue (i.e., moving μ_{0} to the eigenvalue E). However, one can use the following three-step procedure to generate a prescribed degeneracy at an eigenvalue E of H:

(i). Use the Dirichlet deformation method to move μ to a discrete eigenvalue E of H. (This removes both the discrete eigenvalue E of H and the (Dirichlet) eigenvalue μ of H_{−} ⊕ H_{+}).

(ii) As a consequence of step (i), there is now another eigenvalue μ̃ of H_{−}⊕H_{+} in the resulting larger spectral gap of H. Move μ̃ to E using the Dirichlet deformation method.

(iii) Use the double commutation method to insert an eigenvalue of H at E and to change σ_{0} into any allowed value.

Theorem 11.16 then shows that the resulting operator is unitarily equivalent to the original operator H, and Theorem 11.21 proves that the remaining Dirichlet eigenvalues remain invariant.

Finally, we briefly comment on how to iterate the Dirichlet deformation method. All ingredients have already been derived in Section 11.3. Suppose

(11.177)  (E_{0,j}, E_{1,j}), (μ_{0,j}, σ_{0,j}), (μ_{j}, σ_{j}) ∈ [E_{0,j}, E_{1,j}] × {±}

satisfy (H.11.31) for each j = 1, …, N, N ∈ ℕ. Then the Dirichlet deformation result after N steps, denoted by H_{(μ_{1},σ_{1}),...,(μ_{N} ,σ_{N})}, is associated with the sequences

a_{(μ_{1},σ_{1}),...,(μ_{N} ,σ_{N})}(n) = \sqrt{a(n − N)a(n + N)}

×\frac{\sqrt{C_{n−N}(U_{1,...,N})C_{n−N+2}(U_{1,...,N})}}{C_{n−N+1}(U_{1,...,N})},

b_{(μ_{1},σ_{1}),...,(μ_{N} ,σ_{N})}(n) = b(n) − ∂^{∗} a(n)\frac{D̃_{n−N+1}(U_{1,...,N})}{C_{n−N+1}(U_{1,...,N})}, (11.178)

where (U_{1,...,N}) = (u_{σ_{0,1}}(μ_{0,1}), u_{σ_{1}}(μ_{1}), …, u_{σ_{0,N}} (μ_{0,N}), u_{σ_{N}} (μ_{N})).

Notes on literature

Chapter 1. The literature on second order difference equations is enormous and tracing all results would require a historian. Hence I can only give some hints which serve as starting points. Classical references are the books by Atkinson [19], Berezanskiĭ [27] (Chapter 7), and Fort [87]. A more recent reference would be the book by Carmona and Lacroix [40] which focuses on random operators. Clearly, there are far more books on this topic available, but these are the ones I found most useful for my purpose. However, note that not everything in the first chapter can be found in these references. I have added some results, needed in later chapters, which I could not find in the literature (e.g., I don’t know a reference for Lemma 1.2).

In addition, recent and easy introductions to difference equations can be found in, for instance, Jordan [141], Kelley and Peterson [147], and Mickens [171]. A book which contains numerous examples and special results is the one by Agarwal [4].

Chapter 2. The theory presented in this chapter is the discrete analog of what is known as Weyl-Titchmarsh-Kodaira theory for Sturm-Liouville operators. The continuous theory originated in the celebrated work of Weyl [244] with subsequent extensions by Titchmarsh ([227], [228]), Kodaira and numerous others (see any book on Sturm-Liouville operators for further details). The origins of the discrete theory go back to Hellinger [131] and Nevanlinna [182]. Many results have been rediscovered several times and the literature on this subject is far to extensive to be quoted in full.

Again classical references are Atkinson [19], Berezanskiĭ [27] (Chapter 7), Fort [87], and the more recent monograph by Carmona and Lacroix [40]. My personal favorite though is Berezanskiĭ [27]. For connections with spectral theory on the half line and the moment problem I suggest the classical book by Akhiezer [9]. There is also a very good and recent survey article on the moment problem by Simon [209]. For the view point of orthogonal polynomials, the standard reference is the book of Szegö [217]. You might also have a look at [211], which has emphasis on applications, or [57], which uses a Riemann-Hilbert approach.

Especially for relations between Weyl m-functions and the theory of continued fractions see Atkinson [9] and also [24], [240]. A paper which closely follows the original ideas of Titchmarsh is [18]. There is a large bulk of papers on the question of essential self-adjointness of Jacobi operators (i.e., limit point, limit circle criteria). See for instance [9] (addenda and problems to Chapter 1), [19], [27], [110] (Appendix E), [132], [135], [188]. The characterization of boundary conditions in the limit circle case can be found in [242], [243] (however, a much more cumbersome approach is used there). A paper which focuses on the contributions by Krein is [186].

The investigation of positive operators respectively positive solutions has attracted quite some interest and many results have been discovered more than once. Our section compiles results from [26], [111], and [188]. I have streamlined most arguments and simplified the proofs. Further references are [6], [133], and [191].

The section on inverse problems is taken from [222]. Additional results can be found in [88], [89], [92] and, in the case of finite Jacobi matrices, [104].

There are extensions of many results to block Jacobi matrices (i.e., Jacobi matrices whose coefficients are again matrices). The interested reader might look at (e.g.) [5], [7], or [245].

Chapter 3. The first section summarizes some well-known results, which however are often hard to find. Some of them don’t seem to be written up at all and I can’t give a good reference for this section. For the continuous case you should look at the beautiful paper of Simon [204] and, again, at the book of Berezanskiĭ [27].

The second section extends results of [215] to the case a(n) ≠ 1 and provides a few basic facts for easy reference.

The absolutely (resp. singularly) continuous spectrum is one of the most investigated object in spectral theory. The contents of this section clearly has its origin in the beautiful theory of Gilbert and Pearson [120] which is known as the principle of subordinacy. This theory has since then been simplified and extended by various people (see e.g., [119], [48], [145]). I follow ideas of [68], [139], [163], and [207]. For some general results on singular continuous spectra see [208].

Bounds on the number of eigenvalues are considered in [95] and [96] (see also [195], Theorem XIII.10).

Additional topics like embedded eigenvalues, slowly decreasing potentials, or some explicitly solvable problems can be found in [179], [180], [58], [149], [215], or [77]. A different approach to find classes of operators with absolutely continuous spectral measures is presented in [71] (see also [72]).

Chapter 4. Oscillation theory originates in the work of Sturm [216]. Numerous papers have been written on this subject in the case of differential equations and several monographs are devoted solely to this topic. In the discrete case there is a large amount of papers investigating whether a given Jacobi equation is oscillatory or not (e.g., [128], [132], [133], [134], [188], and [189]). The case of finite matrices is treated (e.g.) in the textbook by Fort [87]. For the case of infinite matrices only the connection between (non)oscillation and the infimum of the essential spectrum was known ([121], Theorem 32). The precise connection between the number of oscillations and the spectrum was only clarified recently in [223]. This article is also the main reference for this chapter. Additional references are [19], [26], and [147].

Chapter 5. The first main theorem, that the spectrum is a.s. the same for all operators is due to Pastur [187]. The results concerning the Lyapunov exponent are mainly taken from [49], the ones for the integrated density of states respectively the rotation number are mainly collected from [69], [70], and [22]. It is sometimes claimed that the process (5.48) is subadditve (which is true for differential operators) so I tried to clarify this point. The results on the absolutely continuous spectrum originate from work of Kotani and are taken from [205]. The section on almost periodic operators summarizes some results from [21] and [122]. There have recently been some efforts to prove uniform spectral results (i.e, for all operators in the hull), see, for example, [51], [52] and the references therein. In this respect, it should be mentioned that the absolutely continuous spectrum is constant for all operators in the hull ([163], Theorem 1.5). It is also known that the other parts need not be constant (see [140]). If you want to find out almost everything about the almost Mathieu operator, I suggest [138] and [162].

The reader who wants to learn more about this subject should consult the monograph by Carmona and Lacroix [40] which is devoted entirely to this subject. Another brief introduction can be found in Cycon et al. [50] (Chapters 9 and 10). For some recent generalizations to orthogonal polynomials on the unit circle see [100].

Chapter 6. The first trace formulas for Schrödinger operators on a finite interval were given by Gel’fand and Levitan [93]. Since then numerous extensions have been made. In particular, recently Gesztesy and Simon showed [105] that all main results can be viewed as special cases of Krein’s spectral shift theory [155]. Most parts of this section are taken from [222].

Chapter 7. Floquet (resp. Bloch) theory is one of the classical results found in most textbooks on differential equations. In the case of periodic Jacobi operators the first spectral characterization was given by [173]. The first two sections contain the analogs of several results from periodic Sturm-Liouville operators. The third section is somewhat inspired by [169] (see also Chapter 8). Further results on perturbed periodic operators can be found in [98], [121] (Section 67), [150], and [197]. For extensions to complex or operator valued coefficients see [176] or [165], respectively.

Chapter 8. Reflectionless operators have attracted several mathematicians since they allow an explicit treatment. In the first two sections I follow closely ideas of [113] and [222]. However, I have added some additional material. Complementary results to this sections, namely for the case of Cantor spectra, can be found in [12] – [14], and [212], [213].

The remaining sections are of importance for the second part. The approach is inspired by ideas of McKean [169]. Some parts are taken from [35] (see also [173]), but again the material has been extended to a larger class of operators and augmented.

Chapter 9. The use of Riemann surfaces usually appears in connection with the Toda lattice. The first results seem due to Krichever (see, e.g., [156] – [160]). The main reference for this chapter is again [35]. Additional references are [74], [169], [173], [230], and [175]. This technique can also be extended to surfaces with infinitely many branch points [12] – [14], [212] and [213].

For the explicit calculations in the genus one case, I have used the tables for elliptic integrals in [38]. In addition, you might also find [10], [123], and [127] useful.

Chapter 10. Discrete scattering theory has been first developed on an informal level by Case in a series of papers [42] – [47]. The first rigorous results were given by Guseinov [124], but unfortunately without proof. Additional results are given in [125], [126], and [224]. This chapter extends the results of [124] and gives (to the best of my knowledge) the first proof in print. Most of the other results are also new and have not previously appeared elsewhere. Further references are [80] (Chap. 2.III), [94], [95], [97], [102], and [230]. For generalizations to polynomials on the unit circle see [99].

Chapter 11. The single commutation method for Jacobi operators first appeared in Gesztesy et al. [117]. A special case of the double commutation method can be found in [235]. The general case of the double commutation method seems due to [110] which is one of the the main sources for this chapter. It also contains the analogue of the FITformula (Section 11.4) derived in [83] for Schrödinger operators. The second main source is [221] where the discrete Dirichlet deformation method is introduced.

Part 2

Completely Integrable Nonlinear Lattices The Toda system

This chapter is devoted to the Toda hierarchy. The first section gives an informal introduction and is mainly for background and motivation. The following sections contain a rigorous treatment based on the Lax pair formalism. The basic existence and uniqueness theorem for solutions of the initial value problem is proven and the connection with hyperelliptic Riemann surfaces is established.

12.1. The Toda lattice

The Toda lattice is a simple model for a nonlinear one-dimensional crystal. It describes the motion of a chain of particles with nearest neighbor interaction. The equation of motion for such a system is given by

(12.1)  m\frac{d^{2}}{dt^{2}} x(n, t) = V′(x(n + 1, t) − x(n, t)) − V′(x(n, t) − x(n − 1, t)),

where m denotes the mass of each particle, x(n, t) is the displacement of the n-th particle from its equilibrium position, and V (r) (V′(r) = \frac{dV}{dr} (r)) is the interaction potential. As discovered by M. Toda, this system gets particularly interesting if one chooses an exponential interaction,

(12.2)  V (r) = \frac{mρ^{2}}{τ^{2}} (e−r/ρ + \frac{r}{ρ} − 1) = \frac{mρ^{2}}{2τ^{2}} ((\frac{r}{ρ})^{2} + O(\frac{r}{ρ})^{3}), τ, ρ ∈ ℝ.

This model is of course only valid as long as the relative displacement is not too large (i.e., at least smaller than the distance of the particles in the equilibrium position). For small displacements it is equal to a harmonic crystal with force constant \frac{m}{τ^{2}} (cf. Section 1.5).

After a scaling transformation, t ↦ t/τ, x ↦ x/ρ, we can assume m = τ = ρ = 1. If we suppose x(n, t) − x(n − 1, t) → 0, ẋ(n, t) → 0 sufficiently fast as |n| → ∞, we can introduce the Hamiltonian (q = x, p = ẋ)

(12.3)  H(p, q) = ∑_{n∈ℤ} (\frac{p(n, t)^{2}}{2} + (e−(q(n + 1, t) − q(n, t)) − 1))

and rewrite the equations of motion in the Hamiltonian form

\frac{d}{dt}p(n, t) = −\frac{∂H(p, q)}{∂q(n, t)}
= e−(q(n, t) − q(n − 1, t)) − e−(q(n + 1, t) − q(n, t)),
(12.4) \frac{d}{dt}q(n, t) = \frac{∂H(p, q)}{∂p(n, t)} = p(n, t).

We remark that these equations are invariant under the transformation

(12.5) p(n, t) → p(n, t) + p_{0} and q(n, t) → q(n, t) + q_{0} + p_{0}t, (p_{0}, q_{0}) ∈ ℝ^{2},

which reflects the fact that the dynamics remains unchanged by a uniform motion of the entire crystal.

The fact which makes the Toda lattice particularly interesting is the existence of soliton solutions. These are pulslike waves traveling through the crystal without changing their shape. Such solutions are rather special since from a generic linear equation one would expect spreading of wave packets (see Section 1.5) and from a generic nonlinear wave equation one would expect that solutions only exist for a finite time (breaking of waves).

The simplest example of such a solitary wave is the one-soliton solution

(12.6)  q_{1}(n, t) = q_{0} − ln \frac{1 + γ exp(−2κn ± 2 sinh(κ)t)}{1 + γ exp(−2κ(n − 1) ± 2 sinh(κ)t)}, κ, γ > 0.

It describes a single bump traveling trough the crystal with speed ± sinh(κ)/κ and width proportional to 1/κ. That is, the smaller the soliton the faster it propagates. It results in a total displacement

(12.7)  lim_{n→∞}(q_{1}(n, t) − q_{1}(−n, t)) = 2κ

of the crystal, which can equivalently be interpreted as the total compression of the crystal around the bump. The total moment and energy are given by

∑_{n∈ℤ} p_{1}(n, t) = 2 sinh(κ),
(12.8)  H(p_{1}, q_{1}) = 2(sinh(κ) cosh(κ) − κ).

Existence of such solutions is usually connected to complete integrability of the system which is indeed the case here. To see this, we introduce Flaschka’s variables

(12.9)  a(n, t) = \frac{1}{2}e−(q(n + 1, t) − q(n, t))/2,  b(n, t) = −\frac{1}{2}p(n, t)

and obtain the form most convenient for us

ȧ(n, t) = a(n, t)(b(n + 1, t) − b(n, t)),
(12.10)  ḃ(n, t) = 2(a(n, t)^{2} − a(n − 1, t)^{2}).

The inversion is given by

p(n, t) = −2b(n, t),
q(n, t) = q(n, 0) − 2 ∫_{0}^{t} b(n, s)ds

= q(0, 0) − 2 ∫_{0}^{t} b(0, s)ds − 2\begin{matrix}n−1 \\ ∑ & ∗ \\ j=0 \end{matrix} ln(2a(j, t))

(12.11)  = q(0, 0) − 2 ∫_{0}^{t} b(n, s)ds − 2\begin{matrix}n−1 \\ ∑ & ∗ \\ j=0 \end{matrix} ln(2a(j, 0)).

To show complete integrability it suffices to find a so-called Lax pair, that is, two operators H(t), P (t) such that the Lax equation

(12.12)  \frac{d}{dt}H(t) = P (t)H(t) − H(t)P (t)

is equivalent to (12.10). One can easily convince oneself that the choice

\begin{aligned}H(t) : & ℓ^{2}(ℤ) & → & ℓ^{2}(ℤ) \\ & f(n) & ↦ & a(n, t)f(n + 1) + a(n − 1, t)f(n − 1) + b(n, t)f(n) \end{aligned} ,

\begin{aligned}P (t) : & ℓ^{2}(ℤ) & → & ℓ^{2}(ℤ) \\ & f(n) & ↦ & a(n, t)f(n + 1) − a(n − 1, t)f(n − 1) \end{aligned} (12.13)

does the trick. Now the Lax equation implies that the operators H(t) for different t ∈ ℝ are all unitarily equivalent and that

(12.14)  tr(H(t)^{j} − H_{0}^{j}), j ∈ ℕ,

are conserved quantities, where H_{0} is the operator corresponding to the constant solution a_{0}(n, t) = \frac{1}{2}, b_{0}(n, t) = 0 (it is needed to make the trace converge). For example,

tr(H(t) − H_{0}) = ∑_{n∈ℤ} b(n, t) = −\frac{1}{2} ∑_{n∈ℤ} p(n, t) and

(12.15) tr(H(t)^{2} − H_{0}^{2}) = ∑_{n∈ℤ} b(n, t)^{2} + 2(a(n, t)^{2} − \frac{1}{4}) = \frac{1}{2}H(p, q)

correspond to conservation of the total momentum and the total energy, respectively.

This reformulation of the Toda equations as a Lax pair is the key to methods of solving the Toda equations based on spectral and inverse spectral theory for the Jacobi operator H.

Using these methods one can find the general N-soliton solution

(12.16)  q_{N}(n, t) = q_{0} − ln \frac{det(1l + C_{N}(n, t))}{det(1l + C_{N}(n − 1, t))},

where (12.17)

C_{N}(n, t) = (\frac{\sqrt{γ_{i}γ_{j}}}{1 − e^{−(κ_{i}+κ_{j})}} e−(κ^{i} + κ^{j})n − (σ^{i} sinh(κ^{i}) + σ^{j} sinh(κ^{j}))t)_{1≤i,j≤N}

with κ_{j}, γ_{j} > 0 and σ_{j} ∈ {±1}. One can also find (quasi-)periodic solutions using techniques from Riemann surfaces (respectively algebraic curves). Each such solution is associated with an hyperelliptic curve of the type

(12.18)  w^{2} = ∏_{j=0}^{2g+2}(z − E_{j}), E_{j} ∈ ℝ,

where E_{j}, 0 ≤ j ≤ 2g + 1, are the band edges of the spectrum of H (which is independent of t and hence determined by the initial conditions). One obtains

(12.19) q(n, t) = q_{0} − 2(tb̃ + n ln(2ã)) − ln \frac{θ(\underline{z}_{0} − 2n\underline{A}_{p_{0}}(∞_{+}) − 2t\underline{c}(g))}{θ(\underline{z}_{0} − 2(n − 1)\underline{A}_{p_{0}}(∞_{+}) − 2t\underline{c}(g))},

where \underline{z}_{0} ∈ ℝ^{g}, θ : ℝ^{g} → ℝ is the Riemann theta function associated with the hyperelliptic curve (12.18), and ã, b̃ ∈ ℝ, \underline{A}_{p_{0}}(∞_{+}), \underline{c}(g) ∈ ℝ^{g} are constants depending only on the curve (i.e., on E_{j}, 0 ≤ j ≤ 2g + 1). If q(n, 0), p(n, 0) are (quasi-) periodic with average 0, then ã = \frac{1}{2}, b̃ = 0.

The rest of this monograph is devoted to a rigorous mathematical investigation of these methods.

12.2. Lax pairs, the Toda hierarchy, and hyperelliptic curves

In this section we introduce the Toda hierarchy using a recursive approach for the standard Lax formalism and derive the Burchnall-Chaundy polynomials in connection with the stationary Toda hierarchy.

We let the sequences a, b depend on an additional parameter t ∈ ℝ and require

Hypothesis H. 12.1. Suppose a(t), b(t) satisfy

(12.20)  a(t), b(t) ∈ ℓ^{∞}(ℤ, ℝ), a(n, t) ≠ 0, (n, t) ∈ ℤ × ℝ,

and let t ∈ ℝ ↦ (a(t), b(t)) ∈ ℓ^{∞}(ℤ) ⊕ ℓ^{∞}(ℤ) be differentiable.

We introduce the corresponding operator H(t) as usual, that is,

(12.21) \begin{aligned}H(t) : & ℓ^{2}(ℤ) & → & ℓ^{2}(ℤ) \\ & f(n) & ↦ & a(n, t)f(n + 1) + a(n − 1, t)f(n − 1) + b(n, t)f(n) \end{aligned} .

The idea of the Lax formalism is to find a finite, skew-symmetric operator P_{2r+2}(t) such that the Lax equation

(12.22)  \frac{d}{dt}H(t) − [P_{2r+2}(t), H(t)] = 0, t ∈ ℝ,

(here [. , ..] denotes the commutator, i.e., [P, H] = P H −HP) holds. More precisely, we seek an operator P_{2r+2}(t) such that the commutator with H(t) is a symmetric difference operator of order at most two. Equation (12.22) will then give an evolution equation for a(t) and b(t). Our first theorem tells us what to choose for P_{2r+2}(t).

Theorem 12.2. Suppose P (t) is of order at most 2r + 2 and the commutator with H(t) is of order at most 2. Then P (t) is of the form

(12.23)  P (t) = ∑_{j=0}^{r} (c_{r−j}P̃_{2j+2}(t) + d_{r−j}H(t)^{j+1}) + d_{r+1}1l,

where c_{j}, d_{j} ∈ ℂ, 0 ≤ j ≤ r, d_{r+1} ∈ ℓ(ℤ), and

(12.24)  P̃_{2j+2}(t) = [H(t)^{j+1}]_{+} − [H(t)^{j+1}]_{−}

(cf. the notation in (1.11)) is called homogeneous Lax operator. Moreover, denote by P_{2r+2}(t) the operator P (t) with c_{0} = 1 and d_{j} = 0, 0 ≤ j ≤ r + 1. Then we have

(12.25) P_{2r+2}(t) = −H(t)^{r+1} + ∑_{j=0}^{r}(2a(t)g_{j}(t)S^{+} − h_{j}(t))H(t)^{r−j} + g_{r+1}(t),

where (g_{j}(n, t))_{0≤j≤r+1} and (h_{j}(n, t))_{0≤j≤r+1} are given by

g_{j}(n, t) = ∑_{ℓ=0}^{j} c_{j−ℓ}⟨δ_{n}, H(t)^{ℓ}δ_{n}⟩,

(12.26)  h_{j}(n, t) = 2a(n, t) ∑_{ℓ=0}^{j} c_{j−ℓ}⟨δ_{n+1}, H(t)^{ℓ}δ_{n}⟩ + c_{j+1}

and satisfy the recursion relations

g_{0} = 1, h_{0} = c_{1},
2g_{j+1} − h_{j} − h_{j}^{−} − 2bg_{j} = 0, 0 ≤ j ≤ r,
(12.27) h_{j+1} − h_{j+1}^{−} − 2(a^{2}g_{j}^{+} − (a^{−})^{2}g_{j}^{−}) − b(h_{j} − h_{j}^{−}) = 0, 0 ≤ j < r.

For the commutator we obtain

[P_{2r+2}(t), H(t)] = a(t)(g_{r+1}^{+}(t) − g_{r+1}(t))S^{+} + a^{−}(t)(g_{r+1}(t) − g_{r+1}^{−}(t))S^{−}

+ (h_{r+1}(t) − h_{r+1}^{−}(t)). (12.28)

Proof. By Lemma 1.2 we can write

(12.29) P (t) = −h_{−1}(t)H(t)^{r+1} + ∑_{j=0}^{r}(2a(t)g_{j}(t)S^{+} − h_{j}(t))H(t)^{r−j} + g_{r+1}(t),

where g_{r+1}(t) is only added for convenience and hence can be chosen arbitrarily. Now we insert this ansatz into [P, H]. Considering the term (S^{−})^{r+2} we see that h_{−1}(t) must be independent of n, say h_{−1}(t) = c_{0} − d_{0}. Next, we obtain after a long but straightforward calculation

[P, H] = 2a(g_{0}^{+} − g_{0})S^{+}H^{r+1} − (h_{0} − h_{0}^{−})H^{r+1}

− ∑_{j=0}^{r−1} a(∂(2g_{j+1} − h_{j} − h_{j}^{−} − 2bg_{j}))S^{+}H^{r−j}

− ∑_{j=0}^{r−1} (h_{j+1} − h_{j+1}^{−} − 2(a^{2}g_{j}^{+} − (a^{−})^{2}g_{j}^{−}) − b(h_{j} − h_{j}^{−}))H^{r−j}

+ a(g_{r+1}^{+} − g_{r+1})S^{+} + S^{−}a(g_{r+1}^{+} − g_{r+1}) + (h_{r+1} − h_{r+1}^{−}), (12.30)

where g_{r+1}, h_{r+1} have been chosen according to

∂(2g_{r+1} − h_{r} − h_{r}^{−} − 2bg_{r}) = 0,
(12.31)  h_{r+1} − h_{r+1}^{−} − 2(a^{2}g_{r}^{+} − (a^{−})^{2}g_{r}^{−}) − b(h_{r} − h_{r}^{−}) = 0.

(Recall ∂f = f^{+} − f.) But (12.30) is of order 2 if and only if (compare Lemma 1.2)

g_{0} = c_{0}, h_{0} = c_{1} − d_{1},
2g_{j+1} − h_{j} − h_{j}^{−} − 2bg_{j} = 2d_{j+1}, 0 ≤ j ≤ r − 1,
(12.32) h_{j+1} − h_{j+1}^{−} − 2(a^{2}g_{j}^{+} − (a^{−})^{2}g_{j}^{−}) − b(h_{j} − h_{j}^{−}) = 0, 0 ≤ j < r − 2,

where c_{0}, c_{1}, d_{j}, 1 ≤ j ≤ r, are constants. By Lemma 6.4

(12.33)  g̃_{j}(n, t) = ⟨δ_{n}, H(t)^{j}δ_{n}⟩, h̃_{j}(n, t) = 2a(n, t)⟨δ_{n+1}, H(t)^{j}δ_{n}⟩

is a solution of this system for c_{0} = 1, c_{j} = d_{j} = 0, 1 ≤ j ≤ r. It is called the homogeneous solution for if we assign the weight one to a and b, then g̃_{j}(n, t), h̃_{j}(n, t) are homogeneous of degree j, j + 1, respectively. The general solution of the above system (12.32) is hence given by g_{j} = ∑_{ℓ=0}^{j} c_{ℓ}g̃_{j−ℓ}, 1 ≤ j ≤ r, and h_{j} = ∑_{ℓ=0}^{j} c_{ℓ}h̃_{j−ℓ} + c_{j+1} − d_{j+1}, 1 ≤ j ≤ r − 1. Introducing another arbitrary sequence d_{r+1} it is no restriction to assume that the formula for h_{j} also holds for j = r.

It remains to verify (12.24). We use induction on r. The case r = 0 is easy. By (12.25) we need to show

(12.34)  P̃_{2r+2} = P̃_{2r}H + (2ag̃_{r}S^{+} − h̃_{r}) − g̃_{r}H + g̃_{r+1}.

This can be done upon considering ⟨δ_{m}, P̃_{2r+2}δ_{n}⟩ and making case distinctions m < n − 1, m = n − 1, m = n, m = n + 1, m > n + 1. Explicitly, one verifies, for instance, in the case m = n,

⟨δ_{n}, P̃_{2r+2}δ_{n}⟩

= ⟨δ_{n}, P̃_{2r}(aδ_{n−1} + a^{−}δ_{n+1} + bδ_{n})⟩ − h̃_{r}(n) − b(n)g̃_{r}(n) + g̃_{r+1}(n)

= ⟨δ_{n}, ([H^{r}]_{+} − [H^{r}]_{−})(aδ_{n−1} + a^{−}δ_{n+1} + bδ_{n})⟩ − h̃_{r}(n) − b(n)g̃_{r}(n) + g̃_{r+1}(n)

= ⟨δ_{n}, [H^{r}]_{+} a^{−}δ_{n+1}⟩ − ⟨δ_{n}, [H^{r}]_{−} aδ_{n−1}⟩ − h̃_{r}(n) − b(n)g̃_{r}(n) + g̃_{r+1}(n)

= a(n)⟨δ_{n}, H^{r}δ_{n+1}⟩ − a(n − 1)⟨δ_{n}, H^{r}δ_{n−1}⟩ − h̃_{r}(n) − b(n)g̃_{r}(n) + g̃_{r+1}(n)

= −\frac{h̃_{r}(n) + h̃_{r}(n − 1)}{2} − b(n)g̃_{r}(n) + g̃_{r+1}(n) = 0

(12.35)

using (12.27),

(12.36)  ⟨δ_{m}, [H^{j}]_{±}δ_{n}⟩ = \{ \begin{matrix}⟨δ_{m}, H^{j}δ_{n}⟩, & ±(n − m) > 0 \\ 0, & ±(n − m) ≤ 0 \end{matrix} ,

and Hδ_{m} = a(m)δ_{m+1} +a(m−1)δ_{m−1} +b(m)δ_{m}. This settles the case m = n. The remaining cases are settled one by one in a similar fashion. □

Remark 12.3. It is also easy to obtain (12.28) from (12.23) and (12.24). In fact, simply evaluate

⟨δ_{m}, [P̃_{2j+2}, H]δ_{n}⟩ = ⟨Hδ_{m}, [H^{j+1}]_{+}δ_{n}⟩ + ⟨Hδ_{n}, [H^{j+1}]_{+}δ_{m}⟩

− ⟨Hδ_{m}, [H^{j+1}]_{−}δ_{n}⟩ − ⟨Hδ_{n}, [H^{j+1}]_{−}δ_{m}⟩ (12.37)

as in the proof of Theorem 12.2.

Since the self-adjoint part of P (t) does not produce anything interesting when inserted into the Lax equation, we will set d_{j} = 0, 0 ≤ j ≤ r + 1, and take

(12.38)  P_{2r+2}(t) = ∑_{j=0}^{r} c_{r−j}P̃_{2j+2}(t)

as our Lax operator in (12.22). Explicitly we have

P_{2}(t) = a(t)S^{+} − a^{−}(t)S^{−}
P_{4}(t) = a(t)a^{+}(t)S^{++} + a(t)(b^{+}(t) + b(t))S^{+} − a^{−}(t)(b(t) + b^{−}(t))S^{−}
− a^{−}(t)a^{−−}(t)S^{−−} + c_{1}(a(t)S^{+} − a^{−}(t)S^{−})

etc. . (12.39)

Clearly, H(t) and iP_{2r+2}(t) are bounded, self-adjoint operators.

Even though the expression (12.25) for P_{2r+2}(t) looks much more complicated and clumsy in comparison to (12.24), we will see that this ruse of expanding P_{2r+2}(t) in powers of H(t) will turn out most favorable for our endeavor. But before we can see this, we need to make sure that the Lax equation is well-defined, that is, that H(t) is differentiable.

First of all, please recall the following facts. Denote by 𝔅(ℓ^{2}(ℤ)) the C^{∗}-algebra of bounded linear operators. Suppose A, B : ℝ → 𝔅(ℓ^{2}(ℤ)) are differentiable with derivative Ȧ, Ḃ, respectively, then we have

• A + B is differentiable with derivative Ȧ + Ḃ,

• AB is differentiable with derivative ȦB + AḂ,

• A^{∗} is differentiable with derivative Ȧ^{∗},

• A^{−1} (provided A is invertible) is differentiable with derivative −A^{−1}ȦA^{−1}. In addition, f : ℝ → ℓ^{∞}(ℤ) is differentiable if and only if the associated multi-

plication operator f : ℝ → 𝔅(ℓ^{2}(ℤ)) is (since the embedding ℓ^{∞}(ℤ) ↪ 𝔅(ℓ^{2}(ℤ)) is isometric).

For our original problem this implies that H(t) and P_{2r+2}(t) are differentiable since they are composed of differentiable operators. Hence the Lax equation (12.22) is well-defined and by (12.30) it is equivalent to

TL_{r}(a(t), b(t))_{1} = ȧ(t) − a(t)(g_{r+1}^{+}(t) − g_{r+1}(t)) = 0,
(12.40)  TL_{r}(a(t), b(t))_{2} = ḃ(t) − (h_{r+1}(t) − h_{r+1}^{−}(t)) = 0,

where the dot denotes a derivative with respect to t. Or, in integral form we have

a(t) = a(0) exp ( ∫_{0}^{t} (g_{r+1}^{+}(s) − g_{r+1}(s))ds),
(12.41)  b(t) = b(0) + ∫_{0}^{t} (h_{r+1}(s) − h_{r+1}^{−}(s))ds.

Varying r ∈ ℕ_{0} yields the Toda hierarchy (TL hierarchy)

(12.42)  TL_{r}(a, b) = (TL_{r}(a, b)_{1}, TL_{r}(a, b)_{2}) = 0, r ∈ ℕ_{0}.

Notice that multiplying P_{2r+2}(t) with c_{0} ≠ 0 gives only a rescaled version of the Toda hierarchy which can be reduced to the original one by substituting t → t/c_{0}. Hence our choice c_{0} = 1.

Explicitly, one obtains from (12.27),

g_{1} = b + c_{1},
h_{1} = 2a^{2} + c_{2},
g_{2} = a^{2} + (a^{−})^{2} + b^{2} + c_{1}b + c_{2},
h_{2} = 2a^{2}(b^{+} + b) + c_{1}2a^{2} + c_{3},
g_{3} = a^{2}(b^{+} + 2b) + (a^{−})^{2}(2b + b^{−}) + b^{3}
+ c_{1}(a^{2} + (a^{−})^{2} + b^{2}) + c_{2}b + c_{3},
h_{3} = 2a^{2}((a^{+})^{2} + a^{2} + (a^{−})^{2} + b^{2} + b^{+}b + (b^{+})^{2})
+ c_{1}2a^{2}(b^{+} + b) + c_{2}2a^{2} + c_{4},

etc. (12.43)

and hence

TL_{0}(a, b) = \binom{ȧ − a(b^{+} − b)}{ḃ − 2(a^{2} − (a^{−})^{2})},
TL_{1}(a, b) = \binom{ȧ − a((a^{+})^{2} − (a^{−})^{2} + (b^{+})^{2} − b^{2})}{ḃ − 2a^{2}(b^{+} + b) + 2(a^{−})^{2}(b + b^{−})}

− c_{1}\binom{a(b^{+} − b)}{2(a^{2} − (a^{−})^{2})},

TL_{2}(a, b) = \binom{ȧ−a((a^{+})^{2}(b^{++}+2b^{+})+a^{2}(2b^{+}+b)+(b^{+})^{3}−a^{2}(b^{+}+2b)−(a^{−})^{2}(2b+b^{−})−b^{3})}{ḃ+2((a^{−})^{4}+(a^{−−}a^{−})^{2}−a^{2}(a^{2}+(a^{+})^{2}+(b^{+})^{2}+bb^{+}+b^{2})+(a^{−})^{2}(b^{2}+b^{−}b+(b^{−})^{2}))}

− c_{1}\binom{a((a^{+})^{2}−(a^{−})^{2}+(b^{+})^{2}−b^{2})}{2a^{2}(b^{+}+b)−2(a^{−})^{2}(b+b^{−})} − c_{2}\binom{a(b^{+}−b)}{2(a^{2}−(a^{−})^{2})},

etc. (12.44)

represent the first few equations of the Toda hierarchy. We will require c_{j} ∈ ℝ even though c_{j} could depend on t. The corresponding homogeneous Toda equations obtained by taking all summation constants equal to zero, c_{ℓ} ≡ 0, 1 ≤ ℓ ≤ r, are then denoted by

(12.45)  T̃L_{r}(a, b) = TL_{r}(a, b)|_{c_{ℓ}≡0, 1≤ℓ≤r}.

We are interested in investigating the initial value problem associated with the Toda equations, that is,

(12.46)  TL_{r}(a, b) = 0, (a(t_{0}), b(t_{0})) = (a_{0}, b_{0}),

where (a_{0}, b_{0}) are two given (bounded) sequences. Since the Toda equations are autonomous, we will choose t_{0} = 0 without restriction.

In order to draw a number of fundamental consequences from the Lax equation (12.22), we need some preparations.

Let P (t), t ∈ ℝ, be a family of bounded skew-adjoint operators in ℓ^{2}(ℤ). A two parameter family of operators U(t, s), (t, s) ∈ ℝ^{2}, is called a unitary propagator for P (t), if

(1) U(t, s), (s, t) ∈ ℝ^{2}, is unitary.

(2) U(t, t) = 1l for all t ∈ ℝ.
(3) U(t, s)U(s, r) = U(t, r) for all (r, s, t) ∈ ℝ^{3}.

(4) The map t ↦ U(t, s) is differentiable in the Banach space 𝔅(ℓ^{2}(ℤ)) of bounded linear operators and

(12.47)  \frac{d}{dt}U(t, s) = P (t)U(t, s), (t, s) ∈ ℝ^{2}.

Note U(s, t) = U(t, s)^{−1} = U(t, s)^{∗} and d/dt U(s, t) = −U(s, t)P (t).

With this notation the following well-known theorem from functional analysis holds:

Theorem 12.4. Let P (t), t ∈ ℝ, be a family of bounded skew-adjoint operators such that t ↦ P (t) is differentiable. Then there exists a unique unitary propagator U(t, s) for P (t).

Proof. Consider the equation U̇(t) = P (t)U(t). By standard theory of differential equations, solutions for the initial value problem exist locally and are unique (cf., e.g., Theorem 4.1.5 of [1]). Moreover, since ∥P (t)∥ is uniformly bounded on compact sets, all solutions are global. Hence we have a unique solution U(t, s), (t, s) ∈ ℝ^{2} such that U(s, s) = 1l. It remains to verify that this propagator U(t, s) is unitary. Comparing the adjoint equation

(12.48)  \frac{d}{dt}U(t, s)^{∗} = (\frac{d}{dt}U(t, s))^{∗} = (P (t)U(t, s))^{∗} = −U(t, s)^{∗}P (t)

and

(12.49)  \frac{d}{dt}U(t, s)^{−1} = −U(t, s)^{−1}(\frac{d}{dt}U(t, s))U(t, s)^{−1} = −U(t, s)^{−1}P (t)

we infer U(t, s)^{∗} = U(t, s)^{−1} by unique solubility of the initial value problem and U(s, s)^{∗} = U(s, s)^{−1} = 1l. □

If P (t) = P is actually time-independent (stationary solutions), then the unitary propagator is given by Stone’s theorem, that is, U(t, s) = exp((t − s)P).

The situation for unbounded operators is somewhat more difficult and requires the operators P (t), t ∈ ℝ, to have a common dense domain (cf. [218], Corollary on page 102, [193], Theorem X.69).

Now we can apply this fact to our situation.

Theorem 12.5. Let a(t), b(t) satisfy TL_{r}(a, b) = 0 and (H.12.1). Then the Lax equation (12.22) implies existence of a unitary propagator U_{r}(t, s) for P_{2r+2}(t) such that

(12.50)  H(t) = U_{r}(t, s)H(s)U_{r}(t, s)^{−1}, (t, s) ∈ ℝ^{2}.

Thus all operators H(t), t ∈ ℝ, are unitarily equivalent and we might set

(12.51)  σ(H) ≡ σ(H(t)) = σ(H(0)), ρ(H) ≡ ρ(H(t)) = ρ(H(0)).

(Here σ(.) and ρ(.) = ℂ\σ(.) denote the spectrum and resolvent set of an operator, respectively.)

In addition, if ψ(s) ∈ ℓ^{2}(ℤ) solves H(s)ψ(s) = zψ(s), then the function

(12.52)  ψ(t) = U_{r}(t, s)ψ(s)

fulfills

(12.53)  H(t)ψ(t) = zψ(t),  \frac{d}{dt}ψ(t) = P_{2r+2}(t)ψ(t).

Proof. Let U_{r}(t, s) be the unitary propagator for P_{2r+2}(t). We need to show that H̃(t) = U_{r}(t, s)^{−1}H(t)U_{r}(t, s) is equal to H(s). Since H̃(s) = H(s) it suffices to show that H̃(t) is independent of t, which follows from

(12.54)  \frac{d}{dt}H̃(t) = U_{r}(t, s)^{−1}(\frac{d}{dt}H(t) − [P_{2r+2}(t), H(t)])U_{r}(t, s) = 0.

The rest is immediate from the properties of the unitary propagator. □

To proceed with our investigation of the Toda equations, we ensure existence and uniqueness of global solutions next. To do this, we consider the Toda equations as a flow on the Banach space M = ℓ^{∞}(ℤ) ⊕ ℓ^{∞}(ℤ).

Theorem 12.6. Suppose (a_{0}, b_{0}) ∈ M. Then there exists a unique integral curve t ↦ (a(t), b(t)) in C^{∞}(ℝ, M) of the Toda equations, that is, TL_{r}(a(t), b(t)) = 0, such that (a(0), b(0)) = (a_{0}, b_{0}).

Proof. The r-th Toda equation gives rise to a vector field X_{r} on M, that is,

(12.55)  \frac{d}{dt}(a(t), b(t)) = X_{r}(a(t), b(t)) ⇔ TL_{r}(a(t), b(t)) = 0.

Since this vector field has a simple polynomial dependence in a and b it is differentiable and hence (cf. again [1], Theorem 4.1.5) solutions of the initial value problem exist locally and are unique. In addition, by equation (12.50) we have ∥a(t)∥_{∞} + ∥b(t)∥_{∞} ≤ 2∥H(t)∥ = 2∥H(0)∥ (at least locally). Thus any integral curve (a(t), b(t)) is bounded on finite t-intervals implying global existence (see, e.g., [1], Proposition 4.1.22). □

Let τ(t) denote the differential expression associated with H(t). If Ker(τ(t)−z), z ∈ ℂ, denotes the two-dimensional nullspace of τ(t) − z (in ℓ(ℤ)), we have the following representation of P_{2r+2}(t) restricted to Ker(τ(t) − z),

(12.56)  P_{2r+2}(t)|_{Ker(τ(t)−z)} = 2a(t)G_{r}(z, t)S^{+} − H_{r+1}(z, t),

where G_{r}(z, n, t) and H_{r+1}(z, n, t) are monic (i.e. the highest coefficient is one) polynomials given by

G_{r}(z, n, t) = ∑_{j=0}^{r} g_{r−j}(n, t)z^{j},

(12.57)  H_{r+1}(z, n, t) = z^{r+1} + ∑_{j=0}^{r} h_{r−j}(n, t)z^{j} − g_{r+1}(n, t).

One easily obtains

ȧ = a(H_{r+1}^{+} + H_{r+1} − 2(z − b^{+})G_{r}^{+}),
(12.58)  ḃ = 2(a^{2}G_{r}^{+} − (a^{−})^{2}G_{r}^{−}) + (z − b)^{2}G_{r} − (z − b)H_{r+1}.

As an illustration we record a few of the polynomials G_{r} and H_{r+1},

G_{0} = 1 = G̃_{0},
H_{1} = z − b = H̃_{1},
G_{1} = z + b + c_{1} = G̃_{1} + c_{1}G̃_{0},
H_{2} = z^{2} + a^{2} − (a^{−})^{2} − b^{2} + c_{1}(z − b) = H̃_{2} + c_{1}H̃_{1},
G_{2} = z^{2} + bz + a^{2} + (a^{−})^{2} + b^{2} + c_{1}(z + b) + c_{2} = G̃_{2} + c_{1}G̃_{1} + c_{2}G̃_{0},
H_{3} = z^{3} + 2a^{2}z − 2(a^{−})^{2}b − b^{3} + a^{2}b^{+} − (a^{−})^{2}b^{−}
+ c_{1}(z^{2} + a^{2} − (a^{−})^{2} − b^{2}) + c_{2}(z − b) = H̃_{3} + c_{1}H̃_{2} + c_{2}H̃_{1},

etc. . (12.59)

Here G̃_{r}(z, n) and H̃_{r+1}(z, n) are the homogeneous quantities corresponding to G_{r}(z, n) and H_{r+1}(z, n), respectively. By (12.38) we have

(12.60)  G_{r}(z, n) = ∑_{ℓ=0}^{r} c_{r−ℓ}G̃_{ℓ}(z, n), H_{r+1}(z, n) = ∑_{ℓ=0}^{r} c_{r−ℓ}H̃_{ℓ+1}(z, n).

Remark 12.7. (i). Since, by (12.27), a(t) enters quadratically in g_{j}(t), h_{j}(t), respectively G_{r}(z, ., t), H_{r+1}(z, ., t), the Toda hierarchy (12.42) is invariant under the substitution

(12.61)  a(n, t) → ε(n)a(n, t),

where ε(n) ∈ {+1, −1}. This result should be compared with Lemma 1.6. (ii). If a(n_{0}, 0) = 0 we have a(n_{0}, t) = 0 for all t ∈ ℝ (by (12.41)). This implies H = H_{−,n_{0}+1} ⊕ H_{+,n_{0}} with respect to the decomposition ℓ^{2}(ℤ) = ℓ^{2}(−∞, n_{0} + 1) ⊕ ℓ^{2}(n_{0}, ∞). Hence P_{2r+2} = P_{−,n_{0}+1,2r+2} ⊕ P_{+,n_{0},2r+2} decomposes as well and we see that the Toda lattice also splits up into two independent parts TL_{±,n_{0},r}. In this way, the half line Toda lattice follows from our considerations as a special case. Similarly, we can obtain Toda lattices on finite intervals.

12.3. Stationary solutions

In this section we specialize to the stationary Toda hierarchy characterized by ȧ = ḃ = 0 in (12.42) or, equivalently, by commuting difference expressions

(12.62)  [P_{2r+2}, H] = 0

of order 2r + 2 and 2, respectively. Equations (12.58) then yield the equivalent conditions

(z − b)(H_{r+1} − H_{r+1}^{−}) = 2a^{2}G_{r}^{+} − 2(a^{−})^{2}G_{r},
(12.63)  H_{r+1}^{+} + H_{r+1} = 2(z − b^{+})G_{r}^{+}.

Comparison with Section 8.3 suggests to define

(12.64)  R_{2r+2} = (H_{r+1}^{2} − 4a^{2}G_{r}G_{r}^{+}).

A simple calculation using (12.63)

(z − b)(R_{2r+2} − R_{2r+2}^{−})
= (z − b)((H_{r+1} + H_{r+1}^{−})(H_{r+1} − H_{r+1}^{−}) − 4G_{r}(a^{2}G_{r}^{+} − (a^{−})^{2}G_{r}^{−}))

= 2(H_{r+1} + H_{r+1}^{−} − 2(z − b)G_{r})(a^{2}G_{r}^{+} − (a^{−})^{2}G_{r}^{−}) = 0 (12.65) then proves that R_{2r+2} is independent of n. Thus one infers

(12.66)  R_{2r+2}(z) = ∏_{j=0}^{2r+1}(z − E_{j}), {E_{j}}_{0≤j≤2r+1} ⊂ ℂ.

The resulting hyperelliptic curve of (arithmetic) genus r obtained upon compactification of the curve

(12.67)  w^{2} = R_{2r+2}(z) = ∏_{j=0}^{2r+1}(z − E_{j})

will be the basic ingredient in our algebro-geometric treatment of the Toda hierarchy in Section 13.1.

Equations (12.63), (12.64) plus Theorem 2.31 imply

(12.68)  g(z, n) = \frac{G_{r}(z, n)}{R_{2r+2}^{1/2}(z)}, h(z, n) = \frac{H_{r+1}(z, n)}{R_{2r+2}^{1/2}(z)},

where g(z, n) = ⟨δ_{n}, (H − z)^{−1}δ_{n}⟩, h(z, n) = 2a(n)⟨δ_{n+1}, (H − z)^{−1}δ_{n}⟩ − 1 as usual.

Despite these similarities we need to emphasize that the numbers E_{m}, 0 ≤ m ≤ 2r + 1, do not necessarily satisfy (8.54). This is no contradiction but merely implies that there must be common factors in the denominators and numerators of (12.68) which cancel. That is, if the number of spectral gaps of H is s + 2, then there is a monic polynomial Q_{r−s}(z) (independent of n) such that G_{r}(z, n) = Q_{r−s}(z)G_{s}(z, n), H_{r+1}(z, n) = Q_{r−s}(z)H_{s+1}(z, n), and R_{2r+2}(z) = Q_{r−s}(z)^{2}R_{2s+2}(z).

We have avoided these factors in Section 8.3 (which essentially correspond to closed gaps (cf. Remark 7.6)). For example, in case of the constant solution a(n, t) = 1/2, b(n, t) = 0 of TL_{r}(a, b) = 0 we have for r = 0, 1, …

G_{0}(z) = 1, H_{1}(z) = z, R_{2}(z) = z^{2} − 1,
G_{1}(z) = z + c_{1}, H_{2}(z) = (z + c_{1})z, R_{4}(z) = (z + c_{1})^{2}(z^{2} − 1),

etc. . (12.69)

Conversely, any given reflectionless finite gap sequences (a, b) satisfy (12.63) and hence give rise to a stationary solution of some equation in the Toda hierarchy and we obtain

Theorem 12.8. The stationary solutions of the Toda hierarchy are precisely the reflectionless finite-gap sequences investigated in Section 8.3.

In addition, with a little more work, we can even determine which equation (i.e., the constants c_{j}). This will be done by relating the polynomials G_{r}(z, n), H_{r+1}(z, n) to the homogeneous quantities G̃_{r}(z, n), H̃_{r+1}(z, n). We introduce the constants c_{j}(\underline{E}), \underline{E} = (E_{0}, …, E_{2r+1}), by

(12.70)  R_{2r+2}^{1/2}(z) = −z^{r+1} ∑_{j=0}^{∞} c_{j}(\underline{E})z^{−j}, |z| > ∥H∥,

implying

(12.71)  c_{0}(\underline{E}) = 1, c_{1}(\underline{E}) = −\frac{1}{2} ∑_{j=0}^{2r+1} E_{j}, etc. .

Lemma 12.9. Let a(n), b(n) be given reflectionless finite-gap sequences (see Section 8.3) and let G_{r}(z, n), H_{r+1}(z, n) be the associated polynomials (see (8.66) and (8.70)). Then we have (compare (12.60))

(12.72) G_{r}(z, n) = ∑_{ℓ=0}^{r} c_{r−ℓ}(\underline{E})G̃_{ℓ}(z, n), H_{r+1}(z, n) = ∑_{ℓ=0}^{r} c_{r−ℓ}(\underline{E})H̃_{ℓ+1}(z, n).

In addition, g̃_{ℓ}(n) can be expressed in terms of E_{j} and μ_{j}(n) by

(12.73) g̃_{1}(n) = b^{(1)}(n), g̃_{ℓ}(n) = \frac{1}{ℓ}(b^{(ℓ)}(n) + ∑_{j=1}^{ℓ−1} \frac{j}{ℓ} g̃_{ℓ−j}(n)b^{(j)}(n)), ℓ > 1,

where

(12.74)  b^{(ℓ)}(n) = \frac{1}{2} ∑_{j=0}^{2r+1} E_{j}^{ℓ} − ∑_{j=1}^{r} μ_{j}(n)^{ℓ}.

Proof. From (8.96) we infer for |z| > ∥H∥, using Neumann’s expansion for the resolvent of H and the explicit form of g̃, h̃ given in Theorem 12.2, that

G_{r}(z, n) = −\frac{R_{2r+2}^{1/2}(z)}{z} ∑_{ℓ=0}^{∞} g̃_{ℓ}(n)z^{−ℓ},

(12.75)  H_{r+1}(z, n) = R_{2r+2}^{1/2}(z)(1 − \frac{1}{z} ∑_{ℓ=0}^{∞} h̃_{ℓ}(n)z^{−ℓ}).

This, together with (12.70), completes the first part. The rest follows from (6.59) and Theorem 6.10. □

Corollary 12.10. Let a(n), b(n) be given reflectionless finite-gap sequences with corresponding polynomial R_{2s+2}^{1/2}(z). Then (a, b) is a stationary solution of TL_{r} if and only if there is a constant polynomial Q_{r−s}(z) of degree r − s such that c_{j} = c_{j}(\underline{E}), where \underline{E} is the vector of zeros of R_{2r+2}(z) = Q_{r−s}(z)^{2}R_{2s+2}(z).

It remains to show how all stationary solutions for a given equation of the Toda hierarchy can be found. In fact, up to this point we don’t even know whether the necessary conditions (12.63) can be satisfied for arbitrary choice of the constants (c_{j})_{1≤j≤r}.

Suppose (c_{j})_{1≤j≤r} is given and define d_{0} = c_{0} = 1,

(12.76)  d_{j} = 2c_{j} + ∑_{ℓ=1}^{j−1} c_{ℓ}c_{j−ℓ}, 1 ≤ j ≤ r.

Choose d_{j} ∈ ℝ, r + 1 ≤ j ≤ 2r + 2, and define (E_{j})_{0≤j≤2r+1} by

(12.77)  R_{2r+2}(z) = ∑_{j=0}^{2r+2} d_{2r+2−j}z^{j} = ∏_{j=0}^{2r+1}(z − E_{j}).

Note that our choice of d_{j} implies c_{j}(\underline{E}) = c_{j}, 1 ≤ j ≤ r, which is a necessary condition by Lemma 12.9. Since g(z, n) cannot be meromorphic, R_{2r+2}(z) must not be a complete square. Hence those choices of d_{j} ∈ ℝ, r+1 ≤ j ≤ 2r+2, have to be discarded. For any other choice we obtain a list of band edges satisfying (8.54) after throwing out all closed gaps. In particular, setting

(12.78)  Σ(\underline{d}) = ⋃_{j=0}^{r}[E_{2j}, E_{2j+1}]

any operator in Iso_{R}(Σ(\underline{d})) (see Section 8.3) produces a stationary solution. Thus, the procedure of Section 8.3 can be used to compute all corresponding stationary gap solutions.

Theorem 12.11. Fix TL_{r}, that is, fix (c_{j})_{1≤j≤r} and r. Let (d_{j})_{1≤j≤r} be given by (12.76). Then all stationary solutions of TL_{r} can be obtained by choosing (d_{j})_{r+1≤j≤2r+2} such that R_{2r+2}(z) defined as in (12.77) is not a complete square and then choosing any operator in the corresponding isospectral class Iso_{R}(Σ(\underline{d})).

Remark 12.12. The case where R_{2r+2}(z) is a complete square corresponds to stationary solutions with a(n) = 0 for some n. For instance, G_{0}(n) = 1, H_{1}(n) = z − b(n), R_{2r+2}(z) = (z − b(n))^{2} corresponds to a(n) = 0, n ∈ ℤ.

Finally, let us give a further interpretation of the polynomial R_{2r+2}(z).

Theorem 12.13. The polynomial R_{2r+2}(z) is the Burchnall-Chaundy polynomial relating P_{2r+2} and H, that is,

(12.79)  P_{2r+2}^{2} = R_{2r+2}(H) = ∏_{j=0}^{2r+1}(H − E_{j}).

Proof. Because of (12.62) one computes

(P_{2r+2}|_{Ker(τ−z)})^{2} = ((2aG_{r}S^{+} − H_{r+1})|_{Ker(τ−z)})^{2}
= (2aG_{r}(2(z − b^{+})G_{r}^{+} − H_{r+1}^{+} − H_{r+1})S^{+} + H_{r+1}^{2} − 4a^{2}G_{r}G_{r}^{+})|_{Ker(τ−z)}
= R_{2r+2}(z)|_{Ker(τ−z)}

(12.80)

and since z ∈ ℂ is arbitrary, the rest follows from Corollary 1.3. □

This result clearly shows again the close connection between the Toda hierarchy and hyperelliptic curves of the type M = {(z, w)|w^{2} = ∏_{j=0}^{2r+2}(z − E_{j})}.

12.4. Time evolution of associated quantities

For our further investigations in the next two chapters, it will be important to know how several quantities associated with the Jacobi operator H(t) vary with respect to t.

First we will try to calculate the time evolution of the fundamental solutions c(z, ., t), s(z, ., t) if a(t), b(t) satisfy TL_{r}(a, b) = 0. For simplicity of notation we will not distinguish between H(t) and its differential expression τ(t). To emphasize that a solution of H(t)u = zu is not necessarily in ℓ^{2}(ℤ) we will call such solutions weak solutions. Similarly for P_{2r+2}(t).

First, observe that (12.22) implies

(12.81)  (H(t) − z)(\frac{d}{dt} − P_{2r+2}(t))Φ(z, ., t) = 0,

where Φ(z, n, t) is the transfer matrix from (1.30). But this means

(12.82)  (\frac{d}{dt} − P_{2r+2}(t))Φ(z, ., t) = Φ(z, ., t)C_{r}(z, t)

for a certain matrix C_{r}(z, t). If we evaluate the above expression at n = 0, using Φ(z, 0, t) = 1l, we obtain

C_{r}(z, t) = (P_{2r+2}(t)Φ(z, ., t))(0)

= ( \begin{matrix}−H_{r+1}(z, 0, t) & 2a(0, t)G_{r}(z, 0, t) \\ −2a(0, t)G_{r}(z, 1, t) & 2(z − b(1, t))G_{r}(z, 1, t) − H_{r+1}(z, 1, t) \end{matrix} ). (12.83)

The time evolutions of c(z, n, t) and s(z, n, t) now follow from

(12.84)  Φ̇(z, ., t) = P_{2r+2}Φ(z, ., t) + Φ(z, ., t)C_{r}(z, t)

or more explicitly

ċ(z, n, t) = 2a(n, t)G_{r}(z, n, t)c(z, n + 1, t) − (H_{r+1}(z, n, t)
+ H_{r+1}(z, 0, t))c(z, n, t) − 2a(0, t)G_{r}(z, 1, t)s(z, n, t),
ṡ(z, n, t) = 2a(n, t)G_{r}(z, n, t)s(z, n + 1, t) − (H_{r+1}(z, n, t) + H_{r+1}(z, 1, t)

− 2(z − b(1, t))G_{r}(z, 1, t))s(z, n, t) + 2a(0, t)G_{r}(z, 0, t)c(z, n, t). (12.85)

Remark 12.14. In case of periodic coefficients, this implies for the time evolution of the monodromy matrix M(z, t)

(12.86)  \frac{d}{dt}M(z, t) = [M(z, t), C_{r}(z, n, t)],

where

(12.87)  C_{r}(z, n, t) = ( \begin{matrix}−H_{r+1} & 2aG_{r} \\ −2aG_{r}^{+} & 2(z − b^{+})G_{r}^{+} − H_{r+1} \end{matrix} ) .

This shows (take the trace) that the discriminant is time independent and that {E_{j}}_{j=1}^{2N}, A, and B (cf. Section 7.1) are time independent.

Evaluating (12.86) explicitly and expressing everything in terms of our polynomials yields (omitting some dependencies)

\frac{d}{dt} ( \begin{matrix}−H/2 & aG \\ −aG^{+} & H/2 \end{matrix} ) =

( \begin{matrix}2a^{2}(G_{r}G^{+} − G_{r}^{+}G) & −2a(G_{r}H − (z − b^{+})G_{r}^{+}G) \\ −2a(G_{r}^{+}H − (z − b^{+})G_{r}^{+}G^{+}) & −2a^{2}(G_{r}G^{+} − G_{r}^{+}G) \end{matrix} ) . (12.88)

Equation (12.84) enables us to prove

Lemma 12.15. Assume (H.12.1) and suppose TL_{r}(a, b) = 0. Let u_{0}(z, n) be a weak solution of H(0)u_{0} = zu_{0}. Then the system

(12.89)  H(t)u(z, n, t) = zu(z, n, t), \frac{d}{dt}u(z, n, t) = P_{2r+2}(t)u(z, n, t)

has a unique weak solution fulfilling the initial condition

(12.90)  u(z, n, 0) = u_{0}(z, n).

If u_{0}(z, n) is continuous (resp. holomorphic) with respect to z, then so is u(z, n, t). Furthermore, if u_{1,2}(z, n, t) both solve (12.89), then

W (u_{1}(z), u_{2}(z)) = W_{n}(u_{1}(z, t), u_{2}(z, t)) =

a(n, t)(u_{1}(z, n, t)u_{2}(z, n + 1, t) − u_{1}(z, n + 1, t)u_{2}(z, n, t)) (12.91)

depends neither on n nor on t.

Proof. Any solution u(z, n, t) of the system (12.89) can be written as

(12.92)  u(z, n, t) = u(z, 0, t)c(z, n, t) + u(z, 1, t)s(z, n, t)

and from (12.84) we infer that (12.89) is equivalent to the ordinary differential equation

(12.93)  \binom{u̇(z, 0, t)}{u̇(z, 1, t)} = −C_{r}(z, t)\binom{u(z, 0, t)}{u(z, 1, t)}, \binom{u(z, 0, 0)}{u(z, 1, 0)} = \binom{u_{0}(z, 0)}{u_{0}(z, 1)},

which proves the first assertion. The second is a straightforward calculation using (12.56) and (12.58). □

The next lemma shows that solutions which are square summable near ±∞ for one t ∈ ℝ remain square summable near ±∞ for all t ∈ ℝ, respectively.

Lemma 12.16. Let u_{±,0}(z, n) be a solution of H(0)u = zu which is square summable near ±∞. Then the solution u_{±}(z, n, t) of the system (12.89) with initial data u_{±,0}(z, n) ∈ ℓ_{±}^{2}(ℤ) is square summable near ±∞ for all t ∈ ℝ, respectively.

Denote by G(z, n, m, t) the Green function of H(t). Then we have (z ∈ ρ(H))

(12.94) G(z, m, n, t) = \frac{1}{W (u_{−}(z), u_{+}(z))} \{ \begin{matrix}u_{+}(z, n, t)u_{−}(z, m, t) for & m ≤ n \\ u_{+}(z, m, t)u_{−}(z, n, t) for & n ≤ m \end{matrix} .

Especially, if z < σ(H) and a(n, t) < 0 we can choose u_{±,0}(z, n) > 0, implying u_{±}(z, n, t) > 0.

Proof. We only prove the u_{−} case (the u_{+} case follows from reflection) and drop z for notational simplicity. By Lemma 12.15 we have a solution u(n, t) of (12.89) with initial condition u(n, 0) = u_{+,0}(n) and hence

(12.95)  S(n, t) = S(n, 0) + 2 ∫_{0}^{t} Re ∑_{j=−n}^{0} \overline{u(j, s)}P_{2r+2}(s)u(j, s)ds,

where S(n, t) = ∑_{j=−n}^{0} |u(j, t)|^{2}. Next, by boundedness of a(t), b(t), we can find a constant C > 0 such that 4|H_{r+1}(n, t)| ≤ C and 8|a(n, t)G_{r}(n, t)| ≤ C. Using (12.56) and the Cauchy-Schwarz inequality yields

(12.96)  | ∑_{j=−n}^{0} \overline{u(j, s)}P_{2r+2}(s)u(j, s)| ≤ \frac{C}{2} (|u(1, s)|^{2} + S(n, s)).

Invoking Gronwall’s inequality shows

(12.97)  S(n, t) ≤ (S(n, 0) + C ∫_{0}^{t} |u(1, s)|^{2}e^{−Cs}ds)e^{Ct}

and letting n → ∞ implies that u(z, ., t) ∈ ℓ_{−}^{2}(ℤ).

Since u(z, n, t) = 0, z < σ(H), is not possible by Lemma 2.6, positivity follows as well and we are done. □

12.4. Time evolution of associated quantities 239

Using the Lax equation (12.22) one infers (z ∈ ρ(H))

(12.98)  \frac{d}{dt}(H(t) − z)^{−1} = [P_{2r+2}(t), (H(t) − z)^{−1}].

Furthermore, using (12.94) we obtain for m ≤ n

\frac{d}{dt}G(z, n, m, t) =

= \frac{u_{−}(z, m)(P_{2r+2}u_{+}(z, .))(n) + u_{+}(z, n)(P_{2r+2}u_{−}(z, .))(m)}{W (u_{−}(z), u_{+}(z))}. (12.99)

As a consequence (use (12.56) and (12.58)) we also have

\frac{d}{dt}g(z, n, t) = 2(G_{r}(z, n, t)h(z, n, t) − g(z, n, t)H_{r+1}(z, n, t)),
\frac{d}{dt}h(z, n, t) = 4a(n, t)^{2}(G_{r}(z, n, t)g(z, n + 1, t)

− g(z, n, t)G_{r}(z, n + 1, t)) (12.100) and

\frac{d}{dt}g̃_{j}(t) = −2g̃_{r+j+1}(t) + 2 ∑_{ℓ=0}^{r}(g_{r−ℓ}(t)h̃_{ℓ+j}(t) − g̃_{ℓ+j}(t)h_{r−ℓ}(t))
+ 2g_{r+1}(t)g̃_{j}(t),
(12.101) \frac{d}{dt}h̃_{j}(t) = 4a(t)^{2} ∑_{ℓ=0}^{r}(g_{r−ℓ}(t)g̃_{ℓ+j}^{+}(t) − g̃_{ℓ+j}(t)g_{r−ℓ}^{+}(t)).

The initial value problem for the Toda system

In the previous chapter we have seen that the initial value problem associated with the Toda equations has a unique (global) solution. In this section we will consider certain classes of initial conditions and derive explicit formulas for solutions in these special cases.

13.1. Finite-gap solutions of the Toda hierarchy

In this section we want to construct reflectionless finite-gap solutions (see Section 8.3) for the Toda hierarchy. Our starting point will be an r-gap stationary solution (a_{0}, b_{0}) of the type (8.83). This r-gap stationary solution (a_{0}, b_{0}) represents the initial condition for our Toda flow (cf. (12.42)),

(13.1)  T̂L_{s}(a(t), b(t)) = 0, (a(0), b(0)) = (a_{0}, b_{0})

for some s ∈ ℕ_{0}, whose explicit solution we seek. That is, we take a stationary solution of the Toda hierarchy and consider the time evolution with respect to a (in general) different equation of the Toda hierarchy. To stress this fact, we use a hat for the Toda equation (and all associated quantities) which gives rise to the time evolution.

From our treatment in Section 8.3 we know that (a_{0}, b_{0}) is determined by the band edges (E_{j})_{0≤j≤2r+1} and the Dirichlet eigenvalues (μ̂_{0,j}(n_{0}))_{1≤j≤r} at a fixed point n_{0} ∈ ℤ. Since we do not know whether the reflectionless property of the initial condition is preserved by the Toda flow, (a(t), b(t)) might not be reflectionless. However, if we suppose that (a(t), b(t)) is reflectionless for all t, we have

(13.2)  g(z, n_{0}, t) = \frac{∏_{j=1}^{r} z − μ_{j}(n_{0}, t)}{R_{2r+2}^{1/2}(z)}.

Plugging (13.2) into (12.100) and evaluating at μ_{j}(n_{0}, t) we arrive at the following time evolution for the Dirichlet eigenvalues μ̂_{j}(n_{0}, t)

\frac{d}{dt}μ_{j}(n_{0}, t) = −2Ĝ_{s}(μ_{j}(n_{0}, t), n_{0}, t)σ_{j}(n_{0}, t)R_{j}(n_{0}, t),
(13.3)  μ̂_{j}(n_{0}, 0) = μ̂_{0,j}(n_{0}), 1 ≤ j ≤ r, t ∈ ℝ,

(recall (8.67)). Here Ĝ_{s}(z, n_{0}, t) has to be expressed in terms of μ̂_{j}(n_{0}, t), 1 ≤ j ≤ r (cf. Lemma 12.9).

Now our strategy consists of three steps.

(1) Show solvability of the system (13.3).

(2) Construct a(n, t), b(n, t) (and associated quantities) from (μ̂_{j}(n_{0}, t))_{1≤j≤r} as in Section 8.3.

(3) Show that a(n, t), b(n, t) solve T̂L_{r}(a, b) = 0.

In order to show our first step, that is, solubility of the system (13.3) we will first introduce a suitable manifold M^{D} for the Dirichlet eigenvalues μ̂_{j}. For the sake of simplicity we will only consider the case r = 2 with E_{0} = E_{1} < E_{2} = E_{3} < E_{4} < E_{5}, the general case being similar. We define M^{D} as the set of all points (μ̂_{1}, μ̂_{2}) satisfying (H.8.12) together with the following charts (compare Section A.7). (i). Set U_{σ̃_{1},σ̃_{2}} = {(μ̂_{1}, μ̂_{2}) ∈ M^{D}|μ_{1,2} ≠ E_{2}, μ_{2} ≠ E_{4}, σ_{1,2} = σ̃_{1,2}}, U′_{σ̃_{1},σ̃_{2}} = (E_{0}, E_{2}) × (E_{2}, E_{4}).

(13.4)  \begin{matrix}U_{σ̃_{1},σ̃_{2}} & → & U′_{σ̃_{1},σ̃_{2}} & U′_{σ̃_{1},σ̃_{2}} & → & U_{σ̃_{1},σ̃_{2}} \\ (μ̂_{1}, μ̂_{2}) & ↦ & (μ_{1}, μ_{2}) & (μ_{1}, μ_{2}) & ↦ & ((μ_{1}, σ̃_{1}), (μ_{2}, σ̃_{2})) \end{matrix} .

(ii). Set U_{E_{2}} = {(μ̂_{1}, μ̂_{2}) ∈ M^{D}|μ_{2} ≠ E_{4}, σ_{1} = −σ_{2}}, U′_{E_{2}} = (E_{0} − E_{4}, E_{4} − E_{0}) × (0, ∞).

\begin{matrix}U_{E_{2}} & → & U′_{E_{2}} \\ (μ̂_{1}, μ̂_{2}) & ↦ & (σ_{1}(μ_{2} − μ_{1}), −(\frac{μ_{2}−E_{2}}{μ_{1}−E_{2}} )^{σ_{1}}) \\ ((E_{2}, σ), (E_{2}, σ)) & ↦ & (0, \frac{1−σ}{1+σ}) \end{matrix} ,

\begin{matrix}U′_{E_{2}} & → & U_{E_{2}} \\ (ν_{1} > 0, ν_{2}) & ↦ & ((E_{2} − \frac{ν_{1}}{1+ν_{2}} , +), (E_{2} + \frac{ν_{1}ν_{2}}{1+ν_{2}} , −)) \\ (0, ν_{2}) & ↦ & ((E_{2}, \frac{1−ν_{2}}{1+ν_{2}} ), (E_{2}, \frac{1−ν_{2}}{1+ν_{2}} )) \\ (ν_{1} < 0, ν_{2}) & ↦ & ((E_{2} + \frac{ν_{1}ν_{2}}{1+ν_{2}} , −), (E_{2} − \frac{ν_{1}}{1+ν_{2}} , +)) \end{matrix} . (13.5)

(iii). Set U_{E_{4},σ} = {(μ̂_{1}, μ̂_{2}) ∈ M^{D}|μ_{1,2} ≠ E_{2}, σ_{1} = σ}, U′_{E_{4},σ} = (E_{0}, E_{2}) × (−\sqrt{E_{4} − E_{2}}, \sqrt{E_{4} − E_{2}}).

\begin{matrix}U_{E_{4},σ} & → & U′_{E_{4},σ} \\ (μ̂_{1}, μ̂_{2}) & ↦ & (μ_{1}, σ_{2}\sqrt{E_{4} − μ_{2}}) \end{matrix} ,

\begin{matrix}U′_{E_{4},σ} & → & U_{E_{4},σ} \\ (ζ_{1}, ζ_{2}) & ↦ & ((ζ_{1}, σ), (E_{4} − ζ_{2}^{2}, sgn(ζ_{2}))) \end{matrix} . (13.6)

Next, we look at our system (13.3). In the chart (i) it reads (omitting the dependence on t and n_{0})

μ̇_{1} = −2Ĝ_{s}(μ_{1})\frac{σ_{1}R_{6}^{1/2}(μ_{1})}{μ_{1} − μ_{2}},

(13.7)  μ̇_{2} = −2Ĝ_{s}(μ_{2})\frac{σ_{2}R_{6}^{1/2}(μ_{2})}{μ_{2} − μ_{1}}.

In the chart (ii) it reads (ν_{1} ≠ 0)

ν̇_{1} = \frac{1}{1 + ν_{2}} (ν_{2}F_{s}(\frac{ν_{1}ν_{2}}{1 + ν_{2}} ) − F_{s}(\frac{ν_{1}}{1 + ν_{2}} )) ,
(13.8)  ν̇_{2} = ν_{2} \frac{F_{s}(\frac{ν_{1}ν_{2}}{1+ν_{2}} ) − F_{s}(\frac{ν_{1}}{1+ν_{2}} )}{ν_{1}},

where

(13.9)  F_{s}(z) = \frac{Ĝ_{s}(z)R_{6}^{1/2}(z)}{z − E_{2}} = −2Ĝ_{s}(z)(z − E_{0})\sqrt{z − E_{4}}\sqrt{z − E_{5}}.

Note that F_{s}(z) remains invariant if μ_{1} and μ_{2} are exchanged. Moreover, the singularity at ν_{1} = 0 in the second equation is removable and hence this vector field extends to the whole of U′_{E_{2}}.

Finally, in the chart (iii) it reads

ζ̇_{1} = −2Ĝ_{s}(ξ_{1})\frac{σ_{1}R_{6}^{1/2}(ξ_{1})}{ξ_{1} − E_{4} + ξ_{2}^{2}} ,
(13.10) ζ̇_{2} = −2Ĝ_{s}(E_{4} − ξ_{2}^{2})\frac{(E_{4} − E_{0} + ξ_{2}^{2})(E_{4} − E_{2} + ξ_{2}^{2})\sqrt{E_{5} − E_{4} − ξ_{2}^{2}}}{ξ_{1} − E_{4} + ξ_{2}^{2}} .

Hence our system (13.3) is a C^{∞} vector field on the manifold M^{D}. Since M^{D} is not compact, this implies unique solubility only locally. Thus we need to make sure that any solution with given initial conditions in M^{D} stays within M^{D}. Due to the factor (μ_{1} − E_{0}) (hidden in R_{6}^{1/2}(μ_{1})) in (13.7), μ_{1} cannot reach E_{0}. If μ_{2} hits E_{4} it simply changes its sign σ_{2} and moves back in the other direction. Because of the factor ν_{2} in (13.8), ν_{2} remains positive. Finally, we have to consider the case where μ_{1,2} is close to E_{2} and σ_{1} = σ_{2}. In this case we can make the coordinate change ν_{1} = μ_{1} + μ_{2}, ν_{2} = (E_{2} − μ_{1})(μ_{2} − E_{2}). Again we get ν̇_{2} = ν_{2}(…) and hence ν_{2} remains positive or equivalently neither μ_{1} nor μ_{2} can reach E_{2} if σ_{1} = σ_{2}.

Concluding, for given initial conditions (μ̂_{1}(0), μ̂_{2}(0)) = (μ̂_{0,1}, μ̂_{0,2}) ∈ M^{D} we get a unique solution (μ̂_{1}(t), μ̂_{2}(t)) ∈ C^{∞}(ℝ, M^{D}). Due to the requirement μ_{1} ∈ (E_{0}, E_{2}] and μ_{2} ∈ [E_{2}, E_{4}], the functions μ_{1,2}(t) are not C^{∞}(ℝ, ℝ)! However, note that the transformation (13.5) amounts to exchanging μ_{1} and μ_{2} whenever μ_{1} and μ_{2} cross E_{2}. Hence if F (μ_{1}, μ_{2}) ∈ C^{∞}(ℝ^{2}, ℝ) is symmetric with respect to μ_{1} and μ_{2}, then we have F (μ_{1}(.), μ_{2}(.)) ∈ C^{∞}(ℝ, ℝ).

In summary,

Theorem 13.1. The system (13.3) has a unique global solution

(13.11)  (μ̂_{j}(n_{0}, .))_{1≤j≤r} ∈ C^{∞}(ℝ, M^{D})

for each initial condition satisfying (H.8.12).

If s = r and ĉ_{ℓ} = c_{ℓ}, 1 ≤ ℓ ≤ s, we obtain, as expected from the stationary r-gap outset, μ̂_{j}(n_{0}, t) = μ̂_{0,j}(n_{0}) from (13.17) since G_{r}(μ̂_{0,j}(n_{0}), n_{0}, 0) = 0.

Now we come to the second step. Using (13.11) we can define a(n, t), b(n, t) and associated polynomials G_{r}(z, n_{0}, t), H_{r+1}(z, n_{0}, t) as in Section 8.3 (cf. (8.83) and (8.66), (8.70)).

Rather than trying to compute the time derivative of a(n, t) and b(n, t) directly, we consider G_{r}(z, n, t) and H_{r+1}(z, n, t) first. We start with the calculation of the time derivative of G_{r}(z, n_{0}, t). With the help of (13.3) we obtain

(13.12) \frac{d}{dt}G_{r}(z, n_{0}, t)|_{z=μ_{j}(n_{0},t)} = 2Ĝ_{s}(μ_{j}(n_{0}, t), n_{0}, t)H_{r+1}(μ_{j}(n_{0}, t), n_{0}, t),

1 ≤ j ≤ r. Since two polynomials of (at most) degree r − 1 coinciding at r points are equal, we infer

(13.13) \frac{d}{dt}G_{r}(z, n_{0}, t) = 2(Ĝ_{s}(z, n_{0}, t)H_{r+1}(z, n_{0}, t) − G_{r}(z, n_{0}, t)Ĥ_{s+1}(z, n_{0}, t)),

provided we can show that the right-hand side of (13.13) is a polynomial of degree less or equal to r − 1. By (12.60) it suffices to consider the homogeneous case. We divide (13.13) by R_{2r+2}^{1/2}(z) and use (12.75) to express G_{r}(z, n_{0}, t) and H_{r+1}(z, n_{0}, t) respectively (12.57) to express G̃_{s}(z, n_{0}, t) and H̃_{s+1}(z, n_{0}, t). Collecting powers of z shows that the coefficient of z^{j}, 0 ≤ j ≤ r, is

(13.14)  g̃_{s−j} − ∑_{ℓ=j+1}^{s} g̃_{ℓ−j−1}h̃_{s−ℓ} − g̃_{s−j} + ∑_{ℓ=j+1}^{s} h̃_{s−ℓ}g̃_{ℓ−j−1} = 0,

which establishes the claim.

To obtain the time derivative of H_{r+1}(z, n_{0}, t) we use

(13.15)  H_{r+1}(z, n_{0}, t)^{2} − 4a(n_{0}, t)^{2}G_{r}(z, n_{0}, t)G_{r}(z, n_{0} + 1, t) = R_{2r+2}^{1/2}(z)

as in (8.73). Again, evaluating the time derivative first at μ_{j}(n_{0}, t), one obtains

\frac{d}{dt}H_{r+1}(z, n_{0}, t) = 4a(n_{0}, t)^{2}(Ĝ_{s}(z, n_{0}, t)G_{r}(z, n_{0} + 1, t)

− G_{r}(z, n_{0}, t)Ĝ_{s}(z, n_{0} + 1, t)) (13.16)

for those t ∈ ℝ such that μ_{j}(n_{0}, t) ∉ {E_{2j−1}, E_{2j}}, provided the right-hand side of (13.16) is of degree at most r − 1. This can be shown as before. Since the exceptional set is discrete, the identity follows for all t ∈ ℝ by continuity.

Similarly, differentiating (13.15) and evaluating (d/dt)G_{r}(z, n_{0} + 1, t) at z = μ_{j}(n_{0} + 1, t) (the zeros of G_{r}(z, n_{0} + 1, t)), we see that (13.13) also holds with n_{0} replaced by n_{0} + 1 and finally that (μ̂_{j}(n_{0} + 1, t))_{1≤j≤r} satisfies (13.3) with initial condition μ̂_{j}(n_{0} + 1, 0) = μ̂_{0,j}(n_{0} + 1). Proceeding inductively, we obtain this result for all n ≥ n_{0} and with a similar calculation (cf. Section 8.3) for all n ≤ n_{0}.

Summarizing, we have constructed the points (μ̂_{j}(n, t))_{1≤j≤r} for all (n, t) ∈ ℤ × ℝ such that

\frac{d}{dt}μ_{j}(n, t) = −2Ĝ_{s}(μ_{j}(n, t), n, t)σ_{j}(n, t)R_{j}(n, t),
(13.17)  μ̂_{j}(n, 0) = μ̂_{0,j}(n), 1 ≤ j ≤ r, (n, t) ∈ ℤ × ℝ,

with (μ̂_{j}(n, .))_{1≤j≤r} ∈ C^{∞}(ℝ, M^{D}).

Furthermore, we have corresponding polynomials G_{r}(z, n, t) and H_{r+1}(z, n, t) satisfying

\frac{d}{dt}G_{r}(z, n, t) = 2(Ĝ_{s}(z, n, t)H_{r+1}(z, n, t)

− G_{r}(z, n, t)Ĥ_{s+1}(z, n, t)) (13.18)

and

\frac{d}{dt}H_{r+1}(z, n, t) = 4a(n, t)^{2}(Ĝ_{s}(z, n, t)G_{r}(z, n + 1, t)

− G_{r}(z, n, t)Ĝ_{s}(z, n + 1, t)). (13.19)

It remains to verify that the sequences a(t), b(t) solve the Toda equation T̂L_{s}(a, b) = 0.

Theorem 13.2. The solution (a(t), b(t)) of the T̂L_{s} equations (13.1) with r-gap initial conditions (a_{0}, b_{0}) of the type (8.83) is given by

a(n, t)^{2} = \frac{1}{2} ∑_{j=1}^{r} R̂_{j}(n, t) + \frac{1}{8} ∑_{j=0}^{2r+1} E_{j}^{2} − \frac{1}{4} ∑_{j=1}^{r} μ_{j}(n, t)^{2} − \frac{1}{4}b(n, t)^{2},

(13.20) b(n, t) = \frac{1}{2} ∑_{j=0}^{2r+1} E_{j} − ∑_{j=1}^{r} μ_{j}(n, t),

where (μ̂_{j}(n, t))_{1≤j≤r} is the unique solution of (13.17).

Proof. Differentiating (13.15) involving (13.18), (13.19) and (8.82) yields the first equation of (12.58). Differentiating (8.82) using (13.18), (13.19) and (8.101) yields the second equation of (12.58). □

Observe that this algorithm is constructive and can be implemented numerically (it only involves finding roots of polynomials and integrating ordinary differential equations).

Next, let us deduce some additional information from the results obtained thus far.

Clearly (13.18) and (13.19) imply

\frac{d}{dt}K_{r+1}^{β}(z, n, t) = 2a(n, t)((β^{−1}G_{r}(z, n + 1, t) − βG_{r}(z, n, t))K̂_{s+1}^{β}(z, n, t)

− (β^{−1}Ĝ_{s}(z, n + 1, t) − βĜ_{s}(z, n, t))K_{r+1}^{β}(z, n, t))

+ \frac{ȧ(n, t)}{a(n, t)}(K_{r+1}^{β}(z, n, t) + β(β^{−1}G_{r}(z, n + 1, t) − βG_{r}(z, n, t))) (13.21)

and as a consequence we note (use (8.113))

Lemma 13.3. The time evolution of the zeros λ_{j}^{β}(n, t), β ∈ ℝ\{0}, of the polynomial K_{r+1}^{β}(., n, t) is given by

\frac{d}{dt}λ_{j}^{β}(n, t) = (\frac{a(n, t)}{β}K̂_{s+1}^{β}(λ_{j}^{β}(n, t), n, t) + \frac{ȧ(n, t)}{2a(n, t)})

×\frac{R_{2r+2}^{1/2}(λ_{j}^{β}(n, t))}{∏_{ℓ≠j}^{r+1} λ_{j}^{β}(n, t) − λ_{ℓ}^{β}(n, t)}. (13.22)

If λ_{j}^{β}(n, t) = λ_{j+1}^{β}(n, t) the right-hand side has to be replaced by a limit as in (8.67). Finally, let us compute the time dependent Baker-Akhiezer function. Using

(13.23)  ϕ(p, n, t) = \frac{H_{r+1}(p, n, t) + R_{2r+2}^{1/2}(p)}{2a(n, t)G_{r}(p, n, t)} = \frac{2a(n, t)G_{r}(p, n + 1, t)}{H_{r+1}(p, n, t) − R_{2r+2}^{1/2}(p)}

we define for ψ(p, n, n_{0}, t)

ψ(p, n, n_{0}, t) = exp ( ∫_{0}^{t} (2a(n_{0}, x)Ĝ_{s}(p, n_{0}, x)ϕ(p, n_{0}, x)

− Ĥ_{s+1}(p, n_{0}, x))dx)\begin{matrix}n−1 \\ ∏ & ∗ \\ m=n_{0} \end{matrix} ϕ(p, m, t). (13.24)

Straightforward calculations then imply

(13.25)  a(n, t)ϕ(p, n, t) + a(n − 1, t)ϕ(p, n − 1, t)^{−1} = π(p) − b(n, t),
\frac{d}{dt} ln ϕ(p, n, t) = −2a(n, t)(Ĝ_{s}(p, n, t)ϕ(p, n, t) + Ĝ_{s}(p, n + 1, t)ϕ(p, n, t)^{−1})
+ 2(π(p) − b(n + 1, t))Ĝ_{s}(p, n + 1, t)
− Ĥ_{s+1}(p, n + 1, t) + Ĥ_{s+1}(p, n, t)
= 2a(n + 1, t)Ĝ_{s}(p, n + 1, t)ϕ(p, n + 1, t)
− 2a(n, t)Ĝ_{s}(p, n, t)ϕ(p, n, t)

− Ĥ_{s+1}(p, n + 1, t) + Ĥ_{s+1}(p, n, t). (13.26) Similarly, (compare Lemma 12.15)

a(n, t)ψ(p, n + 1, n_{0}, t) + a(n − 1, t)ψ(p, n − 1, n_{0}, t)

= (π(p) − b(n, t))ψ(p, n, n_{0}, t), (13.27) and

\frac{d}{dt}ψ(p, n, n_{0}, t) = 2a(n, t)Ĝ_{s}(p, n, t)ψ(p, n + 1, n_{0}, t)
− Ĥ_{s+1}(p, n, t)ψ(p, n, n_{0}, t)
(13.28)  = P̂_{2s+2}(t)ψ(p, ., n_{0}, t)(n).

The analogs of all relations found in Section 8.3 clearly extend to the present time dependent situation.

13.2. Quasi-periodic finite-gap solutions and the time-dependent Baker-Akhiezer function

In this section we again consider the case with no eigenvalues, that is r = g, since we do not want to deal with singular curves. The manifold for our Dirichlet eigenvalues is in this case simply the submanifold ⊗_{j=1}^{g}π^{−1}([E_{2j−1}, E_{2j}]) ⊂ M_{g} introduced in Lemma 9.1.

For simplicity we set n_{0} = 0 and omit it in the sequel. In order to express ϕ(p, n, t) and ψ(p, n, t) in terms of theta functions, we need a bit more of notation. Let ω_{∞_{±},j} be the normalized Abelian differential of the second kind with a single pole at ∞_{±} of the form (using the chart (A.102))

(13.29)  ω_{∞_{±},j} = (w^{−2−j} + O(1))dw near ∞_{±}, j ∈ ℕ_{0}.

For ω_{∞_{+},j} − ω_{∞_{−},j} we can make the following ansatz

(13.30)  ω_{∞_{+},j} − ω_{∞_{−},j} = ∑_{k=0}^{g+j+1} c̃_{g+j−k+1} \frac{π^{k}dπ}{R_{2g+2}^{1/2}} .

From the asymptotic requirement (13.29) we obtain c̃_{k} = c_{k}(\underline{E}), 0 ≤ k ≤ j + 1, where c_{k}(\underline{E}) is defined in (12.70). The remaining g constants c̃_{k+j+1}, 1 ≤ k ≤ g, have to be determined from the fact that the a-periods must vanish.

Given the summation constants ĉ_{1}, …, ĉ_{s} in Ĝ_{s}, see (12.38), we then define

(13.31)  Ω_{s} = ∑_{j=0}^{s}(j + 1)ĉ_{s−j}(ω_{∞_{+},j} − ω_{∞_{−},j}), ĉ_{0} = 1.

Since the differentials ω_{∞_{±},j} were supposed to be normalized, we have

(13.32)  ∫_{a_{j}} Ω_{s} = 0, 1 ≤ j ≤ g.

Next, let us write down the expansion for the abelian differentials of the first kind ζ_{j},

(13.33)  ζ_{j} = (∑_{ℓ=0}^{∞} η_{j,ℓ}(∞_{±})w^{ℓ}) dw = ± (∑_{ℓ=0}^{∞} η_{j,ℓ}(∞_{+})w^{ℓ}) dw near ∞_{±}.

Using (A.116) we infer

(13.34)  \underline{η}_{ℓ} = ∑_{k=max\{1,g−ℓ\}}^{g} \underline{c}(k)d_{ℓ−g+k}(\underline{E}),

where d_{k}(\underline{E}) is defined by

(13.35) \frac{1}{R_{2g+2}^{1/2}(z)} = \frac{−1}{z^{g+1}} ∑_{k=0}^{∞} \frac{d_{k}(\underline{E})}{z^{k}}, d_{0}(\underline{E}) = 1, d_{1}(\underline{E}) = \frac{1}{2} ∑_{k=0}^{2g+1} E_{k}, etc. .

In addition, relation (A.19) yields

(13.36)  U_{s,j} = \frac{1}{2πi} ∫_{b_{j}} Ω_{s} = 2 ∑_{ℓ=0}^{s} ĉ_{s−ℓ}η_{j,ℓ}(∞_{+}), 1 ≤ j ≤ g.

As in Lemma 9.7 we will show uniqueness of the Baker-Akhiezer function.

Lemma 13.4. Let ψ(., n, t), (n, t) ∈ ℤ×ℝ, be meromorphic on M\{∞_{+}, ∞_{−}} with essential singularities at ∞_{±} such that

(13.37)  ψ̃(p, n, t) = ψ(p, n, t) exp ( − t ∫_{p_{0}}^{p} Ω_{s})

is multivalued meromorphic on M and its divisor satisfies

(13.38)  (ψ̃(., n, t)) ≥ −D_{\underline{μ̂}(0,0)} + n(D_{∞_{+}} − D_{∞_{−}}).

Define a divisor D_{zer}(n, t) by

(13.39)  (ψ̃(., n, t)) = D_{zer}(n, t) − D_{\underline{μ̂}(0,0)} + n(D_{∞_{+}} − D_{∞_{−}}).

If D_{\underline{μ̂}(0,0)} is nonspecial, then so is D_{zer}(n, t) for all (n, t) ∈ ℤ × ℝ, that is,

(13.40)  i(D_{zer}(n, t)) = 0, (n, t) ∈ ℤ × ℝ,

and ψ(., n, t) is unique up to a constant multiple (which may depend on n and t).

Proof. By the Riemann-Roch theorem for multivalued functions (cf. [81], III.9.12) there exists at least one Baker-Akhiezer function. The rest follows as in Lemma 9.7.

□

Given these preparations we obtain the following characterization of ϕ(p, n, t) and ψ(p, n, t) in (13.23) and (13.24).

Theorem 13.5. Introduce

\underline{z}(p, n, t) = \underline{Â}_{p_{0}}(p) − \underline{α̂}_{p_{0}}(D_{\underline{μ̂}(0,0)}) + 2n\underline{Â}_{p_{0}}(∞_{+}) + t\underline{U}_{s} − \underline{Ξ̂}_{p_{0}},
(13.41)  \underline{z}(n, t) = \underline{z}(∞_{+}, n, t).

Then we have

ϕ(p, n, t) = C(n, t)\frac{θ(\underline{z}(p, n + 1, t))}{θ(\underline{z}(p, n, t))} exp ( ∫_{p_{0}}^{p} ω_{∞_{+},∞_{−}}),
(13.42) ψ(p, n, t) = C(n, 0, t)\frac{θ(\underline{z}(p, n, t))}{θ(\underline{z}(p, 0, 0))} exp (n ∫_{p_{0}}^{p} ω_{∞_{+},∞_{−}} + t ∫_{p_{0}}^{p} Ω_{s}),

where C(n, t), C(n, 0, t) are real-valued,

(13.43)  C(n, t)^{2} = \frac{θ(\underline{z}(n − 1, t))}{θ(\underline{z}(n + 1, t))}, C(n, 0, t)^{2} = \frac{θ(\underline{z}(0, 0))θ(\underline{z}(−1, 0))}{θ(\underline{z}(n, t))θ(\underline{z}(n − 1, t))},

and the sign of C(n, t) is opposite to that of a(n, t). Moreover,

(13.44)  \underline{α}_{p_{0}}(D_{\underline{μ̂}(n,t)}) = \underline{α}_{p_{0}}(D_{\underline{μ̂}(0,0)}) − 2n\underline{A}_{p_{0}}(∞_{+}) − [t\underline{U}_{s}],

where [.] denotes the equivalence class in J(M) (cf. (A.53)).

Hence the flows (13.11) are linearized by the Abel map

(13.45)  \frac{d}{dt}\underline{α̂}_{p_{0}}(D_{\underline{μ̂}(n,t)}) = −\underline{U}_{s}.

Proof. First of all note that both functions in (13.42) are well-defined due to (9.34), (13.32), (13.36), and (A.69).

Denoting the right-hand side of (13.42) by Ψ(p, n, t), our goal is to prove ψ = Ψ. By Theorem 9.2 it suffices to identify

(13.46)  ψ(p, 0, t) = Ψ(p, 0, t).

We start by noting that (13.23), (13.24), and (13.18) imply

ψ(p, 0, t) = exp (∫_{0}^{t} (2a(0, x)Ĝ_{s}(z, 0, x)ϕ(p, 0, x) − Ĥ_{s+1}(z, 0, x))dx)

= exp (∫_{0}^{t} (\frac{Ĝ_{s}(z, 0, x)(R_{2g+2}^{1/2}(p) + H_{g+1}(z, 0, x))}{G_{g}(z, 0, x)} − Ĥ_{s+1}(z, 0, x))dx) . (13.47)

In order to spot the zeros and poles of ψ on M\{∞_{+}, ∞_{−}} we need to expand the integrand in (13.47) near its singularities (the zeros μ_{j}(0, x) of G_{g}(z, 0, x)). Using (13.17) one obtains

ψ(p, 0, t) = exp (∫_{0}^{t} (\frac{\frac{d}{dx}μ_{j}(0, x)}{μ_{j}(0, x) − π(p)} + O(1))dx)

(13.48) = \{ \begin{matrix}(μ_{j}(0, t) − π(p))O(1) & for p near μ̂_{j}(0, t) ≠ μ̂_{j}(0, 0) \\ O(1) & for p near μ̂_{j}(0, t) = μ̂_{j}(0, 0) \\ (μ_{j}(0, 0) − π(p))^{−1}O(1) & for p near μ̂_{j}(0, 0) ≠ μ̂_{j}(0, t) \end{matrix} ,

with O(1) ≠ 0. Hence all zeros and all poles of ψ(p, 0, t) on M\{∞_{+}, ∞_{−}} are simple and the poles coincide with those of Ψ(p, 0, t). Next, we need to identify the essential singularities of ψ(p, 0, t) at ∞_{±}. For this purpose we use (13.25) and rewrite (13.47) in the form

ψ(p, 0, t) = exp (∫_{0}^{t} (\frac{1}{2} \frac{\frac{d}{dx}G_{g}(z, 0, x)}{G_{g}(z, 0, x)} + R_{2g+2}^{1/2}(p)\frac{Ĝ_{s}(z, 0, x)}{G_{g}(z, 0, x)})dx)

(13.49)  = (\frac{G_{g}(z, 0, t)}{G_{g}(z, 0, 0)})^{1/2} exp (R_{2g+2}^{1/2}(p) ∫_{0}^{t} \frac{Ĝ_{s}(z, 0, x)}{G_{g}(z, 0, x)}dx) .

We claim that

(13.50) R_{2g+2}^{1/2}(p)Ĝ_{s}(z, n, t)/G_{g}(z, n, t) = ∓ ∑_{ℓ=0}^{s} ĉ_{s−ℓ}z^{1+ℓ} + O(1) for p near ∞_{±}.

By (12.38), in order to prove (13.50), it suffices to prove the homogeneous case ĉ_{0} = 1, ĉ_{ℓ} = 0, 1 ≤ ℓ ≤ s, that is Ĝ_{s} = G̃_{s}. Using (8.96), we may rewrite (13.50) in the form

G̃_{s}(z, n, t)z^{−s−1} = z^{−1} ∑_{ℓ=0}^{s} g̃_{s−ℓ}(n, t)z^{ℓ−s}
(13.51)  = −g(z, n, t) + O(z^{−s−1}) as z → ∞.

But this follows from (6.7) and hence we conclude ψ = const(t)Ψ. The remaining constant can be determined as in Theorem 9.2. □

It is instructive to give a direct proof that (13.17) is linearized by Abel’s map. It suffices to assume that all μ_{j}(n, t) are away from the band edges E_{m} (the general case follows from continuity) and that ĉ_{j} = 0, 1 ≤ j ≤ s. Therefore we calculate (in the chart induced by the projection) using (9.17), (13.17)

\frac{d}{dt}\underline{α}_{p_{0}}(D_{\underline{μ̂}(n,t)}) = ∑_{j=1}^{g} μ̇_{j}(n, t) ∑_{k=1}^{g} \underline{c}(k)\frac{μ_{j}(n, t)^{k−1}}{R_{2g+2}^{1/2}(μ̂_{j}(n, t))}

= −2 ∑_{j,k=1}^{g} \underline{c}(k)\frac{G̃_{s}(μ_{j}(n, t), n, t)}{∏_{ℓ≠j}(μ_{j}(n, t) − μ_{ℓ}(n, t))}μ_{j}(n, t)^{k−1}

(13.52)  = \frac{−2}{2πi} ∑_{k=1}^{g} \underline{c}(k) ∫_{Γ} \frac{G̃_{s}(z, n, t)}{∏_{ℓ=1}^{g}(z − μ_{ℓ}(n, t))}z^{k−1}dz,

where Γ is a closed path in ℂ encircling all Dirichlet eigenvalues μ_{j}(n, t). The last integral can be evaluated by calculating the residue at ∞. Using (see (13.51))

(13.53)  \frac{G̃_{s}(z, n, t)}{∏_{j=1}^{g}(z − μ_{k}(n, t))} = \frac{G̃_{s}(z, n, t)}{g(z, n, t)} \frac{1}{R_{2g+2}^{1/2}(z)} = \frac{z^{s+1}(1 + O(z^{−s}))}{−R_{2g+2}^{1/2}(z)}

we infer (cf. (13.34))

(13.54)  \frac{d}{dt}\underline{α}_{p_{0}}(D_{\underline{μ̂}(n,t)}) = −2 ∑_{k=max\{1,g−s\}}^{g} \underline{c}(k)d_{s−g+k}(\underline{E}) = −2\underline{η}_{s} = −\underline{U}_{s}.

Equation (9.13) shows that the flows (13.22) are linearized by the Abel map as well

(13.55)  \frac{d}{dt}\underline{α̂}_{p_{0}}(D_{\underline{λ̂}^{β}(n,t)}) = −\underline{U}_{s}.

Finally, the θ-function representation for the time dependent quasi-periodic g-gap solutions of the Toda hierarchy follows as in Theorem 9.4.

Theorem 13.6. The solution (a(t), b(t)) of the T̂L_{s} equations (13.1) with g-gap initial conditions (a_{0}, b_{0}) is given by

a(n, t)^{2} = ã^{2} \frac{θ(\underline{z}(n − 1, t))θ(\underline{z}(n + 1, t))}{θ(\underline{z}(n, t))^{2}},

b(n, t) = b̃ + ∑_{j=1}^{g} c_{j}(g)\frac{∂}{∂w_{j}} ln (\frac{θ(\underline{w} + \underline{z}(n, t))}{θ(\underline{w} + \underline{z}(n − 1, t))}) |_{\underline{w}=0}

= E_{0} + ã\frac{θ(\underline{z}(n − 1, t))θ(\underline{z}(p_{0}, n + 1, t))}{θ(\underline{z}(n, t))θ(\underline{z}(p_{0}, n, t))}

+ ã\frac{θ(\underline{z}(n, t))θ(\underline{z}(p_{0}, n − 1, t))}{θ(\underline{z}(n − 1, t))θ(\underline{z}(p_{0}, n, t))}, (13.56)

with ã, b̃ introduced in (9.42).

13.3. A simple example – continued

We now continue our simple example of Section 1.3. The sequences

(13.57)  a(n, t) = a_{0} = \frac{1}{2},  b(n, t) = b_{0} = 0,

solve the Toda equation TL_{r}(a_{0}, b_{0}) = 0 since we have

G_{0,r}(z, n, t) = G_{0,r}(z) = ∏_{j=1}^{r}(z − E_{2j}),

H_{0,r+1}(z, n, t) = H_{0,r+1}(z) = z ∏_{j=1}^{r}(z − E_{2j}),

(13.58)  R_{2r+2}(z) = ∏_{j=0}^{2r−1}(z − E_{j}),

where E_{0} = −1, E_{2r+1} = 1, E_{2j−1} = E_{2j}, 1 ≤ j ≤ r. In addition, we have

(13.59)  g̃_{0,2j} = ( \begin{matrix}2j \\ j \end{matrix} ) , g̃_{0,2j+1} = 0, h̃_{0,j} = g̃_{0,j−1}.

Furthermore, the sequences

(13.60)  ψ_{±}(z, n, t) = k^{±n} exp (\frac{±α_{r}(k)t}{2}), z = \frac{k + k^{−1}}{2} ,

where

(13.61)  α_{r}(k) = 2(kG_{0,r}(z) − H_{0,r+1}(z)) = (k − k^{−1})G_{0,r}(z)

satisfy (compare Section 1.3)

H_{0}(t)ψ_{±}(z, n, t) = zψ_{±}(z, n, t),
\frac{d}{dt}ψ_{±}(z, n, t) = P_{0,2r+2}(t)ψ_{±}(z, n, t)
(13.62)  = 2a_{0}G_{0,r}(z)ψ_{±}(z, n + 1, t) − H_{0,r+1}(z)ψ_{±}(z, n, t).

Note α_{r}(k) = −α_{r}(k^{−1}). Explicitly we have (13.63)

G̃_{0,2r}(z) = ∑_{j=0}^{r} \frac{(2j)!}{4^{j}(j!)^{2}} z^{2(r−j)}, G̃_{0,2r+1}(z) = zG̃_{0,2r}(z), H̃_{0,r+1}(z) = zG̃_{0,r}(z)

and hence

α_{0}(k) = k − k^{−1},
α_{1}(k) = k^{2} − k^{−2} + c_{1}(k − k^{−1}),

etc. . (13.64)

13.4. Inverse scattering transform

This section is an application of Chapter 10 to the Toda case. Our strategy is similar to that of the previous section. Again we will look for the time evolution of suitable spectral data, namely the scattering data associated with H(t). We first show that it suffices to check (10.1) for one t_{0} ∈ ℝ.

Lemma 13.7. Suppose a(n, t), b(n, t) is a solution of the Toda system satisfying (10.1) for one t_{0} ∈ ℝ, then (10.1) holds for all t ∈ ℝ, that is,

(13.65)  ∑_{n∈ℤ} |n|(|1 − 2a(n, t)| + |b(n, t)|) < ∞.

Proof. Without loss of generality we choose t_{0} = 0. Shifting a → a − \frac{1}{2} we can consider the norm

(13.66)  ∥(a, b)∥_{∗} = ∑_{n∈ℤ}(1 + |n|)(|1 − 2a(n)| + |b(n)|).

Proceeding as in Theorem 12.6 we conclude local existence of solutions with respect to this norm. Next, we note that by (12.26) we have the estimate

∑_{n∈ℤ}(1 + |n|)|g_{r}(n, t) − g_{0,r}| ≤ C_{r}(∥H(0)∥)∥(a(t), b(t))∥_{∗},
(13.67)  ∑_{n∈ℤ}(1 + |n|)|h_{r}(n, t) − h_{0,r}| ≤ C_{r}(∥H(0)∥)∥(a(t), b(t))∥_{∗},

where C_{r}(∥H(0)∥) is some positive constant. It suffices to consider the homogeneous case c_{j} = 0, 1 ≤ j ≤ r. In this case the claim follows by induction from equations (6.9) and (6.13) (note that we have g_{i}(n, t)g_{j}(m, t) − g_{0,i}g_{0,j} = (g_{i}(n, t) − g_{0,i})g_{j}(m, t) − g_{0,i}(g_{j}(m, t) − g_{0,j})). Hence we infer from (12.40)

|a(n, t) − \frac{1}{2}| ≤ |a(n, 0) − \frac{1}{2}| + ∥H(0)∥ ∫_{0}^{t} |g_{r+1}(n, s) − g_{0,r+1}|
+ |g_{r+1}(n + 1, s) − g_{0,r+1}|ds,
|b(n, t)| ≤ |b(n, 0)| + ∫_{0}^{t} |h_{r+1}(n, s) − h_{0,r+1}|

+ |h_{r+1}(n − 1, s) − h_{0,r+1}|ds (13.68)

and thus

(13.69)  ∥(a(t), b(t))∥_{∗} ≤ ∥(a(0), b(0))∥_{∗} + C̃ ∫_{0}^{t} ∥(a(s), b(s))∥_{∗}ds,

where C̃ = 2(1 + ∥H(0)∥)C_{r+1}(∥H(0)∥). The rest follows from Gronwall’s inequality. □

Thus we can define Jost solutions, transmission and reflection coefficients as in Chapter 10 with the only difference that they now depend on an additional parameter t ∈ ℝ. For example, we set

(13.70)  S_{±}(H(t)) = {R_{±}(k, t), |k| = 1; k_{ℓ}, γ_{±,ℓ}(t), 1 ≤ ℓ ≤ N}.

Clearly we are interested how the scattering data vary with respect to t.

Theorem 13.8. Suppose a(n, t), b(n, t) is a solution of the Toda system satisfying (10.1) for one t_{0} ∈ ℝ. Then the functions

(13.71)  exp(±\frac{α_{r}(k)t}{2})f_{±}(k, n, t)

fulfill (12.89) with z = (k + k^{−1})/2, where f_{±}(k, n, t) are the Jost solutions and α_{r}(k) is defined in (13.61). In addition, we have

T (k, t) = T (k, 0),
R_{±}(k, t) = R_{±}(k, 0) exp(±α_{r}(k)t),
(13.72)  γ_{±,ℓ}(t) = γ_{±,ℓ}(0) exp(±α_{r}(k_{ℓ})t), 1 ≤ ℓ ≤ N.

Proof. As in Lemma 7.10 one shows that f_{±}(k, n, t) is continuously differentiable with respect to t and that lim_{n→±∞} k^{∓n}ḟ_{±}(k, n, t) → 0 (use the estimates (13.67)). Now let (k + k^{−1})/2 ∈ ρ(H), then Lemma 12.16 implies that the solution of (12.89) with initial condition f_{±}(k, n, 0), |k| < 1, is of the form

(13.73)  C_{±}(t)f_{±}(k, n, t).

Inserting this into (12.89), multiplying with k^{∓n}, and evaluating as n → ±∞ yields

(13.74)  C_{±}(t) = exp(±α_{r}(k)t/2).

The general result for all |k| ≤ 1 now follows from continuity. This immediately implies the formulas for T (k, t), R_{±}(k, t). Finally, let k = k_{ℓ}. Then we have

(13.75)  exp(±α_{r}(k_{ℓ})t/2)f_{±}(k_{ℓ}, n, t) = U_{r}(t, 0)f_{±}(k_{ℓ}, n, 0)

(compare (12.52)), which implies

(13.76)  \frac{d}{dt} \frac{exp(±α_{r}(k_{ℓ})t)}{γ_{±,ℓ}(t)} = \frac{d}{dt}∥U_{r}(t, 0)f_{±}(k_{ℓ}, ., 0)∥^{2} = 0

and concludes the proof. □

Thus S_{±}(H(t)) can be expressed in terms of S_{±}(H(0)) as follows

(13.77)  S_{±}(H(t)) = \{ \begin{matrix}R_{±}(k, 0) exp(±α_{r}(k)t), |k| = 1; \\ k_{ℓ}, γ_{±,ℓ}(0) exp(∓α_{r}(k_{ℓ})t), 1 ≤ ℓ ≤ N\} \end{matrix}

and a(n, t), b(n, t) can be computed from a(n, 0), b(n, 0) using the Gel’fand-Levitan-Marchenko equations. This procedure is known as inverse scattering transform and can be summarized as follows:

[figure: (a(0), b(0)); (a(t), b(t)); scattering; theory; Gel’fand-Levitan-; Marchenko equations; S_{±}(H(0)); S_{±}(H(t)); time evolution]

Hence, in the short range situation, the initial value problem for the Toda lattice equations can be reduced to solving three linear problems. The mathematical pillars of this method are formed by Theorem 12.6, Lemma 13.7, Theorem 13.8, and Theorem 10.10.

Since the transmission coefficient is time independent, so is the perturbation determinant (10.36)

(13.78)  α(k(z), t) = α(k(z), 0) = exp ( ∫_{ℝ} \frac{ξ_{α}(λ)dλ}{λ − z}).

Moreover, the traces

(13.79)  tr(H(t)^{ℓ} − (H_{0})^{ℓ}) = tr(H(0)^{ℓ} − (H_{0})^{ℓ}) = ℓ ∫_{ℝ} λ^{ℓ−1}ξ_{α}(λ)dλ.

produce an infinite series of conservation laws for the Toda lattice. Or, one could also take the expansion coefficients K_{m} (see (10.17)) as conserved quantities.

13.5. Some additions in case of the Toda lattice

The rest of this chapter is devoted to the Toda lattice, that is, to the case TL_{0}(a, b) = 0.

Combining

(13.80)  \frac{d}{dt}\underline{α}_{p_{0}}(D_{\underline{μ̂}(n,t)}) = −2\underline{c}(g).

with (9.16) we obtain

(13.81)  \underline{α}_{p_{0}}(D_{\underline{μ̂}(n,t)}) = \underline{α}_{p_{0}}(D\underline{μ̂}) − 2n\underline{A}_{p_{0}}(∞_{+}) − 2[t\underline{c}(g)].

Hence we can rewrite (13.56) as

a(n, t)^{2} = ã^{2} \frac{θ(\underline{z}(n + 1, t))θ(\underline{z}(n − 1, t))}{θ(\underline{z}(n, t))^{2}},
(13.82)  b(n, t) = b̃ + \frac{1}{2} \frac{d}{dt} ln (\frac{θ(\underline{z}(n, t))}{θ(\underline{z}(n − 1, t))}) .

Now let us verify TL_{0}(a, b) = 0 directly. We start with the first equation

\frac{d}{dt}a(n, t)^{2} = ã^{2} \frac{d}{dt} \frac{θ(\underline{z}(n + 1, t))θ(\underline{z}(n − 1, t))}{θ(\underline{z}(n, t))^{2}}

= ã^{2} \frac{θ(\underline{z}(n + 1, t))θ(\underline{z}(n − 1, t))}{θ(\underline{z}(n, t))^{2}} \frac{d}{dt} ln (\frac{θ(\underline{z}(n + 1, t))θ(\underline{z}(n − 1, t))}{θ(\underline{z}(n, t))^{2}})
(13.83) = 2a(n, t)^{2}(b(n + 1, t) − b(n, t)).

The second equation is more complicated. We compare the expansions (6.26) and (9.45), (9.42) to obtain

− a^{2}(n − 1, t) = b̃^{2} − c̃ + b̃ ∑_{j=1}^{g} B_{j}(n, t) + \frac{1}{2}(( ∑_{j=1}^{g} B_{j}(n, t))^{2}

− ∑_{j=1}^{g} ( ∑_{k=1}^{2g+2} \frac{E_{k}}{2} + \frac{c_{j}(g − 1)}{c_{j}(g)})B_{j}(n, t) + ∑_{j,k=1}^{g} BB_{j,k}(n, t)),

a^{2}(n, t) + b^{2}(n, t) = c̃ + b̃ ∑_{j=1}^{g} B_{j}(n, t) + \frac{1}{2}(( ∑_{j=1}^{g} B_{j}(n, t))^{2}

+ ∑_{j=1}^{g} ( ∑_{k=1}^{2g+2} \frac{E_{k}}{2} + \frac{c_{j}(g − 1)}{c_{j}(g)})B_{j}(n, t) + ∑_{j,k=1}^{g} BB_{j,k}(n, t)). (13.84)

13.5. Some additions in case of the Toda lattice 255 Adding these equations and eliminating b(n, t)^{2} yields

a^{2}(n, t) − a^{2}(n − 1, t) = ∑_{j,k=1}^{g} BB_{j,k}(n, t) = \frac{1}{4} \frac{d^{2}}{dt^{2}} ln (\frac{θ(\underline{z}(n, t))}{θ(\underline{z}(n − 1, t))})
(13.85)  = \frac{1}{2} \frac{d}{dt}b(n, t).

The Baker-Akhiezer function for the Toda lattice ψ is given by

(13.86)  ψ(p, n, t) = C(n, 0, t)\frac{θ(\underline{z}(p, n, t))}{θ(\underline{z}(p, 0, 0))} exp (n ∫_{p_{0}}^{p} ω_{∞_{+},∞_{−}} + t ∫_{p_{0}}^{p} Ω_{0}) ,

where

(13.87)  C(n, 0, t)^{2} = \frac{θ(\underline{z}(0, 0))θ(\underline{z}(−1, 0))}{θ(\underline{z}(n, t))θ(\underline{z}(n − 1, t))}

and Ω_{0} = ω_{∞_{+},0} − ω_{∞_{−},0}. Since we have the following expansion near ∞_{±}

(13.88)  ∫_{p_{0}}^{p} Ω_{0} = ∓(z + d̃+ O(\frac{1}{z})), d̃∈ ℝ,

we can make the following ansatz for Ω_{0}

(13.89)  Ω_{0} = \frac{∏_{j=0}^{g}(π − λ̃_{j})}{R_{2g+2}^{1/2}}dπ.

The constants λ̃_{j} have to be determined from the normalization

(13.90)  ∫_{a_{j}} Ω_{0} = 2 ∫_{E_{2j−1}}^{E_{2j}} \frac{∏_{ℓ=0}^{g}(z − λ̃_{ℓ})}{R_{2g+2}^{1/2}(z)}dz = 0,

and the requirement

(13.91) ∑_{j=0}^{g} λ̃_{j} = \frac{1}{2} ∑_{j=0}^{2g+1} E_{j}.

It follows that the b-periods are given by

(13.92)  ∫_{b_{j}} Ω_{0} = −2 ∫_{E_{2j−2}}^{E_{2j−1}} \frac{∏_{ℓ=0}^{g}(z − λ̃_{ℓ})}{R_{2g+2}^{1/2}(z)}dz = 4πi c_{j}(g).

Remark 13.9. Expanding equation (13.86) around ∞_{±} shows that

(13.93)  ∫_{p_{0}}^{p} Ω_{0} = ∓ (z + b̃ + O(z^{−1})) near ∞_{±},

where b̃ is defined in (9.42). Conversely, proving (13.28) as in (9.61) (by expanding both sides in (13.28) around ∞_{±} and using Lemma 13.4) turns out to be equivalent to proving (13.93). Comparison with the expansion (13.88) further yields b̃ = d̃.

13.6. The elliptic case – continued

In this section we extend our calculations for the special case g = 1 started in Section 9.3.

We first express the differential Ω_{0}. We need to determine the constants λ̃_{0}, λ̃_{1} from

(13.94)  λ̃_{0} + λ̃_{1} = \frac{E_{0} + E_{1} + E_{2} + E_{3}}{2}

and ([38] 254.00, 254.10, 340.01, 340.02, 336.02)

∫_{a_{1}} Ω_{0} = 2 ∫_{E_{1}}^{E_{2}} \frac{x^{2} − (λ̃_{0} + λ̃_{1})x + λ̃_{0}λ̃_{1}}{R_{4}^{1/2}(x)} dx
= −\frac{C}{2} ((E_{0}E_{3} + E_{1}E_{2} − 2λ̃_{0}λ̃_{1})K(k)

+ (E_{2} − E_{0})(E_{3} − E_{1})E(k)) = 0, (13.95)

which implies

(13.96)  2λ̃_{0}λ̃_{1} = (E_{2} − E_{0})(E_{3} − E_{1})\frac{E(k)}{K(k)} + (E_{0}E_{3} + E_{1}E_{2}).

The b-period is given by (cf. (13.92))

(13.97)  ∫_{b_{1}} Ω_{0} = −2 ∫_{E_{0}}^{E_{1}} \frac{x^{2} − (λ̃_{0} + λ̃_{1})x + λ̃_{0}λ̃_{1}}{R_{4}^{1/2}(x)} dx = 4πi c(1) = \frac{2πi}{CK(k)}.

And for p = (z, ±R_{4}^{1/2}(z)) near p_{0} we have ([38] 251.00, 251.03, 340.01, 340.02, 336.02)

∫_{p_{0}}^{p} Ω_{0} = ± ∫_{E_{0}}^{z} \frac{x^{2} − (λ̃_{0} + λ̃_{1})x + λ̃_{0}λ̃_{1}}{R_{4}^{1/2}(x)} dx

= \frac{C(E_{2} − E_{0})}{2}((E_{3} − E_{1})\frac{E(k)F (u(z), k) − K(k)E(u(z), k)}{K(k)}

+ (E_{3} − E_{0})\frac{u(z)\sqrt{1 − u(z)^{2}}\sqrt{1 − k^{2}u(z)^{2}}}{1 − \frac{E_{3}−E_{0}}{E_{3}−E_{1}} u(z)^{2}} ). (13.98)

Finally we remark that (9.85) now reads

(13.99)  μ(n, t) = E_{1} \frac{1 − \frac{E_{2}−E_{1}}{E_{2}−E_{0}} \frac{E_{0}}{E_{1}} sn(2K(k)δ − 2nF (\sqrt{\frac{E_{3}−E_{1}}{E_{3}−E_{0}}} , k) − \frac{2t}{C})^{2}}{1 − \frac{E_{2}−E_{1}}{E_{2}−E_{0}} sn(2K(k)δ − 2nF (\sqrt{\frac{E_{3}−E_{1}}{E_{3}−E_{0}}} , k) − \frac{2t}{C})^{2}} ,

since we have

(13.100)  A_{(E_{0},0)}(μ̂(n, t)) = A_{(E_{0},0)}(μ̂(0, 0)) − 2nA_{(E_{0},0)}(∞_{+}) − 2[tc(1)].

The Kac-van Moerbeke system

In this chapter we will introduce the Kac-van Moerbeke (KM) hierarchy. Using a commutation (sometimes also called a supersymmetric) approach we will show that the Kac-van Moerbeke hierarchy is a modified Toda hierarchy precisely in the way that the modified Korteweg-de Vries (mKdV) hierarchy is related to the Korteweg-de Vries (KdV) hierarchy. The connection between the TL hierarchy and its modified counterpart, the KM hierarchy, is given by a Bäcklund transformation (the Miura transformation in case of the KdV and mKdV equation) based on the factorization of difference expressions found in Section 11.1. This will allow us to compute new solutions from known ones. In particular, we will be able to compute the N-soliton solution starting from the trivial solution (see Section 13.3) of the TL equation.

14.1. The Kac-van Moerbeke hierarchy and its relation to the Toda hierarchy

This section is devoted to the Kac-van Moerbeke hierarchy and its connection with the Toda hierarchy.

We will suppose that ρ(t) satisfies the following hypothesis throughout this chapter.

Hypothesis H. 14.1. Let

(14.1)  ρ(t) ∈ ℓ^{∞}(ℤ, ℝ), ρ(n, t) ≠ 0, (n, t) ∈ ℤ × ℝ,

and let t ∈ ℝ ↦ ρ(t) ∈ ℓ^{∞}(ℤ) be differentiable.

Define the “even” and “odd” parts of ρ(t) by

(14.2)  ρ_{e}(n, t) = ρ(2n, t), ρ_{o}(n, t) = ρ(2n + 1, t), (n, t) ∈ ℤ × ℝ,

and consider the bounded operators (in ℓ^{2}(ℤ))

(14.3)  A(t) = ρ_{o}(t)S^{+} + ρ_{e}(t),  A(t)^{∗} = ρ_{o}^{−}(t)S^{−} + ρ_{e}(t).

As in Section 11.1 we have the factorization

(14.4)  H_{1}(t) = A(t)^{∗}A(t),  H_{2}(t) = A(t)A(t)^{∗},

with

(14.5)  H_{k}(t) = a_{k}(t)S^{+} + a_{k}^{−}(t)S^{−} + b_{k}(t),  k = 1, 2,

and

a_{1}(t) = ρ_{e}(t)ρ_{o}(t),  b_{1}(t) = ρ_{e}(t)^{2} + ρ_{o}^{−}(t)^{2},
(14.6)  a_{2}(t) = ρ_{e}^{+}(t)ρ_{o}(t),  b_{2}(t) = ρ_{e}(t)^{2} + ρ_{o}(t)^{2}.

In the sequel it will be important to observe that the transformation ρ̂(n, t) = ρ(n + 1, t) implies

(14.7)  ρ̂_{e}(n, t) = ρ_{o}(n, t),  ρ̂_{o}(n, t) = ρ_{e}(n + 1, t)

and hence â_{1}(t) = a_{2}(t), b̂_{1}(t) = b_{2}(t), â_{2}(t) = a_{1}^{+}(t), b̂_{2}(t) = b_{1}^{+}(t). In addition, we have ĝ_{1,j}(t) = g_{2,j}(t), ĥ_{1,j}(t) = h_{2,j}(t) and ĝ_{2,j}(t) = g_{1,j}^{+}(t), ĥ_{2,j}(t) = h_{1,j}^{+}(t).

Using ⟨δ_{m}, H_{2}^{j+1}δ_{n}⟩ = ⟨δ_{m}, AH_{1}^{j}A^{∗}δ_{n}⟩ we can express g_{2,j}, h_{2,j}, j ∈ ℕ, in terms of g_{1,j}, h_{1,j}, j ∈ ℕ. Explicit evaluation with m = n gives (provided we pick the same summation constant c_{1,ℓ} = c_{2,ℓ} ≡ c_{ℓ}, 1 ≤ ℓ ≤ r)

(14.8)  g_{2,j+1}(t) = ρ_{o}(t)^{2}g_{1,j}^{+}(t) + ρ_{e}(t)^{2}g_{1,j}(t) + h_{1,j}(t).

Similarly, for m = n + 1 we obtain

h_{2,j+1}(t) − h_{1,j+1} = ρ_{o}(t)^{2}(2(ρ_{e}^{+}(t)^{2}g_{1,j}^{+}(t) − ρ_{e}(t)^{2}g_{1,j}(t))

+ (h_{1,j}^{+}(t) − h_{1,j}(t))). (14.9)

By virtue of (14.7) we can even get two more equations

g_{1,j+1}(t) = ρ_{e}(t)^{2}g_{2,j}(t) + ρ_{o}^{−}(t)^{2}g_{2,j}^{−}(t) + h_{2,j}^{−}(t),
h_{2,j+1}^{−}(t) − h_{1,j+1} = ρ_{e}(t)^{2}( − 2(ρ_{o}(t)^{2}g_{2,j}(t) − ρ_{o}^{−}(t)^{2}g_{2,j}^{−}(t))

− (h_{2,j}(t) − h_{2,j}^{−}(t))). (14.10)

These relations will turn out useful below.

Now we define matrix-valued operators D(t), Q_{2r+2}(t) in ℓ^{2}(ℤ, ℂ^{2}) as follows,

D(t) = ( \begin{matrix}0 & A(t)^{∗} \\ A(t) & 0 \end{matrix} ) ,
(14.11)  Q_{2r+2}(t) = ( \begin{matrix}P_{1,2r+2}(t) & 0 \\ 0 & P_{2,2r+2}(t) \end{matrix} ) ,

r ∈ ℕ_{0}. Here P_{k,2r+2}(t), k = 1, 2, are defined as in (12.38), that is,

P_{k,2r+2}(t) = −H_{k}(t)^{r+1} + ∑_{j=0}^{r}(2a_{k}(t)g_{k,j}(t)S^{+} − h_{k,j}(t))H_{k}(t)^{j} + g_{k,r+1},

P_{k,2r+2}(t)|_{Ker(τ_{k}(t)−z)} = 2a_{k}(t)G_{k,r}(z, t)S^{+} − H_{k,r+1}(z, t). (14.12)

The sequences (g_{k,j}(n, t))_{0≤j≤r}, (h_{k,j}(n, t))_{0≤j≤r+1} are defined as in (12.27), and the polynomials G_{k,r}(z, n, t), H_{k,r+1}(z, n, t) are defined as in (12.57). Moreover, we choose the same summation constants in P_{1,2r+2}(t) and P_{2,2r+2}(t) (i.e., c_{1,ℓ} = c_{2,ℓ} ≡ c_{ℓ}, 1 ≤ ℓ ≤ r). Explicitly we have

Q_{2}(t) = ( \begin{matrix}ρ_{e}(t)ρ_{o}(t)S^{+} − ρ_{e}^{−}(t)ρ_{o}^{−}(t)S^{−} & 0 \\ 0 & ρ_{e}^{+}(t)ρ_{o}(t)S^{+} − ρ_{e}(t)ρ_{o}^{−}(t)S^{−} \end{matrix} )

etc. . (14.13)

Analogous to equation (12.22) we now look at (14.14)

\frac{d}{dt}D = [Q_{2r+2}, D] = ( \begin{matrix}0 & P_{1,2r+2}A^{∗} − A^{∗}P_{2,2r+2} \\ P_{2,2r+2}A − AP_{1,2r+2} & 0 \end{matrix} ) .

Using H_{1}^{j}A^{∗} = A^{∗}H_{2}^{j} and S^{−} = \frac{1}{a_{2}^{−}} (H_{2} − a_{2}S^{+} − b_{2}) we obtain after a little calculation

A^{∗}P_{2,2r+2} − P_{1,2r+2}A^{∗} =

∑_{j=1}^{r} \frac{ρ_{o}ρ_{e}^{+}}{ρ_{e}} (2ρ_{e}^{2}(g_{2,j} − g_{1,j}) + (h_{2,j}^{−} − h_{1,j}))S^{+}H_{2}^{r−j}

+ ∑_{j=1}^{r} \frac{1}{ρ_{e}} ((h_{2,j+1}^{−} − h_{1,j+1}) + 2((a_{1})^{2}g_{1,j} − (a_{2}^{−})^{2}g_{2,j}^{−})

− ρ_{o}^{2}(h_{2,j}^{−} − h_{1,j}) + ρ_{e}^{2}(h_{2,j} − h_{2,j}^{−}))H_{2}^{r−j} + (h_{2,r+1}^{−} − h_{1,r+1})

− ρ_{o}^{−}(g_{2,r+1}^{−} − g_{1,r+1})S^{−} + ρ_{e}(g_{2,r+1} − g_{1,r+1}). (14.15)

Evaluating ⟨δ_{n}, (H_{1}^{ℓ}A^{∗} − A^{∗}H_{2}^{ℓ})δ_{n}⟩ = 0 explicitly gives

(14.16)  h_{2,j}^{−} − h_{1,j} + 2ρ_{e}^{2}(g_{2,j} − g_{1,j}) = 0

and the second identity in (14.10) together with (14.16) implies (14.17) (h_{2,j+1}^{−} − h_{1,j+1}) + 2((a_{1})^{2}g_{1,j} − (a_{2}^{−})^{2}g_{2,j}^{−}) − ρ_{o}^{2}(h_{2,j}^{−} − h_{1,j}) + ρ_{e}^{2}(h_{2,j} − h_{2,j}^{−}) = 0.

Hence we obtain

(14.18) P_{1,2r+2}A^{∗} − A^{∗}P_{2,2r+2} = −ρ_{o}^{−}(g_{2,r+1}^{−} − g_{1,r+1})S^{−} + ρ_{e}(g_{2,r+1} − g_{1,r+1}).

Or, in other words,

(14.19)  \frac{d}{dt}D(t) − [Q_{2r+2}(t), D(t)] = 0

is equivalent to

\underline{KM}_{r}(ρ) = (KM_{r}(ρ)_{e}, KM_{r}(ρ)_{o})
(14.20)  = ( \begin{matrix}ρ̇_{e} − ρ_{e}(g_{2,r+1} − g_{1,r+1}) \\ ρ̇_{o} + ρ_{o}(g_{2,r+1} − g_{1,r+1}^{+}) \end{matrix} ) = 0.

Here the dot denotes a derivative with respect to t. One look at the transformation (14.7) verifies that the equations for ρ_{o}, ρ_{e} are in fact one equation for ρ. More explicitly, combining g_{k,j}, respectively h_{k,j}, into one sequence

(14.21)  \begin{aligned}G_{j}(2n) & = & g_{1,j}(n) \\ G_{j}(2n + 1) & = & g_{2,j}(n) \end{aligned} , respectively \begin{aligned}H_{j}(2n) & = & h_{1,j}(n) \\ H_{j}(2n + 1) & = & h_{2,j}(n) \end{aligned} ,

we can rewrite (14.20) as

(14.22)  KM_{r}(ρ) = ρ̇ − ρ(G_{r+1}^{+} − G_{r+1}).

From (12.27) we see that G_{j}, H_{j} satisfy the recursions

G_{0} = 1, H_{0} = c_{1},
2G_{j+1} − H_{j} − H_{j}^{−−} − 2(ρ^{2} + (ρ^{−})^{2})G_{j} = 0, 0 ≤ j ≤ r,
H_{j+1} − H_{j+1}^{−−} − 2((ρρ^{+})^{2}G_{j}^{++} − (ρ^{−}ρ^{−−})^{2}G_{j}^{−−})
(14.23)  −(ρ^{2} + (ρ^{−})^{2})(H_{j} − H_{j}^{−−}) = 0, 0 ≤ j < r.

As in the Toda context (12.42), varying r ∈ ℕ_{0} yields the Kac-van Moerbeke hierarchy (KM hierarchy) which we denote by

(14.24)  KM_{r}(ρ) = 0, r ∈ ℕ_{0}.

Using (14.23) we compute

G_{1} = ρ^{2} + (ρ^{−})^{2} + c_{1},
H_{1} = 2(ρρ^{+})^{2} + c_{2},
G_{2} = (ρρ^{+})^{2} + (ρ^{2} + (ρ^{−})^{2})^{2} + (ρ^{−}ρ^{−−})^{2} + c_{1}(ρ^{2} + (ρ^{−})^{2}) + c_{2},
H_{2} = 2(ρρ^{+})^{2}((ρ^{++})^{2} + (ρ^{+})^{2} + ρ^{2}(ρ^{−})^{2}) + c_{1}2(ρρ^{+})^{2} + c_{3},

etc. . (14.25)

and hence

KM_{0}(ρ) = ρ̇ − ρ((ρ^{+})^{2} − (ρ^{−})^{2}) = 0,
KM_{1}(ρ) = ρ̇ − ρ((ρ^{+})^{4} − (ρ^{−})^{4} + (ρ^{++})^{2}(ρ^{+})^{2} + (ρ^{+})^{2}ρ^{2} − ρ^{2}(ρ^{−})^{2}
− (ρ^{−})^{2}(ρ^{−−})^{2}) + c_{1}(−ρ)((ρ^{+})^{2} − (ρ^{−})^{2}) = 0,

etc. . (14.26)

Again the Lax equation (14.19) implies

Theorem 14.2. Let ρ satisfy (H.14.1) and KM(ρ) = 0. Then the Lax equation (14.19) implies the existence of a unitary propagator V_{r}(t, s) such that we have

(14.27)  D(t) = V_{r}(t, s)D(s)V_{r}(t, s)^{−1},  (t, s) ∈ ℝ^{2},

and thus all operators D(t), t ∈ ℝ, are unitarily equivalent. Clearly, we have

(14.28)  V_{r}(t, s) = ( \begin{matrix}U_{1,r}(t, s) & 0 \\ 0 & U_{2,r}(t, s) \end{matrix} ) .

And as in Theorem 12.6 we infer

Theorem 14.3. Suppose ρ_{0} ∈ ℓ^{∞}(ℤ). Then there exists a unique integral curve t ↦ ρ(t) in C^{∞}(ℝ, ℓ^{∞}(ℤ)) of the Kac-van Moerbeke equations, KM_{r}(ρ) = 0, such that ρ(0) = ρ_{0}.

Remark 14.4. In analogy to Remark 12.7 one infers that ρ_{e} and ρ_{o} enter a_{r}, b_{r} quadratically so that the KM hierarchy (14.24) is invariant under the substitution

(14.29)  ρ(t) → ρ_{ε}(t) = {ε(n)ρ(n, t)}_{n∈ℤ}, ε(n) ∈ {+1, −1}, n ∈ ℤ.

This result should again be compared with Lemma 1.6 and Lemma 14.9 below.

As in Section 12.2 (cf. (12.45)) we use a tilde to distinguish between inhomogeneous and homogeneous Kac-van Moerbeke equations, that is,

(14.30)  K̃M_{r}(ρ) = KM_{r}(ρ)|_{c_{ℓ}≡0, 1≤ℓ≤r},

with c_{ℓ} the summation constants in P_{k,2r+2}.

Clearly, (14.20) can also be rewritten in terms of polynomials

\frac{d}{dt}ρ_{e} = 2ρ_{e}ρ_{o}^{2}(G_{1,r}^{+}(z) − G_{2,r}(z)) + ρ_{e}(H_{1,r+1}(z) − H_{2,r+1}(z)),
(14.31) \frac{d}{dt}ρ_{o} = −2ρ_{o}(ρ_{e}^{+})^{2}(G_{1,r}^{+}(z) − G_{2,r}^{+}(z)) − ρ_{o}(H_{1,r+1}^{+}(z) − H_{2,r+1}(z)),

r ∈ ℕ_{0}. One only has to use (14.16) and

(14.32)  h_{2,j} − h_{1,j} + 2ρ_{o}^{2}(g_{2,j} − g_{1,j}^{+}) = 0,

which follows from (14.16) invoking the transformation (14.7).

The connection between P_{k,2r+2}(t), k = 1, 2, and Q_{2r+2}(t) is clear from (14.11), the corresponding connection between H_{k}(t), k = 1, 2, and D(t) is provided by the elementary observation

(14.33)  D(t)^{2} = ( \begin{matrix}H_{1}(t) & 0 \\ 0 & H_{2}(t) \end{matrix} ) .

Moreover, since we have

(14.34)  \frac{d}{dt}D(t)^{2} = [Q_{2r+2}(t), D(t)^{2}]

we obtain the implication

(14.35)  KM_{r}(ρ) = 0 ⇒ TL_{r}(a_{k}, b_{k}) = 0, k = 1, 2.

Using (14.16) and (14.32) this can also be written as

(14.36)  TL_{r}(a_{k}, b_{k}) = W_{k}\underline{KM}_{r}(ρ), k = 1, 2,

where W_{k}(t) denote the matrix-valued difference expressions

(14.37) W_{1}(t) = ( \begin{matrix}ρ_{o}(t) & ρ_{e}(t) \\ 2ρ_{e}(t) & 2ρ_{o}^{−}(t)S^{−} \end{matrix} ) , W_{2}(t) = ( \begin{matrix}ρ_{o}(t)S^{+} & ρ_{e}^{+}(t) \\ 2ρ_{e}(t) & 2ρ_{o}(t) \end{matrix} ) .

That is, given a solution ρ of the KM_{r} equation (14.24) (respectively (14.31)), one obtains two solutions, (a_{1}, b_{1}) and (a_{2}, b_{2}), of the TL_{r} equations (12.42) related to each other by the Miura-type transformations (14.6). Hence we have found a Bäcklund transformation relating the Toda and Kac-van Moerbeke systems. Note that due to (H.14.1), (a_{1}, b_{1}) and (a_{2}, b_{2}) both fulfill (H.12.1).

In addition, we can define

(14.38)  ϕ_{1}(n, t) = −\frac{ρ_{e}(n, t)}{ρ_{o}(n, t)},  ϕ_{2}(n, t) = −\frac{ρ_{o}(n, t)}{ρ_{e}(n + 1, t)}.

This implies

(14.39)  a_{k}(n, t)ϕ_{k}(n, t) + \frac{a_{k}(n − 1, t)}{ϕ_{k}(n − 1, t)} + b_{k}(n, t) = 0

and using (14.8), the first identity in (14.10), and (12.27) one verifies (cf. (13.26))

\frac{d}{dt} ln ϕ_{k}(n, t) = −2a_{k}(n, t)(g_{k,r}(n, t)ϕ_{k}(n, t) + g_{k,r}(n + 1, t)ϕ_{k}(n, t)^{−1})

− 2b_{k}(n + 1, t)g_{k,r}(n + 1, t) + (g_{k,r+1}(n + 1, t) − g_{k,r+1}(n, t))

− (h_{k,r}(n + 1, t) − h_{k,r}(n, t)). (14.40)

Hence we infer

(14.41)  H_{k}(t)u_{k}(n, t) = 0,  \frac{d}{dt}u_{k}(n, t) = P_{k,2r+2}(t)u_{k}(n, t)

(in the weak sense, i.e., u_{k} is not necessarily square summable), where

u_{k}(n, t) = exp ( ∫_{0}^{t} (2a_{k}(n_{0}, x)g_{k,r}(n_{0}, x)ϕ_{k}(n_{0}, x)

− h_{k,r}(n_{0}, x) + g_{k,r+1}(n_{0}, x))dx)\begin{matrix}n−1 \\ ∏ & ∗ \\ m=n_{0} \end{matrix} ϕ_{k}(m, t). (14.42)

Furthermore, explicitly writing out (14.19) shows that if

(14.43) H_{k}(t)u_{k}(z, n, t) = zu_{k}(z, n, t),  \frac{d}{dt}u_{k}(z, n, t) = P_{k,2r+2}(t)u_{k}(z, n, t)

holds for the solution u_{1}(z, n, t) (resp. u_{2}(z, n, t)), then it also holds for the solution u_{2}(z, n, t) = A(t)u_{1}(z, n, t) (resp. u_{1}(z, n, t) = A(t)^{∗}u_{2}(z, n, t)).

Summarizing,

Theorem 14.5. Suppose ρ satisfies (H.14.1) and KM_{r}(ρ) = 0. Then (a_{k}, b_{k}), k = 1, 2, satisfies (H.12.1) and TL_{r}(a_{k}, b_{k}) = 0, k = 1, 2. In addition, if (14.43) holds for the solution u_{1}(z, n, t) (resp. u_{2}(z, n, t)), then it also holds for the solution u_{2}(z, n, t) = A(t)u_{1}(z, n, t) (resp. u_{1}(z, n, t) = A(t)^{∗}u_{2}(z, n, t)).

Finally, let us extend Lemma 12.16 to the case λ ≤ σ(H).

Lemma 14.6. Suppose λ ≤ σ(H) and a(n, t) < 0. Then u_{0}(λ, n) > 0 implies that the solution u(λ, n, t) of (12.89) with initial condition u_{0}(λ, n) is positive.

Proof. Shifting H(t) → H(t) − λ we can assume λ = 0. Now use u_{0}(0, n) > 0 to define ρ_{0}(n) by

ρ_{0,o}(n) = −\sqrt{\frac{−a(n, 0)u_{0}(0, n)}{u_{0}(0, n + 1)}},

(14.44)  ρ_{0,e}(n) = \sqrt{\frac{−a(n, 0)u_{0}(0, n + 1)}{u_{0}(0, n)}}.

By Theorem 14.3 we have a corresponding solution ρ(n, t) of the KM hierarchy and hence (by (14.35)) two solutions a_{k}(n, t), b_{k}(n, t) of the TL hierarchy. Since a_{1}(n, 0) = a(n, 0) and b_{1}(n, 0) = b(n, 0) we infer a_{1}(n, t) = a(n, t) and b_{1}(n, t) = b(n, t) by uniqueness (Theorem 12.6). Finally, we conclude u(0, n, t) = u_{0}(0, n_{0})u_{1}(n, t) > 0 (with u_{1}(n, t) as in (14.42)) again by uniqueness (Theorem 12.15). □

Another important consequence is Lemma 14.7. Let λ ≤ σ(H(0)) and a(n, t) < 0. Suppose u(λ, n, t) solves (12.89) and is a minimal positive solution near ±∞ for one t = t_{0}, then this holds for all t ∈ ℝ. In particular, H(t) − λ is critical (resp. subcritical) for all t ∈ ℝ if and only if it is critical (resp. subcritical) for one t = t_{0}.

Proof. Since linear independence and positivity is preserved by the system (12.89) (by (12.91) and Lemma 14.6) H(t) − λ is critical (resp. subcritical) for all t ∈ ℝ if and only if it is critical (resp. subcritical) for one t = t_{0}. If H(t) − λ is subcritical, we note that the characterization (2.70) of minimal solutions is independent of t. Hence it could only happen that u_{+}(λ, n, t) and u_{−}(λ, n, t) change place during time evolution. But this would imply u_{+}(λ, n, t) and u_{−}(λ, n, t) are linearly dependent at some intermediate time t contradicting H(t) − λ subcritical. □

In the special case of the stationary KM hierarchy characterized by ρ̇ = 0 in (14.24) (resp. (14.31)), or equivalently, by commuting matrix difference expressions of the type

(14.45)  [Q_{2r+2}, D] = 0,

the analogs of (12.79) and (12.67) then read

(14.46)  Q_{2r+2}^{2} = ∏_{j=0}^{2r+1}(D − \sqrt{E_{j}})(D + \sqrt{E_{j}}) = ∏_{j=0}^{2r+1}(D^{2} − E_{j})

and

(14.47)  y^{2} = ∏_{j=0}^{2r+1}(w − \sqrt{E_{j}})(w + \sqrt{E_{j}}) = ∏_{j=0}^{2r+1}(w^{2} − E_{j}).

Here ±\sqrt{E_{j}} are the band edges of the spectrum of D, which appear in pairs since D and −D are unitarily equivalent (see Lemma 14.9 below).

We note that the curve (14.47) becomes singular if and only if E_{j} = 0 for some 0 ≤ j ≤ 2r + 1. Since we have 0 ≤ E_{0} < E_{1} < ⋯ < E_{2r+1} this happens if and only if E_{0} = 0.

We omit further details at this point since we will show in the following sections how solutions of the KM hierarchy can be computed from the corresponding ones of the TL hierarchy.

14.2. Kac and van Moerbeke’s original equations

The Kac-van Moerbeke equation is originally given by

(14.48)  \frac{d}{dt}R(n, t) = \frac{1}{2}(e^{−R(n−1,t)} − e^{−R(n+1,t)}).

The transformation

(14.49)  ρ(n, t) = \frac{1}{2}e^{−R(n,t)/2},

yields the form most convenient for our purpose

(14.50)  ρ̇(n, t) = ρ(n, t)(ρ(n + 1, t)^{2} − ρ(n − 1, t)^{2}),

which is precisely KM_{0}(ρ) = 0. The form obtained via the transformation c(n, t) = 2ρ(n, t)^{2} is called Langmuir lattice,

(14.51)  ċ(n, t) = c(n, t)(c(n + 1, t) − c(n − 1, t)),

and is used for modeling Langmuir oscillations in plasmas.

14.3. Spectral theory for supersymmetric Dirac-type difference operators

In this section we briefly study spectral properties of self-adjoint ℓ^{2}(ℤ, ℂ^{2}) realizations associated with supersymmetric Dirac-type difference expressions.

Given ρ ∈ ℓ^{∞}(ℤ, ℝ) we start by introducing

(14.52)  A = ρ_{o}S^{+} + ρ_{e},  A^{∗} = ρ_{o}^{−}S^{−} + ρ_{e},

with ρ_{e}(n) = ρ(2n), ρ_{o}(n) = ρ(2n + 1). We denote by D the bounded self-adjoint supersymmetric Dirac operator

(14.53)  \begin{aligned}D : & ℓ^{2}(ℤ, ℂ^{2}) & → & ℓ^{2}(ℤ, ℂ^{2}) \\ & f & ↦ & ( \begin{matrix}0 & A^{∗} \\ A & 0 \end{matrix} ) f \end{aligned}

and by U the corresponding unitary involution

(14.54)  \begin{aligned}U : & ℓ^{2}(ℤ, ℂ^{2}) & → & ℓ^{2}(ℤ, ℂ^{2}) \\ & f & ↦ & ( \begin{matrix}0 & 1l \\ 1l & 0 \end{matrix} ) f \end{aligned} .

Theorem 14.8. Suppose ρ ∈ ℓ^{∞}(ℤ, ℝ), then D is a bounded self-adjoint operator with spectrum

(14.55)  σ(D) = {w ∈ ℝ|w^{2} ∈ σ(H_{1}) ∪ σ(H_{2})}

and resolvent

(14.56) (D − w)^{−1} = ( \begin{matrix}w(H_{1} − w^{2})^{−1} & A^{∗}(H_{2} − w^{2})^{−1} \\ A(H_{1} − w^{2})^{−1} & w(H_{2} − w^{2})^{−1} \end{matrix} ) , w^{2} ∈ ℂ\σ(D),

where H_{1} = A^{∗}A, H_{2} = AA^{∗}.

Proof. The spectrum of D follows from the spectral mapping theorem since D^{2} = H_{1} ⊕ H_{2} and since D and −D are unitarily equivalent (see the lemma below). The formula for the resolvent is easily vindicated using A(H_{1} − w^{2})^{−1} = (H_{2} − w^{2})^{−1}A and (H_{1} − w^{2})^{−1}A^{∗} = A^{∗}(H_{2} − w^{2})^{−1} (since AH_{1} = H_{2}A and H_{1}A^{∗} = A^{∗}H_{2}) (see also Theorem 11.1). □

Hence one can reduce the spectral analysis of supersymmetric Dirac operators D to that of H_{1} and H_{2}. Moreover, since H_{1}|_{Ker(H_{1})^{⊥}} and H_{2}|_{Ker(H_{2})^{⊥}} are unitarily equivalent by Theorem 11.1, a complete spectral analysis of D in terms of that of H_{1} and Ker(H_{2}) can be given.

The analog of Lemma 1.6 reads

Lemma 14.9. Suppose ρ satisfies (H.14.1) and introduce ρ_{ε} ∈ ℓ_{ℝ}^{∞}(ℤ) by

(14.57)  ρ_{ε}(n) = ε(n)ρ(n), ε(n) ∈ {+1, −1}, n ∈ ℤ.

Define D_{ε} in ℓ^{2}(ℤ, ℂ^{2}) as in (14.53) with ρ replaced by ρ_{ε}. Then D and D_{ε} are unitarily equivalent, that is, there exists a unitary operator U_{ε} in ℓ^{2}(ℤ, ℂ^{2}) such that

(14.58)  D = U_{ε}D_{ε}U_{ε}^{−1}.

Especially taking ε(n) = −1 shows that D and −D are unitarily equivalent.

Proof. U_{ε} is explicitly represented by

(14.59)  U_{ε} = ( \begin{matrix}U_{1,ε} & 0 \\ 0 & U_{2,ε} \end{matrix} ) , U_{k,ε} = \{ε̃_{k}(n)δ_{m,n}\}_{m,n∈ℤ}, k = 1, 2,

ε̃_{1}(n + 1)ε̃_{2}(n) = ε(2n + 1), ε̃_{1}(n)ε̃_{2}(n) = ε(2n), n ∈ ℤ. □

14.4. Associated solutions

In Theorem 14.5 we saw, that from one solution ρ of KM_{r}(ρ) = 0 we can get two solutions (a_{1}, b_{1}), (a_{2}, b_{2}) of TL_{r}(a, b). In this section we want to invert this process (cf. Section 11.2).

Suppose (a_{1}, b_{1}) satisfies (H.12.1), a_{1}(n, t) < 0 and TL_{r}(a_{1}, b_{1}) = 0. Suppose H_{1} > 0 and let u_{1}(n, t) > 0 solve

(14.60)  H_{1}(t)u_{1}(n, t) = 0,  \frac{d}{dt}u_{1}(n, t) = P_{1,2r+2}(t)u_{1}(n, t)

(cf. Lemma 12.16). This implies that ϕ_{1}(n, t) = u_{1}(n + 1, t)/u_{1}(n, t) fulfills

(14.61)  a_{1}(n, t)ϕ_{1}(n, t) + \frac{a_{1}(n − 1, t)}{ϕ_{1}(n − 1, t)} = −b_{1}(n, t),

and

\frac{d}{dt} ln ϕ_{1}(n, t) = −2a_{1}(n, t)(g_{1,r}(n, t)ϕ_{1}(n, t) + g_{1,r}(n + 1, t)ϕ_{1}(n, t)^{−1})

+ 2b_{1}(n + 1, t)g_{1,r}(n + 1, t) + (g_{1,r+1}(n + 1, t) − g_{1,r+1}(n, t))

− (h_{1,r}(n + 1, t) − h_{1,r}(n, t)). (14.62)

Now define

(14.63)  ρ_{o}(n) = −\sqrt{−\frac{a_{1}(n, t)}{ϕ_{1}(n, t)}},  ρ_{e}(n, t) = \sqrt{−a_{1}(n, t)ϕ_{1}(n, t)}.

Then, using (12.40), (12.27), and (14.8) a straightforward calculation shows that the sequence

(14.64) ρ(n, t) = \{ \begin{matrix}ρ_{e}(m, t) for & n = 2m \\ ρ_{o}(m, t) for & n = 2m + 1 \end{matrix} = (−1)^{n+1}\sqrt{−a_{1}(n, t)ϕ_{1}(n, t)^{(−1)^{n}}},

fulfills (H.14.1) and KM_{r}(ρ) = 0. Hence by (14.35)

(14.65)  a_{2}(n, t) = ρ_{e}(n + 1, t)ρ_{o}(n, t),  b_{2}(n, t) = ρ_{e}(n, t)^{2} + ρ_{o}(n, t)^{2}

satisfy TL_{r}(a_{2}, b_{2}) = 0.

Since we already know some solutions of TL_{r}(a_{1}, b_{1}) = 0, we can illustrate this process. Taking a finite-gap solution

a_{1}(n, t) = −ã\sqrt{\frac{θ(\underline{z}_{1}(n + 1, t))θ(\underline{z}_{1}(n − 1, t))}{θ(\underline{z}_{1}(n, t))^{2}}},

b_{1}(n, t) = b̃ + ∑_{j=1}^{g} c_{j}(g)\frac{∂}{∂w_{j}} ln (\frac{θ(\underline{w} + \underline{z}_{1}(n, t))}{θ(\underline{w} + \underline{z}_{1}(n − 1, t))})|_{\underline{w}=0},

(14.66) ϕ_{1}(n, t) = \sqrt{\frac{θ(\underline{z}_{1}(n − 1, t))}{θ(\underline{z}_{1}(n + 1, t))}} \frac{θ(\underline{z}_{1}(p_{1}, n + 1, t))}{θ(\underline{z}_{1}(p_{1}, n, t))} exp ( ∫_{p_{0}}^{p_{1}}ω_{∞_{+},∞_{−}}),

with π(p_{1}) ≤ E_{0} yields as in Section 11.4

ρ_{o}(n, t) = −\sqrt{−\frac{a_{1}(n, t)}{ϕ_{1}(n, t)}}

= −\sqrt{ã\frac{θ(\underline{z}_{1}(n + 1, t))}{θ(\underline{z}_{1}(n, t))} \frac{θ(\underline{z}_{1}(p_{1}, n, t))}{θ(\underline{z}_{1}(p_{1}, n + 1, t))}} exp (\frac{−1}{2} ∫_{p_{0}}^{p_{1}}ω_{∞_{+},∞_{−}}),

ρ_{e}(n, t) = \sqrt{−a_{1}(n, t)ϕ_{1}(n, t)}

= \sqrt{ã\frac{θ(\underline{z}_{1}(n − 1, t))}{θ(\underline{z}_{1}(n, t))} \frac{θ(\underline{z}_{1}(p_{1}, n + 1, t))}{θ(\underline{z}_{1}(p_{1}, n, t))}} exp (\frac{1}{2} ∫_{p_{0}}^{p_{1}}ω_{∞_{+},∞_{−}}), (14.67)

and

a_{2}(n, t) = −ã\sqrt{\frac{θ(\underline{z}_{2}(n + 1, t))θ(\underline{z}_{2}(n − 1, t))}{θ(\underline{z}_{2}(n, t))^{2}}},

b_{2}(n, t) = b̃ + ∑_{j=1}^{g} c_{j}(g)\frac{∂}{∂w_{j}} ln (\frac{θ(\underline{w} + \underline{z}_{2}(n, t))}{θ(\underline{w} + \underline{z}_{2}(n − 1, t))})|_{\underline{w}=0},

(14.68) ϕ_{2}(n, t) = \sqrt{\frac{θ(\underline{z}_{2}(n − 1, t))}{θ(\underline{z}_{2}(n + 1, t))}} \frac{θ(\underline{z}_{2}(p_{1}^{∗}, n + 1, t))}{θ(\underline{z}_{2}(p_{1}^{∗}, n, t))} exp ( ∫_{p_{0}}^{p_{1}^{∗}}ω_{∞_{+},∞_{−}}),

where \underline{z}_{2}(n) = \underline{z}_{1}(p_{1}, n + 1) respectively \underline{z}_{2}(p_{1}^{∗}, n) = \underline{z}_{1}(n). We remark that if we interchange the role of a_{1} and a_{2} and use p^{∗} instead of p we get a sequence ρ̂(n, t) which is related to ρ(n, t) via ρ̂(n, t) = ρ(n + 1, t) (cf. (14.7)). This implies that we can assume p to lie on the upper sheet.

In the simplest case, where we start with the constant solution a_{1}(n, t) = a and b_{1}(n, t) = b, we obtain for p = (λ, ±R_{2}^{1/2}(λ))

(14.69)  ρ_{e}(n, t) = ρ_{o}(n, t) = \frac{−1}{\sqrt{2}}\sqrt{|λ − b| ± \sqrt{|λ − b|^{2} − 4a^{2}}}

and hence

(14.70)  ρ(n, t) = \frac{(−1)^{n+1}}{\sqrt{2}} \sqrt{|λ − b| ± \sqrt{|λ − b|^{2} − 4a^{2}}}.

14.5. N-soliton solutions on arbitrary background 267

14.5. N-soliton solutions on arbitrary background

In this section we want to extend the calculations of Section 14.4 and of Chapter 11.

Suppose (a, b) satisfies (H.12.1), a(n, t) < 0, and TL_{r}(a, b) = 0. Suppose λ_{1} ≤ σ(H), let u_{±}(λ_{1}, n, t) > 0 be the minimal positive solutions of (12.89) found in Lemma 14.7, and set

(14.71)  u_{σ_{1}}(λ_{1}, n, t) = \frac{1 + σ_{1}}{2} u_{+}(λ_{1}, n, t) + \frac{1 − σ_{1}}{2} u_{−}(λ_{1}, n, t).

Note that the dependence on σ_{1} will drop out in what follows if u_{+}(λ_{1}, n, t) and u_{−}(λ_{1}, n, t) are linearly dependent (for one and hence for all t). Now define

(14.72) ρ_{σ_{1},o}(n, t) = −\sqrt{−\frac{a(n, t)}{ϕ_{σ_{1}}(λ_{1}, n, t)}},  ρ_{σ_{1},e}(n, t) = \sqrt{−a(n, t)ϕ_{σ_{1}}(λ_{1}, n, t)},

where ϕ_{σ_{1}}(λ_{1}, n, t) = u_{σ_{1}}(λ_{1}, n + 1, t)/u_{σ_{1}}(λ_{1}, n, t).

Then, proceeding as in the proof of Lemma 14.6 shows that the sequence

(14.73)  ρ_{σ_{1}}(n, t) = \{ \begin{matrix}ρ_{σ_{1},e}(m, t) for & n = 2m \\ ρ_{σ_{1},o}(m, t) for & n = 2m + 1 \end{matrix} ,

fulfills (H.14.1) and KM_{r}(ρ) = 0. Hence by (14.35) (14.74)

a_{σ_{1}}(n, t) = ρ_{σ_{1},e}(n + 1, t)ρ_{σ_{1},o}(n, t), b_{σ_{1}}(n, t) = ρ_{σ_{1},e}(n, t)^{2} + ρ_{σ_{1},o}(n, t)^{2}

satisfy TL_{r}(a_{σ_{1}}, b_{σ_{1}}) = 0.

We summarize this result in our first main theorem.

Theorem 14.10. Suppose (a, b) satisfies (H.12.1) and TL_{r}(a, b) = 0. Pick λ_{1} ≤ σ(H), σ_{1} ∈ [−1, 1] and let u_{±}(λ_{1}, n, t) be the minimal positive solutions of (12.89). Then the sequences

a_{σ_{1}}(n, t) = \sqrt{\frac{a(n, t)a(n + 1, t)u_{σ_{1}}(λ_{1}, n, t)u_{σ_{1}}(λ_{1}, n + 2, t)}{u_{σ_{1}}(λ_{1}, n + 1, t)^{2}}},
(14.75)  b_{σ_{1}}(n, t) = b(n, t) + ∂^{∗} \frac{a(n, t)u_{σ_{1}}(λ_{1}, n, t)}{u_{σ_{1}}(λ_{1}, n + 1, t)}

with

(14.76)  u_{σ_{1}}(λ_{1}, n + 1, t) = \frac{1 + σ_{1}}{2} u_{+}(λ_{1}, n, t) + \frac{1 − σ_{1}}{2} u_{−}(λ_{1}, n, t),

satisfy TL_{r}(a_{σ_{1}}, b_{σ_{1}}) = 0. Here ∂^{∗}f(n) = f(n − 1) − f(n). In addition,

(14.77)  \frac{a(n, t)(u_{σ_{1}}(λ_{1}, n, t)u(z, n + 1, t) − u_{σ_{1}}(λ_{1}, n + 1, t)u(z, n, t))}{\sqrt{−a(n, t)u_{σ_{1}}(λ_{1}, n, t)u_{σ_{1}}(λ_{1}, n + 1, t)}}

satisfies H_{σ_{1}}u = zu and d/dt u = P_{σ_{1},2r+2}u (weakly) (in obvious notation).

Remark 14.11. (i). Alternatively, one could give a direct algebraic proof of the above theorem using H_{σ_{1}} = A_{σ_{1}}HA_{σ_{1}}^{∗} to express the quantities g_{σ_{1},j}, h_{σ_{1},j} in terms of g_{j}, h_{j}. (ii). We have omitted the requirement a(n, t) < 0 since the formulas for a_{σ_{1}}, b_{σ_{1}} are actually independent of the sign of a(n, t). In addition, we could even allow λ_{1} ≥ σ(H). However, ρ_{σ_{1},e}(n, t) and ρ_{σ_{1},o}(n, t) would be purely imaginary in this case.

Iterating this procedure (cf. Theorem 11.10) gives

Theorem 14.12. Let a(t), b(t) satisfy (H.12.1) and TL_{r}(a, b) = 0. Let H(t) be the corresponding Jacobi operators and choose

(14.78)  λ_{N} < ⋯ < λ_{2} < λ_{1} ≤ σ(H), σ_{ℓ} ∈ [−1, 1], 1 ≤ ℓ ≤ N, N ∈ ℕ.

Suppose u_{±}(λ, n, t), are the principal solutions of (12.89). Then

a_{σ_{1},...,σ_{N}} (n, t) = −\sqrt{a(n, t)a(n + N, t)}

×\frac{\sqrt{C_{n}(U_{σ_{1},...,σ_{N}} (t))C_{n+2}(U_{σ_{1},...,σ_{N}} (t))}}{C_{n+1}(U_{σ_{1},...,σ_{N}} (t))},

(14.79)  b_{σ_{1},...,σ_{N}} (n, t) = b(n, t) + ∂^{∗}a(n, t)\frac{D_{n}(U_{σ_{1},...,σ_{N}} (t))}{C_{n+1}(U_{σ_{1},...,σ_{N}} (t))}

satisfies TL_{r}(a_{σ_{1},...,σ_{N}} , b_{σ_{1},...,σ_{N}}) = 0. Here C_{n} denotes the n-dimensional Casoratian

C_{n}(u_{1}, …, u_{N}) = det (u_{i}(n + j − 1))_{1≤i,j≤N} ,
(14.80)  D_{n}(u_{1}, …, u_{N}) = det ( \begin{matrix}u_{i}(n), & j = 1 \\ u_{i}(n + j), & j > 1 \end{matrix} )_{1≤i,j≤N} ,

and (U_{σ_{1},...,σ_{N}} (t)) = (u_{σ_{1}}^{1}(t), …, u_{σ_{N}}^{N} (t)) with

(14.81)  u_{σ_{ℓ}}^{ℓ}(n, t) = \frac{1 + σ_{ℓ}}{2} u_{+}(λ_{ℓ}, n, t) + (−1)^{ℓ+1} \frac{1 − σ_{ℓ}}{2} u_{−}(λ_{ℓ}, n, t).

Moreover, for 1 ≤ ℓ ≤ N, λ < λ_{ℓ},

(14.82)  u_{±,σ_{1},...,σ_{ℓ}}(λ, n, t) = \frac{± ∏_{j=0}^{ℓ−1} \sqrt{−a(n + j, t)}C_{n}(U_{σ_{1},...,σ_{ℓ}}(t), u_{±}(λ, t))}{\sqrt{C_{n}(U_{σ_{1},...,σ_{ℓ}}(t))C_{n+1}(U_{σ_{1},...,σ_{ℓ}}(t))}},

are the minimal solutions of τ_{σ_{1},...,σ_{ℓ}}(t)u = λu and satisfy

(14.83)  \frac{d}{dt}u_{±,σ_{1},...,σ_{ℓ}}(λ, n, t) = P_{σ_{1},...,σ_{ℓ},2r+2}(t)u_{±,σ_{1},...,σ_{ℓ}}(λ, n, t).

Defining

ρ_{σ_{1},...,σ_{N} ,o}(n, t) =

−\sqrt{−a(n, t)\frac{C_{n+2}(U_{σ_{1},...,σ_{N−1}}(t))C_{n}(U_{σ_{1},...,σ_{N}} (t))}{C_{n+1}(U_{σ_{1},...,σ_{N−1}}(t))C_{n+1}(U_{σ_{1},...,σ_{N}} (t))}},

ρ_{σ_{1},...,σ_{N} ,e}(n, t) =

\sqrt{−a(n + N − 1, t)\frac{C_{n}(U_{σ_{1},...,σ_{N−1}}(t))C_{n+1}(U_{σ_{1},...,σ_{N}} (t))}{C_{n+1}(U_{σ_{1},...,σ_{N−1}}(t))C_{n}(U_{σ_{1},...,σ_{N}} (t))}}, (14.84)

the corresponding sequence ρ_{σ_{1},...,σ_{N}} (n) solves KM_{r}(ρ_{σ_{1},...,σ_{N}}) = 0.

Clearly, if we drop the requirement λ ≤ σ(H) the solution u_{σ_{1}}(λ_{1}, n, t) used to perform the factorization will no longer be positive. Hence the sequences a_{σ_{1}}(n, t), b_{σ_{1}}(n, t) can be complex valued and singular. Nevertheless there are two situations where a second factorization step produces again real-valued non-singular solutions.

Firstly we perform two steps with λ_{1,2} in the same spectral gap of H(0) (compare Section 11.8).

Theorem 14.13. Suppose (a, b) satisfies (H.12.1) and TL_{r}(a, b) = 0. Pick λ_{1,2}, σ_{1,2} ∈ {±1} and let λ_{1,2} lie in the same spectral gap of H(0) ( (λ_{1}, σ_{1}) ≠ (λ_{2}, −σ_{2}) to make sure we get something new). Then the sequences

a_{σ_{1},σ_{2}}(n, t) = a(n, t)\sqrt{\frac{W_{σ_{1},σ_{2}}(n − 1, t)W_{σ_{1},σ_{2}}(n + 1, t)}{W_{σ_{1},σ_{2}}(n, t)^{2}}},

(14.85) b_{σ_{1},σ_{2}}(n, t) = b(n, t) − ∂^{∗} \frac{a(n, t)u_{σ_{1}}(λ_{1}, n, t)u_{σ_{2}}(λ_{2}, n + 1, t)}{W_{σ_{1},σ_{2}}(n, t)},

where

(14.86)  W_{σ_{1},σ_{2}}(n, t) = \{ \begin{matrix}\frac{W_{n}(u_{σ_{1}} (λ_{1},t),u_{σ_{2}} (λ_{2},t))}{λ_{2}−λ_{1}}, & λ_{1} ≠ λ_{2} \\ ∑_{m=σ_{1}∞}^{n} u_{σ_{1}}(λ_{1}, m, t)^{2}, & (λ_{1}, σ_{1}) = (λ_{2}, σ_{2}) \end{matrix} ,

are real-valued non-singular solutions of TL_{r}(a_{σ_{1},σ_{2}}, b_{σ_{1},σ_{2}}) = 0. In addition, the sequence

(14.87)  \frac{W_{σ_{1},σ_{2}}(n, t)u(z, n, t) − \frac{1}{z−λ_{1}} u_{σ_{2}}(λ_{2}, n, t)W_{n}(u_{σ_{1}}(λ_{1}, t), u(z, t))}{\sqrt{W_{σ_{1},σ_{2}}(n − 1, t)W_{σ_{1},σ_{2}}(n, t)}},

satisfies H_{σ_{1},σ_{2}}(t)u = zu, d/dt u = P_{σ_{1},σ_{2},2r+2}(t)u (weakly).

Proof. Theorem 4.19 implies W_{σ_{1},σ_{2}}(n, t)W_{σ_{1},σ_{2}}(n + 1, t) > 0 and hence the sequences a_{σ_{1},σ_{2}}(t), b_{σ_{1},σ_{2}}(t) satisfy (H.12.1). The rest follows from the previous theorem (with N = 2) as follows. Replace λ_{1} by z ∈ (λ_{1} − ε, λ_{1} + ε) and observe that a_{σ_{1},σ_{2}}(n, t), b_{σ_{1},σ_{2}}(n, t) and ȧ_{σ_{1},σ_{2}}(n, t), ḃ_{σ_{1},σ_{2}}(n, t) are meromorphic with respect to z. From the algebraic structure we have simply performed two single commutation steps. Hence, provided Theorem 14.10 applies to this more general setting of meromorphic solutions, we can conclude that our claims hold except for a discrete set with respect to z where the intermediate operators are ill-defined due to singularities of the coefficients. However, the proof of Theorem 14.10 uses these intermediate operators and in order to see that Theorem 14.10 still holds, one has to resort to the direct algebraic proof outlined in Remark 14.11(i). Continuity with respect to z takes care of the remaining points. □

Secondly, we consider again two commutation steps but now with λ_{1} = λ_{2} (compare Section 11.6).

Theorem 14.14. Suppose (a, b) satisfies (H.12.1) and TL_{r}(a, b) = 0. Pick λ_{1} in a spectral gap of H(0) and γ_{1} ∈ [−∥u_{−}(λ_{1})∥^{−2}, ∞) ∪ {∞}. Then the sequences

a_{γ_{1}}(n, t) = a(n, t)\frac{\sqrt{c_{γ_{1}}(λ_{1}, n − 1, t)c_{γ_{1}}(λ_{1}, n + 1, t)}}{c_{γ_{1}}(λ_{1}, n, t)},
(14.88)  b_{γ_{1}}(n, t) = b(n, t) − ∂^{∗} \frac{a(n, t)u_{−}(λ_{1}, n, t)u_{−}(λ_{1}, n + 1, t)}{c_{γ_{1}}(λ_{1}, n, t)}.

satisfy TL_{r}(a_{γ_{1}}, b_{γ_{1}}) = 0, where

(14.89)  c_{γ_{1}}(λ_{1}, n, t) = \frac{1}{γ_{1}} + ∑_{m=−∞}^{n} u_{−}(λ_{1}, m, t)^{2}.

In addition, the sequence

(14.90)  \frac{c_{γ_{1}}(λ_{1}, n, t)u(z, n, t) − \frac{1}{z−λ_{1}} u_{−}(λ_{1}, n, t)W_{n}(u_{−}(λ_{1}, t), u(z, t))}{\sqrt{c_{γ_{1}}(λ_{1}, n − 1, t)c_{γ_{1}}(λ_{1}, n, t)}},

satisfies H_{γ_{1}}(t)u = zu, d/dt u = P_{γ_{1},2r+2}(t)u (weakly).

Proof. Following Section 11.5 we can obtain the double commutation method from two single commutation steps. We pick σ_{1} = −1 for the first factorization. Considering A_{σ_{1}}u_{−}(z, n + 1, t)/(z − λ_{1}) and performing the limit z → λ_{1} shows that

(14.91)  v(λ_{1}, n, t) = \frac{c_{γ_{1}}(λ_{1}, n, t)}{\sqrt{−a(n, t)u_{−}(λ_{1}, n, t)u_{−}(λ_{1}, n + 1, t)}}

is a solution of the new (singular) operator which can be used to perform a second factorization. The resulting operator is associated with a_{γ_{1}}, b_{γ_{1}}. Now argue as before. □

Again we point out that one can also prove this theorem directly as follows. Without restriction we choose λ_{1} = 0. Then one computes

\frac{d}{dt}c_{γ_{1}}(0, n, t)=2a(n, t)^{2}(g_{r−1}(n + 1, t)u_{−}(0, n, t)^{2} + g_{r−1}(n, t)u_{−}(0, n + 1, t)^{2})

+ 2h_{r−1}(n, t)a(n, t)u_{−}(0, n, t)u_{−}(0, n + 1, t) (14.92)

and it remains to relate g_{γ_{1},j}, h_{γ_{1},j} and g_{j}, h_{j}. Since these quantities arise as coefficients of the Neumann expansion of the respective Green functions it suffices to relate the Green functions of H_{γ_{1}} and H. This can be done using (11.108).

Iterating this procedure (cf. Theorem 11.27) gives

Theorem 14.15. Let a(n, t), b(n, t) satisfy (H.12.1). Suppose TL_{r}(a, b) = 0 and let H(t) be the corresponding Jacobi operators. Choose λ_{j} ∈ ρ(H) and γ_{j} ∈ [−∥u_{−}(λ_{j})∥^{−2}, ∞) ∪ {∞}, 1 ≤ j ≤ N, and assume

(14.93)  \frac{d}{dt}u_{−}(λ_{j}, n, t) = P_{2r+2}(t)u_{−}(λ_{j}, n, t).

We define the following matrices

C^{N}(n, t) = (\frac{δ_{ij}}{γ_{i}} + ∑_{m=−∞}^{n} u_{−}(λ_{i}, m, t)u_{−}(λ_{j}, m, t))_{1≤i,j≤N} ,

(14.94) D^{N}(n, t) = ( \begin{matrix}C^{N}(n, t)_{i,j}, & i,j≤N \\ u_{−}(λ_{j}, n − 1, t), & j≤N,i=N+1 \\ u_{−}(λ_{i}, n, t), & i≤N,j=N+1 \\ 0, & i=j=N+1 \end{matrix} )_{1≤i,j≤N+1} .

Then the sequences

a_{γ_{1},...,γ_{N}} (n, t) = a(n, t)\frac{\sqrt{det C^{N}(n − 1, t) det C^{N}(n + 1, t)}}{det C^{N}(n, t)},
(14.95)  b_{γ_{1},...,γ_{N}} (n, t) = b(n, t) − ∂^{∗}a(n, t)\frac{det D^{N}(n + 1, t)}{det C^{N}(n, t)}

satisfy TL_{r}(a_{γ_{1},...,γ_{N}} , b_{γ_{1},...,γ_{N}}) = 0. Moreover, (1 ≤ ℓ ≤ N)

(14.96)  u_{γ_{1},...,γ_{ℓ}}(z, n, t) = \frac{det U^{N}(u(z, n, t))}{\sqrt{det C^{ℓ}(n − 1, t) det C^{ℓ}(n, t)}}

satisfies

(14.97)  H_{γ_{1},...,γ_{N}} u = λ_{j}u,  \frac{d}{dt}u = P_{γ_{1},...,γ_{N} ,2r+2}(t)u

(in the weak sense) if u(z, n, t) satisfies (12.89).

Remark 14.16. In the case r = 0 we even obtain

(14.98)  \frac{d}{dt}c_{γ_{1}}(λ_{1}, n, t) = 2a(n, t)u_{−}(λ_{1}, n, t)u_{−}(λ_{1}, n + 1, t)

and hence

(14.99)  b_{γ_{1}}(n, t) = b(n, t) + \frac{1}{2} \frac{d}{dt} ln \frac{c_{γ_{1}}(λ_{1}, n, t)}{c_{γ_{1}}(λ_{1}, n − 1, t)},

where c_{γ_{1}}(λ_{1}, n, t) = γ_{1}^{−1} + ∑_{m=−∞}^{n} u_{−}(λ_{1}, m, t). Or, by induction,

(14.100)  b_{γ_{1},...,γ_{N}} (n, t) = b(n, t) + \frac{1}{2} \frac{d}{dt} ln \frac{det C^{N}(n, t)}{det C^{N}(n − 1, t)}.

We conclude this section with an example; the N-soliton solution of the TL and KM hierarchies. We take the constant solution of the Toda hierarchy

(14.101)  a_{0}(n, t) = \frac{1}{2},  b_{0}(n, t) = 0,

as our background. Let H_{0}, P_{0,2r+2} denote the associated Lax pair and recall (14.102)

H_{0}(t)u_{0,±}(z, n, t) = zu_{0,±}(z, n, t), \frac{d}{dt}u_{0,±}(z, n, t) = P_{0,2r+2}(t)u_{0,±}(z, n, t),

where

(14.103)  u_{0,±}(z, n, t) = k^{±n} exp ( ± \frac{α_{r}(k)t}{2}), k = z − \sqrt{z^{2} − 1}, |k| ≤ 1,

and

(14.104)  α_{r}(k) = 2(kG_{0,r}(z) − H_{r+1,0}(z)) = (k − k^{−1})G_{0,r}(z).

Then the N-soliton solution of the Toda hierarchy is given by

a_{0,σ_{1},...,σ_{N}} (n, t) = \frac{\sqrt{C_{n}(U_{σ_{1},...,σ_{N}} (t))C_{n+2}(U_{σ_{1},...,σ_{N}} (t))}}{2C_{n+1}(U_{σ_{1},...,σ_{N}} (t))},
(14.105)  b_{0,σ_{1},...,σ_{N}} (n, t) = ∂^{∗} \frac{D_{n}(U_{σ_{1},...,σ_{N}} (t))}{2C_{n+1}(U_{σ_{1},...,σ_{N}} (t))},

where (U_{0,σ_{1},...,σ_{N}} (t)) = (u_{σ_{1}}^{1}(t), …, u_{σ_{N}}^{N} (t)) with

(14.106) u_{0,σ_{j}}^{j}(n, t) = k_{j}^{n} + (−1)^{j+1} \frac{1 − σ_{j}}{1 + σ_{j}} exp(α_{r}(k_{j})t)k_{j}^{−n}, k_{j} = λ_{j} − \sqrt{λ_{j}^{2} − 1}.

The corresponding N-soliton solution ρ_{σ_{1},...,σ_{N}} (n) of the Kac-van Moerbeke hierarchy reads

ρ_{0,σ_{1},...,σ_{N} ,o}(n, t) =

−\sqrt{−\frac{C_{n+2}(U_{0,σ_{1},...,σ_{N−1}}(t))C_{n}(U_{0,σ_{1},...,σ_{N}} (t))}{2C_{n+1}(U_{0,σ_{1},...,σ_{N−1}}(t))C_{n+1}(U_{0,σ_{1},...,σ_{N}} (t))}},

ρ_{0,σ_{1},...,σ_{N} ,e}(n, t) =

\sqrt{−\frac{C_{n}(U_{0,σ_{1},...,σ_{N−1}}(t))C_{n+1}(U_{0,σ_{1},...,σ_{N}} (t))}{2C_{n+1}(U_{0,σ_{1},...,σ_{N−1}}(t))C_{n}(U_{0,σ_{1},...,σ_{N}} (t))}}. (14.107)

Introducing the time dependent norming constants

(14.108)  γ_{j}(t) = γ_{j} exp(α_{r}(k_{j})t)

we obtain the following alternate expression for the N-soliton solution of the Toda hierarchy

a_{0,γ_{1},...,γ_{N}} (n, t) = \frac{\sqrt{det C_{0}^{N}(n − 1, t) det C_{0}^{N}(n + 1, t)}}{2 det C_{0}^{N}(n, t)},
(14.109)  b_{0,γ_{1},...,γ_{N}} (n, t) = −∂^{∗} \frac{det D_{0}^{N}(n + 1, t)}{2 det C_{0}^{N}(n, t)},

where

C_{0}^{N}(n, t) = (\frac{δ_{rs}}{γ_{r}(t)} + \frac{(k_{r}k_{s})^{−n}}{1 − k_{r}k_{s}} )_{1≤r,s≤N} ,

(14.110)  D_{0}^{N}(n, t) = ( \begin{matrix}C_{0}^{N}(n, t)_{r,s}, & r,s≤N \\ k_{s}^{1−n}, & s≤N,r=N+1 \\ k_{r}^{−n}, & r≤N,s=N+1 \\ 0, & r=s=N+1 \end{matrix} )_{1≤r,s≤N+1} .

The sequences a_{0,γ_{1},...,γ_{N}}, b_{0,γ_{1},...,γ_{N}} coincide with a_{0,σ_{1},...,σ_{N}}, b_{0,σ_{1},...,σ_{N}} provided (cf. Remark 11.30)

(14.111)  γ_{j} = (\frac{1 − σ_{j}}{1 + σ_{j}} )^{−1} |k_{j}|^{−1−N} \frac{∏_{ℓ=1}^{N} |1 − k_{j}k_{ℓ}|}{∏N_{{ℓ=1 \atop ℓ≠j}} |k_{j} − k_{ℓ}|}, 1 ≤ j ≤ N.

Chapter 12. The Toda lattice was first introduced by Toda in [233]. For further material see his selected papers [232] or his monograph [230].

The Toda hierarchy was first introduced in [237] and [238] (see also [236]). Our introduction uses a recursive approach for the standard Lax formalism ([164]) which was first advocated by Al’ber [11], Jacobi [137], McKean [169], and Mumford [178] (Sect. III a). It follows essentially the monograph [35] (with some extensions). An existence and uniqueness theorem for the semi infinite case seems to appear first in [65], the form stated here is taken from [220]. The Burchnall-Chaundy polynomial in the case of commuting differential expressions was first obtained in [36], [37] (cf. also [236]).

Further references I refer to a few standard monographs such as [78], [79] [184], and [190]. Chapter 13. Section 1 and 2 closely follow [35] with several additions made. The Baker-Akhiezer function, the fundamental object of our approach, goes back to the work of Baker [23], Burchnall and Chaundy [36], [37], and Akhiezer [8]. The modern approach was initiated by Its and Matveev [136] in connection with the Korteweg-de Vries equation and further developed into a powerful machinery by Krichever (see, e.g., [156]–[158]) and others. Gesztesy and Holden are also working on a monograph on hierarchies of soliton equations and their algebro-geometric solutions [103]. We refer, in particular, to the extensive treatments in [25], [74], [75], [76], [168], [177], and [184]. In the special context of the Toda equations we refer to [3], [75], [76], [156], [159], [168], [170], and [175].

The periodic case was first investigated by Kac and van Moerbeke [143], [144]. Further references for the periodic case are [3], [33], [53], [56], [146], [168], and [173]–[175].

In the case of the Toda lattice (r = 0) the formulas for the quasi periodic finite gap case in terms of Riemann theta functions seem due to [210]. However, some norming constants and the fact b̃ = d̃ (see Remark 13.9) are not established there.

In case of the Toda equation on the half line a different approach using the spectral measure of H_{+} is introduced and extended in [28] – [32] (see also [15]). It turns out that the time evolution of the spectral measure can be computed explicitly as the Freud transform of the initial measure.

The inverse scattering transform for the case r = 0 has been formally developed by Flaschka [85] (see also [230] and [80] for the case of rapidly decaying sequences) for the Toda lattice. In addition, Flaschka also worked out the inverse procedure in the reflectionless case (i.e., R_{±}(k, t) = 0). His formulas clearly apply to the entire Toda hierarchy upon using the t dependence of the norming constants given in (13.72) and coincide with those obtained using the double commutation method (i.e., the last example in Section 14.5). All results from the fourth section are taken from [219] and [224].

For additional results where the Toda equation is driven from one end by a particle see [63], [64] and also [67]. For results on a continuum limit of the Toda lattice see [59]. For connections of the Riemann-Hilbert problem and the inverse scattering method and applications to the Toda lattice see [61] and [62]. In addition, there are also very nice lecture notes by Deift [57] which give an introduction to the Riemann-Hilbert problem and its applications. Chapter 14. The Kac-van Moerbeke equation has been first introduced in [142]. A connection between the Kac-van Moerbeke and Toda systems was already known to Hénon in 1973. The Bäcklund transformation connecting the Toda and the Kac-van Moerbeke equations has first been considered in [234] and [239] (see also [56], [151], [152]). The results presented here are mainly taken from [35], [117] and [220]. An alternative approach to the modified Toda hierarchy, using the discrete analog of the formal pseudo differential calculus, can be found in [161], Ch. 4.

For applications of the Kac-van Moerbeke equation I refer to(e.g.) [25] (Ch. 8), [55], [66], [130], [159], [183], or [246]

Appendix A

Compact Riemann surfaces

– a review

The facts presented in this section can be found in almost any book on Riemann surfaces and their theta functions (cf., e.g. [74], [81], [82], [86], [154], [178], [181]). However, no book contains all results needed. Hence, in order to save you the tedious work of looking everything up from different sources and then translate results from one notation to another, I have included this appendix. In addition, I have worked out some facts on hyperelliptic surfaces in more detail than you will find them in the references. Most of the time we will closely follow [81] in the first sections.

A.1. Basic notation

Let M be a compact Riemann surface (i.e., a compact, second countable, connected Hausdorff space together with a holomorphic structure) of genus g ∈ ℕ. We restrict ourselves to g ≥ 1 since most of our definitions only make sense in this case. We will think of our Riemann surface M often as it’s fundamental polygon (e.g., for g = 2):

[figure: M̂; b_{1}^{−1}; a_{1}^{−1}; b_{1}; a_{1}; b_{2}^{−1}; a_{2}^{−1}; b_{2}; a_{2}]

We denote the closure of the polygon by M̂ and its boundary by ∂M̂ = ∏_{ℓ=1}^{g}(a_{ℓ}b_{ℓ}a_{ℓ}^{−1}b_{ℓ}^{−1}), where {a_{ℓ} , b_{ℓ}}_{1≤ℓ≤g} are (fixed) representatives of a canonical homology basis for M. We get M from M̂ by identifying corresponding points, that is, M = M̂/ ≃.

We denote the set of meromorphic functions on M by M(M), the set of meromorphic differentials by M^{1}(M), and the set of holomorphic differentials by H^{1}(M). If we expand f ∈ M(M) respectively ω ∈ M^{1}(M) in a local chart (U, z) centered at p ∈ M we get (z(p) = 0)

(A.1) f = ∑_{ℓ=m}^{∞} c_{ℓ}z^{ℓ}, respectively ω = ( ∑_{ℓ=m}^{∞} c_{ℓ}z^{ℓ})dz,  c_{m} ≠ 0, m ∈ ℤ.

The number m is independent of the chart chosen and so we can define ord_{p}f = m respectively ord_{p}ω = m. The number m is called the order of f, respectively ω, at p. We will also need the residue res_{p}ω = c_{−1}, which is well-defined and satisfies

(A.2)  ∑_{p∈M} res_{p}ω = 0.

The above sum is finite since ω can only have finitely many poles (M is compact!). Next, we define the branch number b_{f}(p) = ord_{p}(f − f(p)) − 1 if f(p) ≠ ∞ and b_{f}(p) = −ord_{p}(f) − 1 if f(p) = ∞. There is a connection between the total branching number B = ∑_{p∈M} b_{f}(p) (the sum is again finite), the degree n of f

(A.3)  n = degree(f) = ∑_{p∈f^{−1}(c)}(1 + b_{f}(p)),  c ∈ ℂ ∪ {∞}

(we will show that n is independent of c), and the genus g, the Riemann-Hurwitz relation

(A.4)  2g − 2 = B − 2n.

We omit the proof which is not difficult but involves a lot of definitions which are of no further use for our purpose. (Triangulate ℂ ∪ {∞} such that each pole and each zero of f is a vertex. Lift the triangulation to M via f and calculate the Euler characteristic …)

A.2. Abelian differentials

Now consider two closed differentials θ, φ on M and let us try to compute

(A.5)  ∫∫_{M} θ ∧ φ.

We define

(A.6)  Â(p) = ∫_{p_{0}}^{p} φ̂ p_{0}, p ∈ M̂,

where the hat indicates that we regard φ as a differential on to M̂. Â is welldefined since M̂ is simply connected and dφ̂ = 0 (of course we require the path of integration to lie in M̂). Using Stokes theorem we obtain

∫∫_{M} θ ∧ φ = ∫∫M̂ θ ∧ dÂ = − ∫∫M̂ d(θÂ) = − ∫_{∂M̂} θÂ

(A.7)  = − ∑_{ℓ=1}^{g} (∫_{a_{ℓ}} θÂ + ∫_{b_{ℓ}} θÂ + ∫_{a_{ℓ}^{−1}} θÂ + ∫_{b_{ℓ}^{−1}} θÂ) .

Now, what can we say about Â(p) and Â(p̃) if p, p̃ are corresponding points on a_{ℓ}, a_{ℓ}^{−1}, respectively? If γ, γ̃ are paths (inside M̂) from p_{0} to p, p̃, respectively, we have the following situation:

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Thus we have Â(p̃) = Â(p) + ∫_{b_{ℓ}} φ (the parts on a_{ℓ} and a_{ℓ}^{−1} cancel). A similar argument for corresponding points p, p̃ on b_{ℓ}, b_{ℓ}^{−1} yields Â(p̃) = Â(p) − ∫_{a_{ℓ}} φ and we finally get

(A.8)  ∫∫_{M} θ ∧ φ = ∑_{ℓ=1}^{g} (∫_{a_{ℓ}} θ ∫_{b_{ℓ}} φ − ∫_{a_{ℓ}} φ ∫_{b_{ℓ}} θ) .

We will mainly be interested in meromorphic differentials ω ∈ M^{1}(M), also called abelian differentials. We call ω of the first kind if ω is holomorphic, of the second kind if all residues vanish, and of the third kind else.

Unfortunately, formula (A.8) makes only sense in the special case of holomorphic differentials. Nevertheless there is one thing we can do. Let ζ ∈ H^{1}(M), ω ∈ M^{1}(M) and assume that ω has no poles on the boundary – which can always be achieved by slightly (continuously) deforming the (representatives of the) homology basis. The residue theorem now yields

(A.9)  2πi ∑_{p∈M} res_{p}Âω = ∫_{∂M̂} Âω = ∑_{ℓ=1}^{g} (∫_{a_{ℓ}} ζ ∫_{b_{ℓ}} ω − ∫_{a_{ℓ}} ω ∫_{b_{ℓ}} ζ) .

For ζ ∈ H^{1}(M) we define
(A.10)  ∥ζ∥^{2} = i ∫∫_{M} ζ ∧ \overline{ζ} = i ∑_{ℓ=1}^{g} (∫_{a_{ℓ}} ζ ∫_{b_{ℓ}} \overline{ζ} − ∫_{a_{ℓ}} \overline{ζ} ∫_{b_{ℓ}} ζ) ,

(the bar denotes complex conjugation) which is indeed a norm (in local coordinates we get: iζ ∧ \overline{ζ} = 2|η|^{2}dx ∧ dy with ζ = η dz, z = x + iy).

Therefore we conclude that ζ ≡ 0 if and only if (e.g.) all the a-periods are zero and further that the map from the space of holomorphic differentials H^{1}(M) (which is g-dimensional) to the a-periods (≅ ℂ^{g}) is an isomorphism. Thus we may choose a basis {ζ_{j}}_{j=1}^{g} for H^{1}(M) fulfilling

(A.11)  ∫_{a_{k}} ζ_{j} = δ_{jk},  j, k = 1, …, g.

For the b-periods we get

(A.12)  ∫_{b_{k}} ζ_{j} = τ_{j,k},  j, k = 1, …, g,

with τ_{j,k} ∈ ℂ. Now (A.9) implies

(A.13)  τ_{j,k} = τ_{k,j}

and (for ζ = ∑_{ℓ=1}^{g} c_{ℓ}ζ_{ℓ} ∈ H^{1}(M)\{0} with c_{ℓ} ∈ ℂ)

0 < ∥ζ∥^{2} = i ∑_{j,k} c_{j}\overline{c_{k}} ∑_{ℓ} (∫_{a_{ℓ}} ζ_{j} ∫_{b_{ℓ}} \overline{ζ_{k}} − ∫_{b_{ℓ}} ζ_{j} ∫_{a_{ℓ}} \overline{ζ_{k}})
(A.14)  = 2 ∑_{j,k} c_{j}\overline{c_{k}}Im(τ_{j,k}).

Especially for c_{ℓ} = δ_{ℓj} we have Im(τ_{j,j}) > 0. Thus \underline{τ} is symmetric with positive definite imaginary part.

We will now ensure the existence of meromorphic differentials (without proof, cf. [81], II.5.3):

Theorem A.1. For every set {(p_{j}, (c_{j,ℓ})_{1≤ℓ≤n_{j}})}_{1≤j≤m} with distinct p_{j} ∈ M, c_{j,ℓ} ∈ ℂ, and m, n_{j} ∈ ℕ we can find a meromorphic differential ω with poles of the form

(A.15)  ω = (∑_{ℓ=1}^{n_{j}} \frac{c_{j,ℓ}}{z_{j}^{ℓ}} + O(1))dz_{j},

near each p_{j} in a given chart (U_{j}, z_{j}) centered at p_{j}, 1 ≤ j ≤ m, and is holomorphic elsewhere, if and only if

(A.16)  ∑_{p∈M} res_{p}ω = ∑_{j=1}^{m} c_{j,1} = 0.

Hence, for a meromorphic differential we can prescribe the poles and the Laurent principal parts at each pole with the only restriction, that the sum of the residues has to be zero. Two differentials constructed in that way may only differ by a holomorphic differential. From now on we will assume that no poles lie on the boundary – otherwise just deform the homology basis – so that we can make it unique by the following normalization

(A.17)  ∫_{a_{ℓ}} ω = 0,  ℓ = 1, …, g.

We are interested in some special cases. First let ω_{p,n} be holomorphic in M\{p} and let (U, z) be a chart centered at p

(A.18) ω_{p,n} = (\frac{1}{z^{2+n}} + O(1))dz,  ∫_{a_{ℓ}} ω_{p,n} = 0, ℓ = 1, …, g, n ∈ ℕ_{0}.

This definition is obviously depending on the chart around p. The b-periods follow easily from (A.9) with ζ = ζ_{ℓ} and ω = ω_{p,n}

(A.19)  ∫_{b_{ℓ}} ω_{p,n} = 2πi ∑_{p̃∈M} resp̃Âω_{p,n} = \frac{2πi}{(n + 1)!} (\frac{d}{dz} )^{n}η_{ℓ}(z)|_{z=0},

where ζ_{ℓ} = η_{ℓ}(z)dz. Second suppose ω_{pq} is holomorphic in M\{p, q} (with p ≠ q) and let

(A.20)  ord_{p}ω_{pq} = −1 ord_{q}ω_{pq} = −1 res_{p}ω_{pq} = +1 res_{q}ω_{pq} = −1 ,  ∫_{a_{ℓ}} ω_{pq} = 0,  ℓ = 1, …, g.

This definition depends on the specific homology basis chosen. Because if we deform, for example, a_{ℓ} we may cross a pole, thus changing the value of ∫_{a_{ℓ}} ω_{pq}. The b-periods again follow from (A.9) with ζ = ζ_{ℓ} and ω = ω_{pq}

(A.21)  ∫_{b_{ℓ}} ω_{pq} = 2πi ∑_{p̃∈M} resp̃Âω_{pq} = 2πi(Â(p) − Â(q)) = 2πi ∫_{q}^{p} ζ̂_{ℓ}.

Recall that the path of integration in the last integral lies in M̂.

A.3. Divisors and the Riemann-Roch theorem

This section is also valid for g = 0. A divisor D is a map

(A.22)  \begin{aligned}D : & M & → & ℤ \\ & p & ↦ & D(p) \end{aligned}

being nonzero only for a finite number of points in M. The set of all divisors Div(M) becomes a module over the ring ℤ by defining each operation in the natural way (i.e., pointwise). The point divisors D_{p}, p ∈ M,

(A.23)  D_{p}(p) = 1,  D_{p}(q) = 0,  q ∈ M\{p}

form a basis for Div(M). We can also establish a group homomorphism by assigning each divisor a degree

(A.24)

deg : Div(M) → ℤ
D  ↦ deg(D) = ∑_{p∈M} D(p)

(the sum is finite and thus well-defined). We associate a divisor with every nonzero meromorphic function and with every nonzero meromorphic differential on M (A.25)

\begin{aligned}( ) : & M(M)\\{0\} & → & Div(M) \\ & f & ↦ & (f) \end{aligned} , \begin{aligned}( ) : & M^{1}(M)\\{0\} & → & Div(M) \\ & ω & ↦ & (ω) \end{aligned} ,

where (f)(p) = ord_{p}f and (ω)(p) = ord_{p}ω.

It is easy to verify the following formulas

(A.26)  (f_{1} f_{2}) = (f_{1}) + (f_{2}), (\frac{1}{f_{1}} ) = −(f_{1}), f_{1}, f_{2} ∈ M(M)\\{0\},

implying (f_{1}) = (f_{2}) ⇔ (f_{1}/f_{2}) = 0 ⇔ f_{1} = c f_{2}, c ∈ ℂ since the only holomorphic functions on a compact Riemann surface are constants. Similarly,

(A.27)  (f ω_{1}) = (f) + (ω_{1}), (\frac{ω_{1}}{ω_{2}} ) = (ω_{1}) − (ω_{2}), ω_{1}, ω_{2} ∈ M^{1}(M)\\{0\}

implying (ω_{1}) = (ω_{2}) ⇔ (ω_{1}/ω_{2}) = 0 ⇔ ω_{1} = c ω_{2}, c ∈ ℂ.

We claim

deg((f)) = 0,  f ∈ M(M)\{0},
(A.28)  deg((ω)) = 2g − 2,  ω ∈ M^{1}(M)\{0}.

To obtain the first equation observe that deg((f)) = ∑_{p∈M} res_{p}df/f = 0. This implies that f attains every value exactly n times for a certain integer n called the degree of f (counting multiplicities). In fact, if 0 is attained n times the same is true for ∞ (otherwise deg((f)) would not be zero). Now consider the function f −c with c ∈ ℂ. It attains ∞ as often as f and thus zero as often as f.

For the second equation observe deg((ω_{1})) = deg((ω_{2})) which follows from ω_{1}/ω_{2} ∈ M(M)\{0}. Now let f be a meromorphic function of degree n and let us calculate (df/f). At points p with f(p) ∉ {0, ∞}, ord_{p}df/f is just the branch number b_{f}(p) and at each other point it is −1. The desired result now follows from the Riemann-Hurwitz relation (A.4) and

deg((\frac{df}{f})) = ∑_{p∈M} b_{f}(p) − ∑_{p∈f^{−1}(0)}(1 + b_{f}(p)) − ∑_{p∈f^{−1}(∞)}(1 + b_{f}(p))
(A.29)  = B − 2n.

Divisors corresponding to meromorphic functions D = (f) are called principal, divisors corresponding to meromorphic differentials D = (ω) canonical. It follows from the above considerations that the principal divisors form a subgroup. It is clearly normal (since all groups in sight are abelian) and we may consider its factor group induced by the following equivalence relation

(A.30)  D_{1} ≃ D_{2} ⇔ D_{1} − D_{2} = (f).

The factor group is called divisor class group and the equivalence classes [D] divisor classes. Observe, that all canonical divisors lie in the same divisor class (by (A.27)). The subset formed by the divisors of degree zero Div_{0}(M) is a subgroup of Div(M) and it’s factor group is called Picard group Pic(M).

The set Div(M) can be partially ordered by defining

(A.31)  D_{1} ≥ D_{2} ⇔ D_{1}(p) ≥ D_{2}(p), p ∈ M.

We write D_{1} > D_{2} if in addition D_{1} ≠ D_{2}. Note: D_{1} ≥ D_{2} ⇒ deg(D_{1}) ≥ deg(D_{2}) and D_{1} > D_{2} ⇒ deg(D_{1}) > deg(D_{2}).

A divisor is called positive if D ≥ 0 and strictly positive if D > 0. Two divisors D_{1}, D_{2} are called relatively prime if D_{1}(p)D_{2}(p) = 0, p ∈ M. A divisor can always be uniquely written as the difference of two relatively prime positive divisors

(A.32)  D = D^{+} − D^{−},  D^{+} ≥ 0, D^{−} ≥ 0.

For a meromorphic function (f)^{+} and (f)^{−} are called the divisor of zeros and the divisor of poles of f, respectively.

For any divisor D we now define

L(D) = {f ∈ M(M)|f = 0 or (f) ≥ D},
(A.33)  L^{1}(D) = {ω ∈ M^{1}(M)|ω = 0 or (ω) ≥ D}.

Hence L(D) is the set of all meromorphic functions, which are holomorphic at points p ∈ M with D(p) ≥ 0, have zeros of order at least D(p) at points p with D(p) > 0 and poles of order at most −D(p) at points p with D(p) < 0. Similarly for L^{1}(D). Obviously we have

(A.34)  D_{1} ≥ D_{2} ⇒ \{ \begin{matrix}L(D_{1}) ⊂ L(D_{2}) \\ L^{1}(D_{1}) ⊂ L^{1}(D_{2}) \end{matrix}

and L(D), L^{1}(D) inherit the vector space structure from M(M), M^{1}(M), respectively. What can we say about their dimensions

(A.35)  r(D) = dim L(D),  i(D) = dim L^{1}(D).

i(D) is called index of specialty of D and D is called nonspecial if i(D) = 0. First of all we have

(A.36)  r(0) = 1,  i(0) = g,

since a holomorphic function on a compact Riemann surface must be constant and since H^{1}(M) is g-dimensional.

If we take two divisors in the same divisor class, we can easily establish an isomorphism

g ∈ L(D) ⇒ f g ∈ L(D + (f)),
(A.37)  g ∈ L(D + (f)) ⇒ f^{−1}g ∈ L(D),

saying

(A.38)  L(D) ≅ L(D + (f)),  r(D) = r(D + (f)).

Proceeding in a similar manner, we also obtain

L^{1}(D) ≅ L^{1}(D + (f)),  i(D) = i(D + (f)),
(A.39)  L^{1}(D) ≅ L(D − (ω)),  i(D) = r(D − (ω)).

If deg(D) > 0 it follows from (f) ≥ D ⇒ deg((f)) ≥ deg(D) > 0, that r(D) has to be zero. And using (A.39) and (A.28) we get

deg(D) > 0 ⇒ r(D) = 0,
(A.40)  deg(D) > 2g − 2 ⇒ i(D) = 0.

If deg(D) = 0 and f ∈ L(D) it follows (f) = D (since (f) > D is not possible by the same argument as above) and thus

deg(D) = 0 ⇔ \{ \begin{matrix}L(D) = span\{f\} & if D = (f) is principal \\ L(D) = \{0\} & if D is not principal \end{matrix} ,
(A.41) deg(D) = 2g − 2 ⇔ \{ \begin{matrix}L^{1}(D) = span\{ω\} & if D = (ω) is canonical \\ L^{1}(D) = \{0\} & if D is not canonical \end{matrix} .

Next, let us assume r(−D) ≤ 1 + deg(D) for all D with deg(D) ≥ 0 (we already know it for deg(D) = 0). If we subtract a point divisor we get

(A.42)  r(−D) ≤ r(−D − D_{p}) ≤ 1 + r(−D)

by (A.34) and because if f is new in our space, there is for every g ∈ L(−D − D_{p}) a c ∈ ℂ with g − c f ∈ L(D). Hence using induction we infer

r(−D) ≤ 1 + deg(D), deg(D) ≥ 0,

i(−D) ≤ 2g − 1 + deg(D), deg(D) ≥ 2 − 2g. (A.43)

By virtue of Theorem A.1 and (A.36) we have

(A.44)  i(D) = g − 1 − deg(D),  D < 0.

If we add a point divisor D_{p} to D, we obtain from (A.39) and (A.42)

(A.45)  i(D) − 1 ≤ i(D + D_{p}) ≤ i(D).

Thus, using induction we have

(A.46)  i(D) ≥ g − 1 − deg(D),

which is by (A.39) and (A.28) equivalent to (Riemann inequality)

(A.47)  r(−D) ≥ 1 − g + deg(D).

Now we will prove the famous Riemann-Roch Theorem:

Theorem A.2. (Riemann-Roch) Let D ∈ Div(M), then we have

(A.48)  r(−D) = deg(D) + 1 − g + i(D).

Proof. By (A.44) we know that it is true for D ≤ 0. So what happens if we add a point divisor to D? We already know (A.42) and (A.45) but can we have i(D + D_{p}) = i(D) − 1 and r(−D − D_{p}) = 1 + r(−D)? If this were the case, we would have a function f ∈ L(D + D_{p}) with a pole of order (D(p) + 1) at p and a differential ω ∈ L^{1}(D) with a zero of order D(p) at p. Now consider the differential f ω. It would have a pole of order 1 at p and would be holomorphic elsewhere, (f ω) = (f) + (ω) ≥ −D − D_{p} + D = −D_{p}, contradicting our assumption. Thus we have by induction

(A.49)  r(−D) ≤ deg(D) + 1 − g + i(D).

Using (A.39) and (A.28):

i(D) = r(D − (ω)) ≤ deg((ω) − D) + 1 − g + i((ω) − D)
(A.50)  = −deg(D) + g − 1 + r(−D)

completes the proof of the Riemann-Roch theorem. □

Corollary A.3. A Riemann surface of genus 0 is isomorphic to the Riemann sphere (≅ ℂ ∪ {∞}).

Proof. Let p ∈ M. Since r(D_{p}) = 2, there exists a (non constant) meromorphic function with a single pole. It is injective (its degree is 1) and surjective (since every non constant holomorphic map between compact Riemann surfaces is). Hence it provides the desired isomorphism. □

Corollary A.4. (g ≥ 1) There are no meromorphic functions with a single pole of order 1 on M or equivalently, there is no point in M where all ζ ∈ H^{1}(M) vanish.

Proof. If f were such a function with pole at p we would have r(−nD_{p}) = n + 1 for all n ∈ ℕ_{0} (consider f^{n}) and from Riemann-Roch i(nD_{p}) = g. Hence all holomorphic differentials would vanish of arbitrary order at p which is impossible. Thus r(−D_{p}) = 1 for all p ∈ M or equivalently i(D_{p}) = g − 1, p ∈ M. □

If we have a sequence of points {p_{ℓ}}_{ℓ∈ℕ} ⊂ M and define D_{0} = 0 and D_{n} = D_{n−1} + D_{p_{n}} for n ∈ ℕ, then there are exactly g (g ≥ 1) integers {n_{ℓ}}_{ℓ=1}^{g} such that i(D_{n_{ℓ}}) = i(D_{n_{ℓ−1}}) − 1 (have a look at the proof of Riemann-Roch and note i(D_{0}) = g) or equivalently, there exists no meromorphic function f with (f)^{−} = D_{n_{ℓ}}. Moreover,

(A.51)  1 = n_{1} < ⋯ < n_{g} < 2g.

(The first equality follows from our last corollary and the second from (A.41).) The numbers {n_{ℓ}}_{ℓ=1}^{g} are called Noether gaps and in the special case where p_{1} = p_{2} = … Weierstrass gaps.

A surface admitting a function of lowest possible degree (i.e., 2 for g ≥ 1) is called hyperelliptic. Note that if g ≤ 2, M is always hyperelliptic (g = 0, 1 is obvious, for g = 2 choose 0 ≠ ω ∈ L^{1}(0) and calculate r((ω)) = 2 + 1 − 2 + 1 = 2).

A point p ∈ M with i(gD_{p}) > 0 is called a Weierstrass point (there are no Weierstrass points for g ≤ 1). We mention (without proof) that the number of Weierstrass points W is finite and (for g ≥ 2) satisfies 2g + 2 ≤ W ≤ g^{3} − g. The first inequality is attained if and only if at each Weierstrass point the gap sequence is 1, 3, …, 2g − 1 (which is the case if and only if M is hyperelliptic).

A.4. Jacobian variety and Abel’s map

We start with the following (discrete) subset of ℂ^{g}

(A.52)  L(M) = \{\underline{m} + \underline{τ} \underline{n} | \underline{m}, \underline{n} ∈ ℤ^{g}\} ⊂ ℂ^{g}

and define the Jacobian variety of M

(A.53)  J(M) = ℂ^{g}/L(M).

It is a compact, commutative, g-dimensional, complex Lie group. Further we define Abel’s map (with base point p_{0}):

(A.54)  \begin{aligned}\underline{A}_{p_{0}} : & M & → & J(M) \\ & p & ↦ & [∫_{p_{0}}^{p} \underline{ζ}] = [(∫_{p_{0}}^{p} ζ_{1}, …, ∫_{p_{0}}^{p} ζ_{g})] \end{aligned} ,

where [\underline{z}] ∈ J(M) denotes the equivalence class of \underline{z} ∈ ℂ^{g}. It is clearly holomorphic and well-defined as long as we choose the same path of integration for all ζ_{ℓ} ([γ] − [γ′] = \underline{m}[\underline{a}] + \underline{n}[\underline{b}]) since

(A.55)  ∫_{γ} \underline{ζ} − ∫_{γ′} \underline{ζ} = \underline{m} + \underline{τ} \underline{n} ∈ L(M),  \underline{m}, \underline{n} ∈ ℤ^{g}.

Observe that d\underline{A}_{p_{0}} is of maximal rank (i.e., 1) by Corollary A.4 (compare also (A.86)) and that a change of base point results in a global shift of the image. Note: \underline{A}_{p_{0}}(p) = −\underline{A}_{p}(p_{0}).

We will also need a slight modification of \underline{A}_{p_{0}}

(A.56)  \begin{aligned}\underline{Â}_{p_{0}} : & M̂ & → & ℂ^{g} \\ & p & ↦ & ∫_{p_{0}}^{p} ˆ \underline{ζ} = (∫_{p_{0}}^{p} ζ̂_{1}, …, ∫_{p_{0}}^{p} ζ̂_{g}) \end{aligned} ,

with p_{0} ∈ M̂ and the path of integration lying in M̂. We can easily extend Abel’s map to the set of divisors Div(M)

(A.57)  \begin{aligned}\underline{α}_{p_{0}} : & Div(M) & → & J(M) \\ & D & ↦ & ∑_{p∈M} D(p)\underline{A}_{p_{0}}(p) \end{aligned} .

The sum is to be understood in J(M) and multiplication with integers is also welldefined in J(M). A similar extension \underline{α̂}_{p_{0}} is defined for \underline{Â}_{p_{0}}. The natural domain for \underline{α̂}_{p_{0}} would be Div(M̂), if we want to define it on Div(M) we have to make it unique on the boundary. But we will avoid this problem by requiring that divisors in Div(M̂) vanish on the boundary. We can now state Abel’s theorem.

Theorem A.5. (Abel) D ∈ Div(M) is principal if and only if

(A.58)  \underline{α}_{p_{0}}(D) = [0]  and  deg(D) = 0.

Note that a change of the base point p_{0} to p_{1} amounts to adding a constant \underline{A}_{p_{1}}(p) = \underline{A}_{p_{0}}(p) − \underline{A}_{p_{0}}(p_{1}), which yields \underline{α}_{p_{1}}(D) = \underline{α}_{p_{0}}(D) − deg(D) \underline{A}_{p_{0}}(p_{1}). Thus \underline{α}_{p_{0}} is independent of p_{0} for divisors with deg(D) = 0.

Proof. Throughout this whole proof we choose p_{0} ∉ ∂M̂ such that D(p_{0}) = 0. Moreover, we assume that D(p) = 0, p ∈ ∂M̂ (otherwise deform {a_{ℓ}, b_{ℓ}}_{1≤ℓ≤g} slightly).

Let D = (f) be the divisor of the meromorphic function f. Since df/f has only simple poles with residues res_{p}df/f = ord_{p}f ∈ ℤ, we can use (A.20) to write

(A.59)  \frac{df}{f} = ∑_{p∈M} D(p)ω_{pp_{0}} + ∑_{ℓ=1}^{g} c_{ℓ}ζ_{ℓ}.

Note that the poles at p_{0} cancel. Using that the a- and b-periods must be integer multiples of 2πi by the residue theorem and (A.21) we infer

∫_{a_{j}} \frac{df}{f} = c_{j} = 2πim_{j},

∫_{b_{j}} \frac{df}{f} = 2πi ∑_{p∈M} D(p) ∫_{p_{0}}^{p} ζ_{j} + ∑_{ℓ=1}^{g} c_{ℓ}τ_{ℓ,j} = 2πiα̂_{p_{0},j}(D) + ∑_{ℓ=1}^{g} c_{ℓ}τ_{ℓ,j}
(A.60)  = 2πi n_{j},

with m_{j}, n_{j} ∈ ℤ. So we finally end up with

(A.61)  \underline{α}_{p_{0}}(D) = [\underline{α̂}_{p_{0}}(D)] = [\underline{n} − \underline{τ} \underline{m}] = [0].

To prove the converse let D be given and consider (motivated by (A.59))

(A.62) f(q) = exp ( ∑_{p∈M} D(p) ∫_{q_{0}}^{q} ω_{pp_{0}} + ∑_{ℓ=1}^{g} c̃_{ℓ} ∫_{q_{0}}^{q} ζ_{ℓ}) ,  c̃_{ℓ} ∈ ℂ, q ∈ M̂.

where p_{0} ≠ q_{0} ∉ ∂M̂ and D(q_{0}) = 0. The constants c̃_{ℓ} are to be determined. The function f is a well-defined meromorphic function on M̂ since a change of path (within M̂) in the first integral amounts to a factor 2πi n (n ∈ ℤ) which is swallowed by the exponential function. Clearly we have (f) = D. For f to be a meromorphic function on M its value has to be independent of the path (in M) chosen

∑_{p∈M} D(p) ∫_{a_{j}} ω_{pp_{0}} + ∑_{ℓ=1}^{g} c̃_{ℓ} ∫_{a_{j}} ζ_{ℓ} = c̃_{j},

(A.63)  ∑_{p∈M} D(p) ∫_{b_{j}} ω_{pp_{0}} + ∑_{ℓ=1}^{g} c̃_{ℓ} ∫_{b_{j}} ζ_{ℓ} = 2πiα̂_{p_{0},j}(D) + ∑_{ℓ=1}^{g} c̃_{ℓ}ω_{ℓ,j}.

Thus if \underline{α̂}_{p_{0}}(D) = \underline{m} + \underline{τ} \underline{n}, we have to choose c̃_{ℓ} = −2πin_{ℓ} ∈ ℤ. □

Corollary A.6. The Picard group is isomorphic to the Jacobian variety

(A.64)  Pic(M) ≅ J(M).

Proof. The map \underline{α}_{p_{0}} provides an isomorphism since it is injective by Abel’s theorem and surjective by Corollary A.15 below. □

Corollary A.7. \underline{A}_{p_{0}} is injective.

Proof. Let p, q ∈ M with \underline{A}_{p_{0}}(p) = \underline{A}_{p_{0}}(q). By Abel’s theorem D_{p} − D_{q} would be principal which is impossible because of Corollary A.4. □

Thus \underline{A}_{p_{0}} is an embedding of M into J(M). For g = 1 it is even an isomorphism (it is surjective since every non constant map between compact Riemann surfaces is). For g ≥ 1 we mention (without proof) Torelli’s theorem:

Theorem A.8. (Torelli) Two Riemann surfaces M and M′ are isomorphic if and only if J(M) and J(M′) are,

(A.65)  M ≅ M′  ⇔  J(M) ≅ J(M′).

A.5. Riemann’s theta function

We first define the Riemann theta function associated with M

(A.66)  \begin{aligned}θ : & ℂ^{g} & → & ℂ \\ & \underline{z} & ↦ & ∑_{\underline{m}∈ℤ^{g}} exp 2πi (⟨\underline{m}, \underline{z}⟩ + \frac{⟨\underline{m}, \underline{τ} \underline{m}⟩}{2}) \end{aligned} ,

where ⟨\underline{z}, \underline{z}′⟩ = ∑_{ℓ=1}^{g} \overline{z_{ℓ}}z′_{ℓ}. It is holomorphic since the sum converges nicely due to (A.14). We will now show some simple properties of θ(\underline{z}). First observe that

(A.67)  θ(−\underline{z}) = θ(\underline{z})

and second let \underline{n}, \underline{n}′ ∈ ℤ^{g}, then

θ(\underline{z} + \underline{n} + \underline{τ} \underline{n}′) = ∑_{\underline{m}∈ℤ^{g}} exp 2πi (⟨\underline{m}, \underline{z} + \underline{n} + \underline{τ} \underline{n}′⟩ + \frac{⟨\underline{m}, \underline{τ} \underline{m}⟩}{2})

= ∑_{\underline{m}∈ℤ^{g}} exp 2πi(⟨\underline{m} + \underline{n}′, \underline{z}⟩ − ⟨\underline{n}′, \underline{z}⟩ +

+\frac{⟨\underline{m} + \underline{n}′, \underline{τ}(\underline{m} + \underline{n}′)⟩ − ⟨\underline{n}′, \underline{τ} \underline{n}′⟩}{2})

(A.68)  = exp 2πi (−⟨\underline{n}′, \underline{z}⟩ − \frac{⟨\underline{n}′, \underline{τ} \underline{n}′⟩}{2}) θ(\underline{z}).

(One has to use ⟨\underline{m}, \underline{τ} \underline{n}′⟩ = ⟨\underline{n}′, \underline{τ} \underline{m}⟩ which follows from symmetry of \underline{τ}.) As special cases we mention

θ(\underline{z} + δ_{j}) = θ(\underline{z}),
(A.69)  θ(\underline{z} + τ_{j}) = e2πi(−z^{j} − τ^{j,j}/2)θ(\underline{z}).

Here τ_{j} denotes the j’th row of \underline{τ}. Notice that if we choose \underline{z} = −(\underline{n} + \underline{τ} \underline{n}′)/2, we get θ(−\underline{z}) = exp(iπ⟨\underline{n}, \underline{n}′⟩)θ(\underline{z}), implying that θ(−(\underline{n} + \underline{τ} \underline{n}′)/2) = 0 if ⟨\underline{n}, \underline{n}′⟩ is odd.

Let \underline{z} ∈ ℂ^{g}. We will study the (holomorphic) function F̂

(A.70)  \begin{aligned}F̂ : & M̂ & → & ℂ \\ & p & ↦ & θ(\underline{Â}_{p_{0}}(p) − \underline{z}) \end{aligned} .

If F̂ is not identically zero, we can compute the number of zeros in the following manner

\frac{1}{2πi} ∫_{∂M̂} \frac{dF̂}{F̂} = \frac{1}{2πi} ∑_{ℓ=1}^{g} ( ∫_{a_{ℓ}} \frac{dF̂}{F̂} + ∫_{b_{ℓ}} \frac{dF̂}{F̂}

+ ∫_{a_{ℓ}^{−1}} \frac{dF̂}{F̂} + ∫_{b_{ℓ}^{−1}} \frac{dF̂}{F̂} ). (A.71)

Here and in what follows we will always assume that no zero of F̂ ≢ 0 lies on the boundary – otherwise deform the representatives of the homology basis. From (A.69) together with a similar argument as in (A.8) we get

(A.72) Â_{p_{0},j}(p̃) = Â_{p_{0},j}(p) + τ_{ℓ,j} ⇒ F̂(p̃) = e2πi(z^{ℓ} − Â_{p_{0},ℓ}(p) − τ^{ℓ,ℓ}/2)F̂(p)

A.5. Riemann’s theta function 287 if p, p̃ are corresponding points on a_{ℓ}, a_{ℓ}^{−1} and

(A.73) Â_{p_{0},j}(p̃) = Â_{p_{0},j}(p) − δ_{ℓj} ⇒ F̂(p̃) = F̂(p)

if p, p̃ are corresponding points on b_{ℓ}, b_{ℓ}^{−1}. Using this we see

(A.74)  \frac{1}{2πi} ∫_{∂M̂} \frac{dF̂}{F̂} = ∑_{ℓ=1}^{g} ∫_{a_{ℓ}} dÂ_{p_{0},ℓ} = ∑_{ℓ=1}^{g} ∫_{a_{ℓ}} ζ_{ℓ} = g.

Next, let us compute \underline{α̂}_{p_{0}}((F̂)) (Since we have assumed that no zero lies on the boundary we may write (F̂) ∈ Div(M).)

\underline{α̂}_{p_{0}}((F̂)) = \frac{1}{2πi} ∫_{∂M̂} \underline{Â}_{p_{0}} \frac{dF̂}{F̂}
= \frac{1}{2πi} ∑_{ℓ=1}^{g} (∫_{a_{ℓ}} \underline{Â}_{p_{0}} \frac{dF̂}{F̂} + ∫_{b_{ℓ}} \underline{Â}_{p_{0}} \frac{dF̂}{F̂}

+ ∫_{a_{ℓ}^{−1}} \underline{Â}_{p_{0}} \frac{dF̂}{F̂} + ∫_{b_{ℓ}^{−1}} \underline{Â}_{p_{0}} \frac{dF̂}{F̂} ) . (A.75)

Proceeding as in the last integral (using (A.72), (A.73)) yields

(A.76) α̂_{p_{0},j}((F̂)) = \frac{1}{2πi} ∑_{ℓ=1}^{g} (∫_{a_{ℓ}} (2πi(Â_{p_{0},j} + τ_{ℓ,j})ζ_{ℓ} − τ_{ℓ,j} \frac{dF̂}{F̂} ) + δ_{ℓj} ∫_{b_{ℓ}} \frac{dF̂}{F̂} ) .

Hence we have to perform integrals of the type ∫ p̃ ; p d ln(F̂), which is just ln(F̂(p̃)) − ln(F̂(p)). Of course we do not know which branch we have to choose, so we have to add a proper multiple of 2πi. Computing F̂(p), F̂(p̃) as in (A.8) we get

\frac{1}{2πi} ∑_{ℓ=1}^{g} (τ_{ℓ,j} ∫_{a_{ℓ}} \frac{dF̂}{F̂} + δ_{ℓj} ∫_{b_{ℓ}} \frac{dF̂}{F̂} ) =

− Â_{p_{0},j}(p) + z_{j} − \frac{τ_{j,j}}{2} + m_{j} + ∑_{ℓ=1}^{g} τ_{ℓ,j}n_{ℓ}, (A.77)

with \underline{m}, \underline{n} ∈ ℤ^{g} and p is the point in M̂ that a_{ℓ} and b_{ℓ} have in common – we have omitted the dependence of p on ℓ since Â_{p_{0},j}(p) is independent of ℓ. As the last (unknown) factor is in L(M), we can get rid of it by changing to J(M)

(A.78) α_{p_{0},j}((F̂)) = [α̂_{p_{0},j}((F̂))] = [−Â_{p_{0},j}(p) + z_{j} − \frac{τ_{j,j}}{2} + ∑_{ℓ=1}^{g} ∫_{a_{ℓ}} Â_{p_{0},j}ζ_{ℓ}].

There is one more thing we can do,

∫_{a_{ℓ}} Â_{p_{0},ℓ}ζ_{ℓ} = \frac{1}{2}(Â_{p_{0},ℓ}^{2}(p) − Â_{p_{0},ℓ}^{2}(p̃))
(A.79)  = \frac{1}{2}(Â_{p_{0},ℓ}^{2}(p) − (Â_{p_{0},ℓ}(p) − 1)^{2}) = Â_{p_{0},ℓ}(p) − \frac{1}{2}.

(Compare the picture on page 277.) Thus our final result is

(A.80)  \underline{α}_{p_{0}}((F̂)) = [\underline{z}] − \underline{Ξ}_{p_{0}},

with

(A.81)  Ξ_{p_{0},j} = [\frac{1 + τ_{j,j}}{2} − ∑_{ℓ≠j} ∫_{a_{ℓ}} Â_{p_{0},j}ζ_{ℓ}]

the vector of Riemann constants. Finally, we mention that evaluating the integral

(A.82)  \frac{1}{2πi} ∫_{∂M̂} f \frac{dF̂}{F̂} ,

where f ∈ M(M) is an arbitrary meromorphic function yields the useful formula

(A.83)  ∑_{ℓ=1}^{g} ∫_{a_{ℓ}} f ζ_{ℓ} = ∑_{ℓ=1}^{g} f(p_{j}) + ∑_{p∈f^{−1}(∞)} res_{p}(f d ln F̂),

where {p_{j}}_{j=1}^{g} are the zeros of the function F̂ which (for simplicity) are away from the poles of f.

A.6. The zeros of the Riemann theta function

We begin with some preparations. Let M_{n} be the set of positive divisors of degree n ∈ ℕ_{0}. (The case n = 0 is of course trivial, but it is practical to include it.) We can make M_{n} a complex (n-dimensional) manifold if we identify M_{n} with σ^{n}M the n-th symmetric power of M. If D_{n} ∈ M_{n} and if the points {p̃_{j}}_{1≤j≤n} with D_{n}(p) ≠ 0 are distinct (i.e., D_{n}(p) ≤ 1, p ∈ M), a local chart is given by

U = {[(p_{1}, …, p_{n})]|p_{j} ∈ U_{j}}, with U_{j} ∩ U_{k} = ∅ for j ≠ k

[(p_{1}, …, p_{n})] ↦ (z_{1}(p_{1}), …, z_{n}(p_{n})), with p_{j} ∈ U_{j}, (A.84)

where (U_{j}, z_{j}) are charts in M centered at p̃_{j}. If D(p) ≥ 2 for some points, one has to choose more sophisticated charts (elementary symmetric functions). Note, that \underline{α}_{p_{0}} : M_{n} → J(M) is a holomorphic mapping. If D_{n} ∈ M_{n} as above, one can easily compute

(A.85)  \{d\underline{α}_{p_{0}}(D_{n})\}_{jk} = ζ_{j}(p̃_{k}),  1 ≤ j ≤ g, 1 ≤ k ≤ n

(with some abuse of notation: ζ_{j}(p̃_{k}) means the value of η_{j}(z(p̃_{k})) if ζ_{j} = η_{j}(z)dz). What can we say about dim Ker(d\underline{α}_{p_{0}})? If ζ = ∑_{j=1}^{g} c_{j}ζ_{j} ∈ L^{1}(D_{n}) we have ζ(p̃_{k}) = ∑_{j=1}^{g} c_{j}ζ_{j}(p̃_{k}) = 0. But we can even reverse this argument, ending up with

(A.86)  i(D_{n}) = dim Ker(d\underline{α}_{p_{0}}|_{D_{n}}) = g − rank(d\underline{α}_{p_{0}}|_{D_{n}}).

(We remark that this formula remains true for arbitrary D_{n} ∈ M_{n}.)

Now we will prove a small (but useful) lemma:

Lemma A.9. Let D_{1}, D_{2} ∈ M_{g} with \underline{α}_{p_{0}}(D_{1}) = \underline{α}_{p_{0}}(D_{2}) and i(D_{2}) = 0, then D_{1} = D_{2}.

Proof. By Abel’s theorem D_{1} − D_{2} is principal and thus r(D_{1} − D_{2}) = 1. From

r(−D_{2}) = 1 + i(D_{2}) = 1 (Riemann-Roch) we conclude L(D_{1} − D_{2}) = L(−D_{2}) =^{(∼)} ℂ and therefrom D_{1} − D_{2} = 0. □

Next, we will show that i(D) = 0 is no serious restriction for D ∈ M_{g}.

Lemma A.10. Let D_{n} ∈ M_{n} (0 ≤ n ≤ g), then in every neighborhood of D_{n} there is a D′_{n} ∈ M_{n} with i(D′_{n}) = g − n. We may even require D′_{n}(p) ≤ 1, p ∈ M.

Proof. Our lemma is true for n = 1, so let us assume it is also true for some n − 1 ≥ 1. Let D_{n} = D_{n−1} + D_{p} and let {ω_{ℓ}}_{1≤ℓ≤g−n+1} be a basis for L^{1}(D′_{n−1}). We can always find a neighborhood U_{p} of p such that none of the basis elements vanishes on U_{p}\{p} - so just take D′_{n} = D′_{n−1} + Dp̃ with a suitable p̃ ∈ U_{p}\{p}. □

We have shown, that the divisors D′_{n} with i(D′_{n}) = g − n and D′_{n}(p) ≤ 1, p ∈ M, are dense in M_{n}. Hence \underline{α}_{p_{0}} : M_{g} → J(M) has maximal rank for such divisors and θ(\underline{α̂}_{p_{0}}(.) − \underline{z}) does not vanish identically on any open subset of M_{g}. We set W_{n}^{p_{0}} = \underline{α}_{p_{0}}(M_{n}) ⊂ J(M) and Ŵ_{n}^{p_{0}} = \underline{α̂}_{p_{0}}(M_{n}) ⊂ ℂ^{g}. (Our last statement now reads θ(Ŵ_{g}^{p_{0}} − \underline{z}) ≢ 0, where Ŵ_{g}^{p_{0}} − \underline{z} stands for w − \underline{z} with w ∈ Ŵ_{g}^{p_{0}}.) We observe W_{n}^{p_{0}} ⊂ W_{n+1}^{p_{0}} (just add D_{p_{0}} to each divisor in W_{n}^{p_{0}}) and Ŵ_{n}^{p_{0}} ⊂ Ŵ_{n+1}^{p_{0}}.

We will now characterize the set of zeros of θ(\underline{z}):

Theorem A.11. Let \underline{z} ∈ ℂ^{g}. Then θ(\underline{z}) = 0 if and only if [\underline{z}] ∈ W_{g−1}^{p_{0}} + \underline{Ξ}_{p_{0}}.

Note: Though θ is not well-defined on J(M) the set of zeros is a well-defined subset of J(M). Observe also that [\underline{z}] ∈ W_{g−1}^{p_{0}} + \underline{Ξ}_{p_{0}} is not dependent on the base point p_{0} chosen!

Proof. Choose D ∈ M_{g−1} and p̃ ∈ M such that i(D + Dp̃) = 0. Choose a \underline{z} ∈ ℂ^{g} with [\underline{z}] = \underline{α}_{p_{0}}(D + Dp̃) + \underline{Ξ}_{p_{0}} and consider (still assuming that F̂ has no zeros on the boundary)

(A.87)  F̂(p) = θ(\underline{Â}_{p_{0}}(p) − \underline{z}).

If F̂ is not identically zero, we have \underline{α}_{p_{0}}((F̂)) = [\underline{z}] − \underline{Ξ}_{p_{0}} = \underline{α}_{p_{0}}(D + Dp̃). Using Lemma A.9 we get (F̂) = D + Dp̃ and thus F̂(p̃) = 0. As this is trivially true if F̂ vanishes identically, we are ready with the first part because the divisors D under consideration are dense and because \underline{α̂}_{p_{0}} : M_{g−1} → ℂ^{g} is continuous.

Conversely, suppose θ(\underline{z}) = 0 and let s be the integer with θ(Ŵ_{s−1}^{p_{0}}−Ŵ_{s−1}^{p_{0}}−\underline{z}) ≡ 0 and θ(Ŵ_{s}^{p_{0}} − Ŵ_{s}^{p_{0}} − \underline{z}) ≢ 0. (Here Ŵ_{g}^{p_{0}} − Ŵ_{s}^{p_{0}} − \underline{z} stands for w_{1} − w_{2} − \underline{z} with w_{1}, w_{2} ∈ Ŵ_{g}^{p_{0}}.) We have 1 ≤ s ≤ g. Thus we may choose two divisors D_{s}^{1}, D_{s}^{2} ∈ M_{s} such that θ(\underline{α̂}_{p_{0}}(D_{s}^{1}) − \underline{α̂}_{p_{0}}(D_{s}^{2}) − \underline{z}) ≠ 0 and D_{s}^{2}(p) ≤ 1, p ∈ M. Denote D_{s−1}^{1} = D_{s}^{1} − Dp̃ with p̃ such that D_{s}^{1}(p̃) ≠ 0 and consider the function

(A.88)  F̂(p) = θ(\underline{Â}_{p_{0}}(p) + \underline{α̂}_{p_{0}}(D_{s−1}^{1}) − \underline{α̂}_{p_{0}}(D_{s}^{2}) − \underline{z}).

Observe that F̂(p̃) ≠ 0 and that F̂(p) = 0 if D_{s}^{2}(p) = 1. Thus (F̂) may be written as D_{s}^{2} + D_{g−s} with D_{g−s} ∈ M_{g−s}. And calculating \underline{α}_{p_{0}}((F̂)) yields

(A.89) \underline{α}_{p_{0}}((F̂)) = −\underline{α}_{p_{0}}(D_{s−1}^{1}) + \underline{α}_{p_{0}}(D_{s}^{2}) + [\underline{z}] − \underline{Ξ}_{p_{0}} = \underline{α}_{p_{0}}(D_{s}^{2}) − \underline{α}_{p_{0}}(D_{g−s}).

Thus [\underline{z}] = \underline{α}_{p_{0}}(D_{s−1}^{1} + D_{g−s}) + \underline{Ξ}_{p_{0}}. □

Before we can proceed we need:

Lemma A.12. Let D be a positive divisor on M. r(−D) ≥ s ≥ 1 if and only if given any positive divisor D′ of degree < s, there is a positive divisor D′′ such that D′ + D′′ − D is principal. For the if part it suffices to restrict D′ to an open subset U ⊂ M_{s−1}.

Proof. Since −D + D′ + D′′ = (f) we have f ∈ L(−D + D′). But if r(D) < s we can always find a D′ ∈ U with r(−D + D′) = 0 (Let {f_{ℓ}}_{1≤ℓ≤d} be a basis for L(−D). Start with a suitable point p_{1} where none of the functions f_{ℓ} vanishes, construct a new basis f̃_{ℓ} such that f̃_{ℓ}(p_{1}) = 0 for ℓ ≥ 2 – repeat r(D) times and set D′ = ∑_{ℓ=1}^{d} D_{p_{ℓ}}.).

Now assume r(−D) = s and let D′ ∈ M_{s−1}. It follows that r(D′ − D) ≥ s − (s − 1) = 1 (by a simple induction argument – compare (A.42)) and we can choose D′′ = (f) + D − D′ for a nonzero f ∈ L(D − D′). □

Note: D and D′ + D′′ have the same image under Abel’s map and r(−D) = i(D) + 1 for D ∈ M_{g} (by Riemann-Roch).

Now let [\underline{z}] = \underline{α}_{p_{0}}(D) + \underline{Ξ}_{p_{0}} with D ∈ M_{g−1} and i(D) = s(≥ 1). Any point \underline{ξ} ∈ (Ŵ_{s−1}^{p_{0}} −Ŵ_{s−1}^{p_{0}} −\underline{z}) can be written as \underline{α̂}_{p_{0}}(D_{s−1}^{1})−\underline{α̂}_{p_{0}}(D_{s−1}^{2})−\underline{z} with D_{s−1}^{1}, D_{s−1}^{2} ∈ M_{s−1}. Due to our lemma we can also write [\underline{z}] as \underline{α}_{p_{0}}(D_{s−1}^{1} + D′′_{g−s}) + \underline{Ξ}_{p_{0}} and we get [\underline{ξ}] = −\underline{α}_{p_{0}}(D_{s−1}^{2} + D′′_{g−s}) − \underline{Ξ}_{p_{0}} ∈ −(W_{g−1}^{p_{0}} + \underline{Ξ}_{p_{0}}) or equivalently θ(\underline{ξ}) = 0.

On the other hand we may also vary the divisor D_{s}^{1} of the last theorem in a sufficiently small neighborhood getting D_{s−1}^{1}′ and a corresponding D′_{g−s} also fulfilling [\underline{z}] = \underline{α}_{p_{0}}(D_{s−1}^{1}′ + D′_{g−s}) + \underline{Ξ}_{p_{0}}. From Abel’s theorem we conclude that (D_{s−1}^{1} + D_{g−s}) − (D_{s−1}^{1}′ + D′_{g−s}) is principal and from our last lemma: i(D_{s−1}^{1} + D_{g−s}) ≥ s. Thus we have proved most of Riemann’s vanishing theorem:

Theorem A.13. (Riemann) Let \underline{z} ∈ ℂ^{g} and s ≥ 1. θ(Ŵ_{s−1}^{p_{0}} − Ŵ_{s−1}^{p_{0}} − \underline{z}) ≡ 0 but θ(Ŵ_{s}^{p_{0}} − Ŵ_{s}^{p_{0}} − \underline{z}) ≢ 0 is equivalent to [\underline{z}] = \underline{α}_{p_{0}}(D) + \underline{Ξ}_{p_{0}} with D ∈ M_{g−1} and i(D) = s which again is equivalent to the vanishing of all partial derivatives of θ of order less than s at \underline{z} and to the non-vanishing of at least one derivative of order s at \underline{z}.

Note: s ≥ 1 implies θ(\underline{z}) = 0, we always have s ≤ g and Ŵ_{s−1}^{p_{0}} − Ŵ_{s−1}^{p_{0}} is independent of the base point p_{0}.

Proof. We omit the proof of the part concerning the derivatives (cf. [81]), but we will prove the other part: In the one direction we already know the first part and i(D) ≥ s. If i(D) > s we would get θ(Ŵ_{s}^{p_{0}} − Ŵ_{s}^{p_{0}} − \underline{z}) ≡ 0 from the first statement above. In the other direction we would get from θ(Ŵ_{s}^{p_{0}} − Ŵ_{s}^{p_{0}} − \underline{z}) ≡ 0 that i(D) > s using the first direction. □

With some simple changes in the proof of the last theorem, we can also get another one:

Theorem A.14. Let \underline{z} ∈ ℂ^{g}. s is the least integer such that θ(Ŵ_{s+1}^{p_{0}}−Ŵ_{s}^{p_{0}} −\underline{z}) ≢ 0 is equivalent to [\underline{z}] = \underline{α}_{p_{0}}(D) + \underline{Ξ}_{p_{0}} with D ∈ M_{g} and i(D) = s.

Note: We now have 0 ≤ s ≤ g − 1. Since \underline{z} was arbitrary we have solved Jacobi’s inversion problem:

Corollary A.15. (Jacobi) \underline{α}_{p_{0}} : M_{g} → J(M) is surjective.

Using these last theorems we can say a lot about the function

(A.90)  F̂(p) = θ(\underline{Â}_{p_{0}}(p) − \underline{z}),  \underline{z} ∈ ℂ^{g}, p ∈ M̂.

Theorem A.16. Let \underline{z} ∈ ℂ^{g} and let s be the least integer such that θ(Ŵ_{s+1}^{p_{0}} − Ŵ_{s}^{p_{0}} − \underline{z}) ≢ 0. Then F̂ ≢ 0 is equivalent to s = 0. We can write [\underline{z}] = \underline{α}_{p_{0}}(D) + \underline{Ξ}_{p_{0}} with D ∈ M_{g} and i(D) = s. Moreover, if F̂ ≢ 0 we have (F̂) = D (and hence i((F̂)) = 0).

Proof. Observe, that both sides are equivalent to F̂(p) = θ(\underline{Â}_{p_{0}}(p) − \underline{z}) ≢ 0 and D(p_{0}) = 0. Uniqueness follows from Lemma A.9. □

Corollary A.17. θ(\underline{z}) ≠ 0 is equivalent to [\underline{z}] = \underline{α}_{p_{0}}(D) + \underline{Ξ}_{p_{0}} with a unique D ∈ M_{g}, i(D) = 0 and D(p_{0}) = 0.

Finally we prove Abel’s theorem for differentials:

Theorem A.18. D ∈ Div(M) is canonical if and only if

(A.91)  \underline{α}_{p_{0}}(D) = −2 \underline{Ξ}_{p_{0}}  and  deg(D) = 2g − 2.

Proof. Since all canonical divisors lie in the same divisor class their image under Abel’s map is a constant (by Abel’s theorem). To determine this constant let \underline{z} be a zero of θ. Hence −\underline{z} is a zero too, and we conclude that [\underline{z}] = \underline{α}_{p_{0}}(D_{1}) + \underline{Ξ}_{p_{0}} and [−\underline{z}] = \underline{α}_{p_{0}}(D_{2}) + \underline{Ξ}_{p_{0}} with D_{1}, D_{2} ∈ M_{g−1}. Combining these equations we get \underline{α}_{p_{0}}(D_{1} + D_{2}) = −2\underline{Ξ}_{p_{0}}. Since D_{1} is arbitrary (because \underline{z} is) our last lemma yields r(−D_{1} − D_{2}) ≥ g and from Riemann Roch i(D_{1} + D_{2}) ≥ 1. Thus D_{1} + D_{2} is the divisor of a holomorphic differential (since deg(D_{1} + D_{2}) = 2g − 2).

Conversely, suppose D is of degree 2g − 2 and \underline{α}_{p_{0}}(D) = −2\underline{Ξ}_{p_{0}}. Let ω be a meromorphic differential. From Abel’s theorem we get that D − (ω) is the divisor of a meromorphic function f. Thus D is the divisor of fω. □

Corollary A.19. 2 \underline{Ξ}_{p_{0}} = [0] if and only if (2g − 2)D_{p_{0}} is canonical.

A.7. Hyperelliptic Riemann surfaces

Finally, we want to give a constructive approach to hyperelliptic Riemann surfaces. Let

(A.92)  {E_{n}}_{0≤n≤2g+1} ⊂ ℝ,  E_{0} < E_{1} < ⋯ < E_{2g+1}, g ∈ ℕ_{0},

be some fixed points. Using them we may define

(A.93)  Π = ℂ\( ⋃_{j=0}^{g}[E_{2j}, E_{2j+1}]),

which is obviously a domain (connected open subset of ℂ) and a holomorphic function

(A.94)  \begin{aligned}R_{2g+2}^{1/2}(.) : & Π & → & ℂ \\ & z & ↦ & − ∏_{n=0}^{2g+1} \sqrt{z − E_{n}} \end{aligned} ,

where the square root is defined as follows,

(A.95)  \sqrt{z} = |\sqrt{z}|ei arg(z)/2,  arg(z) ∈ (−π, π].

We first extend R_{2g+2}^{1/2}(z) to the whole of ℂ by
(A.96)  R_{2g+2}^{1/2}(z) = lim_{ε↓0} R_{2g+2}^{1/2}(x + iε)

for z ∈ ℂ\Π. This implies R_{2g+2}^{1/2}(z) = i^{2g−n}|R_{2g+2}^{1/2}(z)| for z ∈ (E_{n−1}, E_{n}), 0 ≤ n ≤ 2g + 2, if we set E_{−1} = −∞ and E_{2g+2} = ∞.

We may now define the following set

(A.97)  M = {(z, σR_{2g+2}^{1/2}(z))|z ∈ ℂ, σ ∈ {−1, +1}} ∪ {∞_{+}, ∞_{−}}

and call B = {(E_{n}, 0)}_{0≤n≤2g+1} the set of branch points.

Defining the following charts (ζ_{p_{0}}, U_{p_{0}}), we can make the set M into a Riemann surface. Abbreviate

(A.98)  p_{0} = (z_{0}, σ_{0}R_{2g+2}^{1/2}(z_{0})), p = (z, σR_{2g+2}^{1/2}(z)) ∈ U_{p_{0}} ⊂ M,

U′_{p_{0}} = ζ_{p_{0}}(U_{p_{0}}) ⊂ ℂ.

If p_{0} ∉ B we set:

U_{p_{0}} = {p ∈ M||z −z_{0}| < C and σR_{2g+2}^{1/2}(z) ⟶^{γ} σ_{0}R_{2g+2}^{1/2}(z_{0})}, U′_{p_{0}} = {ζ ∈ ℂ||ζ| <

C}, C = min_{n} |z_{0} − E_{n}| > 0,

(A.99) \begin{aligned}ζ_{p_{0}} : & U_{p_{0}} & → & U′_{p_{0}} & ζ_{p_{0}}^{−1} : & U′_{p_{0}} & → & U_{p_{0}} \\ & p & ↦ & z − z_{0} & & ζ & ↦ & (ζ + z_{0}, σR_{2g+2}^{1/2}(ζ + z_{0})) \end{aligned} ,

where σR_{2g+2}^{1/2}(z) ⟶^{γ} σ_{0}R_{2g+2}^{1/2}(z_{0}) means that σR_{2g+2}^{1/2}(z) is the branch reached by analytic continuation along γ, the straight line from z to z_{0}.

If p_{0} = (E_{m}, 0) we set:

U_{p_{0}} = \{p ∈ M||z − E_{m}| < C\}, U′_{p_{0}} = \{ζ ∈ ℂ||ζ| < \sqrt{C}\}, C = min_{n≠m} |E_{m} − E_{n}| >

0,

\begin{aligned}ζ_{p_{0}} : & U_{p_{0}} & → & U′_{p_{0}} \\ & p & ↦ & σ(z − E_{m})^{1/2} \end{aligned} ,

\begin{aligned}ζ_{p_{0}}^{−1} : & U′_{p_{0}} & → & U_{p_{0}} \\ & ζ & ↦ & (ζ^{2} + E_{m}, ζ ∏_{n≠m}(ζ^{2} + E_{m} − E_{n})^{1/2}) \end{aligned} , (A.100)

where the left root is defined as (z − E_{m})^{1/2} = \sqrt{|z − E_{m}|} exp(i arg(z − E_{m})/2), with 0 ≤ arg(z − E_{m}) < 2π if m is even and −π < arg(z − E_{m}) ≤ π if m is odd. The right root is holomorphic on U_{p_{0}} with the sign fixed by

∏_{n≠m} (ζ^{2} + E_{m} − E_{n})^{1/2} = i^{2g−m−1}| ∏_{n≠m} \sqrt{E_{m} − E_{n}}| ×

×(1 − \frac{1}{2} ∑_{n≠m} \frac{1}{E_{m} − E_{n}} ζ^{2} + O(ζ^{4})). (A.101)

If p_{0} = ∞_{±} we set:

U_{p_{0}} = \{p ∈ M||z| > C\} U′_{p_{0}} = \{ζ ∈ ℂ||ζ| < \frac{1}{C} \} C = max_{n} |E_{n}| < ∞

(A.102)  \begin{aligned}ζ_{p_{0}} : & U_{p_{0}} & → & U′_{p_{0}} & ζ_{p_{0}}^{−1} : & U′_{p_{0}} & → & U_{p_{0}} \\ & p & ↦ & \frac{1}{z} & & ζ & ↦ & (\frac{1}{ζ}, \frac{± ∏_{n}(1 − ζE_{n})^{1/2}}{ζ^{g+1}} ) \\ & ∞_{±} & ↦ & 0 & & 0 & ↦ & ∞_{±} \end{aligned}

where the right root is holomorphic on U_{p_{0}} with the sign fixed by

(A.103)  ∏_{n} (1 − ζE_{n})^{1/2} = −1 + \frac{1}{2} ∑_{n} E_{n}ζ + O(ζ^{2}).

Let us take two subsets

(A.104)  Π_{±} = {(z, ±R_{2g+2}^{1/2}(z))|z ∈ Π} ⊂ M,

and define two more quite useful charts

(A.105)  \begin{aligned}ζ_{±} : & Π_{±} & → & Π \\ & p & ↦ & z \end{aligned} .

It is not hard to verify, that the transition functions of all these charts are indeed holomorphic, for example,

ζ_{±} ∘ ζ_{E_{m}}^{−1} : \begin{matrix}U′_{E_{m}} ∩ ζ_{E_{m}}(Π_{±}) & → & Π ∩ ζ_{±}(U_{E_{m}}) \\ ζ & ↦ & ζ^{2} + E_{m} \end{matrix} ,
(A.106)  ζ_{E_{m}} ∘ ζ_{±}^{−1} : \begin{matrix}Π ∩ ζ_{±}(U_{E_{m}}) & → & U′_{E_{m}} ∩ ζ_{E_{m}}(Π_{±}) \\ ζ & ↦ & ±\sqrt{ζ − E_{m}} \end{matrix} .

Compare the following picture (g = 1, m odd) and note, that the branch cut of the square root does not belong to its domain!

[figure: ℂ; ℂ; ζ_{E_{m}} ∘ ζ_{+}^{−1}; ζ_{+} ∘ ζ_{E_{m}}^{−1}; ζ_{E_{m}}; ζ_{+}; E_{m}; Π_{+}; Π_{−}; M; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; 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The topology induced by these charts is Hausdorff (take the U’s from above and modify C a little if necessary) and second countable (since we only need countably (in particular even finitely) many of these charts to cover M). Consider Π_{±}, they are connected (since they are homeomorphic to Π). Their closures \overline{Π_{±}} are still connected, have points in common and fulfill M = \overline{Π_{+}} ∪ \overline{Π_{−}}. Thus M is connected and a Riemann surface. But they tell us even more: \overline{Π_{±}} is just a sphere with g + 1 holes, and if we want to get M, we have to identify corresponding points on the boundaries ending up with a sphere having g handles.

We will now define three maps on M. First the sheet exchange map,

(A.107)  \begin{aligned}∗ : & M & → & M \\ & (z, σR_{2g+2}^{1/2}(z)) & ↦ & (z, σR_{2g+2}^{1/2}(z))^{∗} = (z, −σR_{2g+2}^{1/2}(z)) \\ & ∞_{±} & ↦ & ∞_{∓} \end{aligned} .

It is clearly holomorphic, since it is locally just the identity (up to a sign – using the charts of the previous page). And second the projection

(A.108)  \begin{aligned}π : & M & → & ℂ ∪ \{∞\} \\ & (z, σR_{2g+2}^{1/2}(z)) & ↦ & z \\ & ∞_{±} & ↦ & ∞ \end{aligned}

and the evaluation map

(A.109)  \begin{aligned}R_{2g+2}^{1/2}(.) : & M & → & ℂ ∪ \{∞\} \\ & (z, σR_{2g+2}^{1/2}(z)) & ↦ & σR_{2g+2}^{1/2}(z) \\ & ∞_{±} & ↦ & ∞ \end{aligned} .

Obviously, both are meromorphic functions (use again the local charts and their inversions). π has poles of order 1 at ∞_{±} and two simple zeros at (0, ±R_{2g+2}^{1/2}(0)) if R_{2g+2}^{1/2}(0) ≠ 0 respectively one double zero at (0, 0) if R_{2g+2}^{1/2}(0) = 0. R_{2g+2}^{1/2}(.) has poles of order g + 1 at ∞_{±} and 2g + 2 simple zeros at (E_{n}, 0). Notice also π(p^{∗}) = π(p) and R_{2g+2}^{1/2}(p^{∗}) = −R_{2g+2}^{1/2}(p).

Thus we may conclude that M is a two-sheeted, ramified covering of the Riemann sphere (≅ ℂ ∪ {∞}), that M is compact (since π is open and ℂ ∪ {∞} is compact), and finally, that M is hyperelliptic (since it admits a function of degree two).

Now consider

(A.110)  \frac{dπ}{R_{2g+2}^{1/2}} .

Again using local charts we see that dπ/R_{2g+2}^{1/2} is holomorphic everywhere and has zeros of order g − 1 at ∞_{±}. So we may conclude that

(A.111)  \frac{π^{j−1}dπ}{R_{2g+2}^{1/2}} ,  1 ≤ j ≤ g,

form a basis for the space of holomorphic differentials. As a consequence we obtain the following result.

Lemma A.20. Let D ≥ 0 and denote by D̃ the divisor on ℂ ∪ {∞} defined by D̃(z) = max_{p∈π^{−1}(z)} D(p) if z ∈ ℂ ∪ \{∞\}\\{E − j\}_{j=0}^{2g+1} and D̃(z) = [[\frac{D(p)+1}{2}]] if z ∈ {E − j}_{j=0}^{2g+1}. Then

(A.112)  i(D) = max{0, g − deg(D̃)}.

Proof. Each ζ ∈ L^{1}(D) must be of the form

(A.113)  ζ = \frac{(P ∘ π)dπ}{R_{2g+2}^{1/2}} ,

where P (z) is a polynomial of degree at most g − D̃(∞) − 1. Moreover, P (z) must vanish of order D̃(z) at z ∈ ℂ. Hence we have max{0, g − deg(D̃)} free constants for P (z). □

Next, we will introduce the representatives {a_{ℓ}, b_{ℓ}}_{ℓ=1}^{g} of a canonical homology basis for M. For a_{ℓ} we start near E_{2ℓ−1} on Π_{+}, surround E_{2ℓ} thereby changing to Π_{−} and return to our starting point encircling E_{2ℓ−1} again changing sheets. For b_{ℓ} we choose a cycle surrounding E_{0}, E_{2ℓ−1} counter-clock-wise (once) on Π_{+}. The cycles are chosen so that their intersection matrix reads

(A.114)  a_{j} ∘ b_{k} = δ_{jk},  1 ≤ j, k ≤ g.

Visualizing for g = 2:

[figure: E_{0}; b_{1}; b_{2}; E_{1}; E_{2}; a_{1}; E_{3}; E_{4}; a_{2}; E_{5}; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; 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.; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .]

Here the solid lines indicate the parts on Π_{+} and the doted ones the parts on Π_{−}.

Now we can construct a canonical basis {ζ_{j}}_{j=1}^{g} for the space of holomorphic differentials in the following way: If we introduce the constants \underline{c}(.) via

(A.115)  c_{j}(k) = C_{jk}^{−1},  C_{jk} = ∫_{a_{k}} \frac{π^{j−1}dπ}{R_{2g+2}^{1/2}} = 2 ∫_{E_{2k−1}}^{E_{2k}} \frac{z^{j−1}dz}{R_{2g+2}^{1/2}(z)} ∈ ℝ.

(C is invertible since otherwise there would be a nonzero differential with vanishing a-periods.) the differentials

(A.116)  \underline{ζ} = ∑_{j=1}^{g} \underline{c}(j)\frac{π^{j−1}dπ}{R_{2g+2}^{1/2}}

fulfill

(A.117)  ∫_{a_{k}} ζ_{j} = δ_{j,k}

and are thus a basis of the required form. The matrix of b-periods

(A.118)  τ_{j,k} = ∫_{b_{k}} ζ_{j} = − ∑_{j=1}^{g} c_{j}(m) ∑_{ℓ=0}^{k−1} 2 ∫_{E_{2ℓ}}^{E_{2ℓ+1}} \frac{z^{m−1}dz}{R_{2g+2}^{1/2}(z)}

satisfies Re(τ_{j,k}) = 0 since we have Im(c_{j}(k)) = 0. In the chart (U_{±}, w) = (U_{∞_{±}}, ζ_{∞_{±}}) we have

\underline{ζ} = ± ∑_{j=1}^{g} \underline{c}(j)\frac{w^{g−j} dw}{\sqrt{∏_{m}(1 − wE_{m})}}

(A.119)  = ±(\underline{c}(g) + (\frac{\underline{c}(g)}{2} ∑_{j=1}^{2g+2} E_{j} + \underline{c}(g − 1))w + O(w^{2}))dw.

Finally, we will show how to evaluate some integrals on M̂. We fix a base point p_{0} = (E_{0}, 0). Then we have, for example,

(A.120)  \underline{Â}_{p_{0}}(∞_{+}) = − ∑_{j=1}^{g} \underline{c}(j) ∫_{−∞}^{E_{0}} \frac{z^{j−1}dz}{R_{2g+2}^{1/2}(z)} − ∑_{j=1}^{g} \underline{δ}_{j}.

To see this, take the straight line segment from E_{0} to −∞ and lift it to Π_{+}. Unfortunately, we cannot take this lift as our path of integration for it intersects all b cycles, whereas on the other hand we are required to stay in M̂. Hence, rather than crossing b_{j} we will follow the boundary of M̂ along b_{j}, a_{j}^{−1}, b_{j}^{−1} until A.7. Hyperelliptic Riemann surfaces 297 we hit the corresponding point on b_{j}^{−1}. The parts on b_{j}, b_{j}^{−1} cancel (compare the picture on page 277) but the part on a_{j}^{−1} contributes. In other words, for each bcycle we cross, we have to subtract the associated a-period. Similarly, \underline{Â}_{p_{0}}(∞_{−}) = −\underline{Â}_{p_{0}}(∞_{+}) − ∑_{j=1}^{g} \underline{δ}_{j}.

Next, let us try to compute \underline{Ξ̂}_{p_{0}}. As before one shows

(A.121) Â_{p_{0},j}((E_{2ℓ−1}, 0)) = −\frac{τ_{j,ℓ}}{2}

and hence

(A.122)  ∫_{a_{ℓ}} Â_{p_{0},j}ζ_{ℓ} = −\frac{τ_{j,ℓ}}{2} + ∫_{a_{ℓ}} Â_{(E_{2ℓ−1},0),j}ζ_{ℓ}.

The last integral can be split up into two parts, one from E_{2ℓ−1} to E_{2ℓ} on Π_{+} and one from E_{2ℓ} to E_{2ℓ−1} on Π_{−} . Since these two parts cancel we infer

(A.123)  Ξ̂_{p_{0},j} = \frac{1}{2} − \frac{1}{2} ∑_{k=1}^{g} τ_{j,k}.

Appendix B

Herglotz functions

The results stated in this appendix are collected from [16], [17], [68], [73], [119], [198], [199], [200], [202], [203], [206].

In this chapter an important role will be played by Borel measures ρ on ℝ. Hence we fix some notation first. For any Borel set B ⊆ ℝ we will denote by ρ(B) = ∫_{B} dρ its ρ-measure. Moreover, to each Borel measure ρ on ℝ corresponds a monotone increasing function (also denoted by ρ for simplicity) which we normalize as

(B.1)  ρ(λ) = \{ \begin{matrix}−ρ((λ, 0]), & λ < 0 \\ 0, & λ = 0 \\ ρ((0, λ]), & λ > 0 \end{matrix} .

By definition, ρ(λ) is right continuous.

Associated to ρ is the (separable) Hilbert space L^{2}(ℝ, dρ) with scalar product

(B.2)  ⟨f, g⟩ = ∫_{ℝ} \overline{f(λ)}g(λ)dρ(λ).

It has the following properties.

Lemma B.1. Suppose ρ(ℝ) < ∞. The set of all continuous functions is dense in L^{2}(ℝ, dρ). Moreover, if ρ is compactly supported (ρ(K) = ρ(ℝ) for a compact set K ⊆ ℝ), then the set of polynomials is also dense.

Proof. First of all, bounded functions are dense. In fact, consider

(B.3)  f_{n}(λ) = \{ \begin{matrix}f(λ), & |f(λ)| ≤ n \\ 0 , & |f(λ)| > n \end{matrix} .

Then |f(λ) − f_{n}(λ)| → 0 a.e. since |f(λ)| < ∞ a.e.. Since |f(λ) − f_{n}(λ)| ≤ |f(λ)| we obtain ∫ |f − f_{n}|^{2}dρ → 0 by the dominated convergence theorem.

So it remains to show that bounded elements can be approximated by continuous ones. By Lusin’s theorem, there are continuous functions f_{n} such that ∥f_{n}∥_{∞} ≤ ∥f∥_{∞} and

(B.4)  ρ(B_{n}) ≤ \frac{1}{n}, B_{n} = \{λ|f(λ) ≠ f_{n}(λ)\}.

Here ∥f∥_{∞} denotes the essential supremum of f, that is, the smallest constant M such that |f(λ)| ≤ M a.e.. Hence

(B.5)  ∥f − f_{n}∥^{2} = ∫_{B_{n}} |f_{n}(λ) − f(λ)|^{2}dρ(λ) ≤ \frac{4∥f∥_{∞}^{2}}{n} → 0

and the first claim follows.

To show the second, it suffices to prove that a continuous element f can be approximated by polynomials. By the Stone-Weierstrass Theorem there is a polynomial P_{n} approximating f in the sup norm on K such that

(B.6)  ∥f − P_{n}∥^{2} ≤ ρ(ℝ)∥f − P_{n}∥_{∞}^{2} ≤ \frac{ρ(ℝ)}{n} → 0

and we are done. □

If ρ(ℝ) < ∞ we can define the Borel transform (also Cauchy transform) of ρ by

(B.7)  F (z) = ∫_{ℝ} \frac{dρ(λ)}{z − λ},  z ∈ ℂ_{±},

where ℂ_{±} = {z ∈ ℂ| ± Im(z) > 0}. In addition, we can also define the Fourier transform

(B.8)  T (λ) = ∫_{ℝ} e^{iλt}dρ(t),  λ ∈ ℝ,

which is a bounded function and is related to the Borel transform via

(B.9)  F (z) = \{ \begin{matrix}i ∫_{−∞}^{0} e^{−izλ} T (λ)dλ, & Im(z) > 0 \\ −i ∫_{0}^{∞} e^{−izλ} T (λ)dλ, & Im(z) < 0 \end{matrix}

as can be verified using Fubini’s theorem.

The Borel transform has the interesting property that it maps the upper half plane ℂ_{+} into itself as can be seen from

(B.10)  Im(F (z)) = Im(z) ∫_{ℝ} \frac{dρ(λ)}{|z − λ|^{2}} .

In general, a holomorphic function F : ℂ_{+} → ℂ_{+} is called a Herglotz function (sometimes also Pick or Nevanlinna-Pick function). It is no restriction to assume that F is defined on ℂ_{−} ∪ ℂ_{+} satisfying F (\overline{z}) = \overline{F (z)}.

The following theorem shows that all Herglotz functions arise as Borel transform (in an extended sense) of some unique measure.

Theorem B.2. F is a Herglotz function if and only if

(B.11)  F (z) = a + b z + ∫_{ℝ} (\frac{1}{λ − z} − \frac{λ}{1 + λ^{2}} ) dρ(λ), z ∈ ℂ_{±},

where a, b ∈ ℝ, b ≥ 0, and ρ is a nonzero measure on ℝ which satisfies ∫_{ℝ}(1 + λ^{2})^{−1}dρ(λ) < ∞.

Moreover, the triple a, b, and ρ is unambiguously determined by F using

(B.12)  a = Re(F (i)), lim_{{z→∞ \atop Im(z)≥ε>0}} \frac{F (z)}{z} = b ≥ 0,

and Stieltjes inversion formula

(B.13)  ρ((λ_{0}, λ_{1}]) = lim_{δ↓0} lim_{ε↓0} \frac{1}{π} ∫_{λ_{0}+δ}^{λ_{1}+δ} Im(F (λ + iε))dλ.

The content of Stieltjes inversion formula can be strengthened.

Lemma B.3. Let F be a Herglotz function with associated measure ρ. Then the measure π^{−1}Im(F (λ + iε))dλ converges weakly to dρ(λ) in the sense that

(B.14)  lim_{ε↓0} \frac{1}{π} ∫_{ℝ} f(λ)Im(F (λ + iε))dλ = ∫_{ℝ} f(λ)dρ(λ)

for all continuous functions f with |f| ≤ const(1 + λ^{2})^{−1}. In addition,

lim_{ε↓0} \frac{1}{π} ∫_{λ_{0}}^{λ_{1}} f(λ)Im(F (λ + iε))dλ = ∫_{(λ_{0},λ_{1})} f(λ)dρ(λ)

+ \frac{f(λ_{1})ρ(\{λ_{1}\}) − f(λ_{0})ρ(\{λ_{0}\})}{2}. (B.15)

The following result shows that a different analytic continuation of F from ℂ_{+} to ℂ_{−} can exist in some cases.

Lemma B.4. Let (λ_{1}, λ_{2}) ⊆ ℝ and suppose lim_{ε↓0} Re(F (λ + iε)) = 0 for a.e. λ ∈ (λ_{1}, λ_{2}). Then F can be analytically continued from ℂ_{+} into ℂ_{−} through the interval (λ_{1}, λ_{2}). The resulting function F̃(z) coincides with F (z) on ℂ_{+} and satisfies

(B.16)  \overline{F̃(z)} = −F̃(\overline{z}).

In addition, Im(F (λ + i0)) > 0, Re(F (λ + i0)) = 0 for all λ ∈ (λ_{1}, λ_{2}).

The measure ρ is called spectral measure of F. The set of all growth points, that is,

(B.17)  σ(ρ) = {λ ∈ ℝ|ρ((λ − ε, λ + ε)) > 0 for all ε > 0},

is called the spectrum of ρ. Invoking Morera’s together with Fubini’s theorem shows that F (z) is holomorphic for z ∈ ℂ\σ(ρ). The converse following from Stieltjes inversion formula. Moreover, we have

(B.18)  \frac{dF (z)}{dz} = − ∫_{ℝ} \frac{dρ(λ)}{(λ − z)^{2}} .

In particular, if ρ((λ_{0}, λ_{1})) = 0, then F (λ), λ ∈ (λ_{0}, λ_{1}), is decreasing and hence tends to a limit (in ℝ ∪ {∞}) as λ ↓ λ_{0} or λ ↑ λ_{1}.

The following result explains why σ(ρ) is called the spectrum of ρ.

Lemma B.5. The set σ(ρ) is precisely the spectrum σ(H̃) of the multiplication operator H̃f(λ) = λf(λ), 𝔇(H̃) = {f ∈ L^{2}(ℝ, dρ)|λf(λ) ∈ L^{2}(ℝ, dρ)}.

Proof. If λ ∈ σ(ρ), then the sequence f_{n} = ρ((λ − \frac{1}{n}, λ + \frac{1}{n}))^{−1/2}χ_{(λ− \frac{1}{n} ,λ+ \frac{1}{n})} satisfies ∥f_{n}∥ = 1, ∥(H̃ − λ)f_{n}∥ → 0 and hence λ ∈ σ(H̃). Conversely, if z ∉ σ(ρ), then (H̃ −z)^{−1}f(λ) = (λ−z)^{−1}f(λ) exists and is bounded, implying z ∉ σ(H̃). □

Our main objective is to characterize σ(ρ), and various decompositions of σ(ρ), in terms of F (z). More precisely, this will be done by investigating the boundary behavior of F (λ + iε), λ ∈ ℝ, as ε ↓ 0.

First we recall the unique decomposition of ρ with respect to Lebesgue measure,

(B.19)  dρ = dρ_{ac} + dρ_{s},

where ρ_{ac} is absolutely continuous with respect to Lebesgue measure (i.e., we have ρ_{ac}(B) = 0 for all B with Lebesgue measure zero) and ρ_{s} is singular with respect to Lebesgue measure (i.e., ρ_{s} is supported, ρ_{s}(ℝ\B) = 0, on a set B with Lebesgue measure zero). The singular part ρ_{s} can be further decomposed into a (singular) continuous and a pure point part,

(B.20)  dρ_{s} = dρ_{sc} + dρ_{pp},

where ρ_{sc} is continuous on ℝ and ρ_{pp} is a step function.

By the Radon-Nikodym Theorem we have

(B.21)  dρ_{ac}(λ) = f(λ)dλ

for a locally integrable positive function f which is unique (a.e.). Moreover, we note that the singular part can be characterized by the following lemma.

Lemma B.6. For any Borel set B we have

(B.22)  ρ_{s}(B) = lim_{n→∞} sup_{I∈I,|I|<1/n} ρ(B ∩ I),

where I is the (countable) family of finite unions of open intervals, each of which has rational endpoints, and |I| denotes the Lebesgue measure of I.

Proof. Since the supremum decreases as n increases, the limit exists. Denote it by ρ̃(B). Using dρ_{ac}(λ) = f(λ)dλ we see

(B.23)  ρ(B ∩ I) ≤ ρ_{s}(B) + R|I| + ρ_{ac}(f^{−1}((R, ∞))).

implying ρ̃(B) ≤ ρ_{s}(B) + lim_{R→∞} ρ_{ac}(f^{−1}((R, ∞))) = ρ_{s}(B).

Conversely, pick a support B_{s} for ρ_{s} of Lebesgue measure |B_{s}| zero. By 0 = |B_{s}| = sup{|O| |B_{s} ⊂ O, O open} there exists a sequence O_{n} such that B_{s} ⊂ O_{n} and |O_{n}| < 1/n. Moreover, we can find a sequence I_{m} ∈ I such that I_{m} ⊆ I_{m+1} and ⋃ I_{m} = O_{n} (hence ρ(I_{m}) → ρ(O_{n})). Using ρ(B ∩ I_{m}) ≥ ρ(B ∩ O_{n}) − ρ(O_{n}\I_{m}) concludes the proof

(B.24)  sup_{I∈I,|I|<1/n} ρ(B ∩ I) ≥ ρ(B ∩ O_{n}) ≥ ρ(B ∩ B_{s}) ≥ ρ_{s}(B).
□

Now we turn to the concept of a minimal support (sometimes also essential support) of a measure. A set M is called a minimal support for ρ if M is a support (i.e., ρ(ℝ\M) = 0) and any subset M_{0} ⊆ M which does not support M (i.e, ρ(M_{0}) = 0) has Lebesgue measure zero. Let

(B.25)  Im(F (λ)) = lim sup_{ε↓0} Im(F (λ + iε))

and let L(ρ) be the set of all λ ∈ ℝ for which lim_{ε↓0} Im(F (λ + iε)) exists (finite or infinite). Then we have Lemma B.7. Minimal supports M, M_{ac}, M_{s} for ρ, ρ_{ac}, ρ_{s}, respectively, are given by

M = {λ ∈ L(ρ)|0 < Im(F (λ)) ≤ ∞},
M_{ac} = {λ ∈ L(ρ)|0 < Im(F (λ)) < ∞},
(B.26)  M_{s} = {λ ∈ L(ρ)|Im(F (λ)) = ∞}.

In particular, the ρ-measure and the Lebesgue measure of ℝ\L(ρ) are both zero. This also says that if Im(F (λ)) < ∞ for all λ ∈ (λ_{1}, λ_{2}), then ρ is purely absolutely continuous on (λ_{1}, λ_{2}). Furthermore, M_{s} has Lebesgue measure zero and might be nonempty even though ρ_{s} = 0. However, note that if M_{s} is a countable set, then ρ_{sc} = 0.

It is interesting to know when the minimal supports determine the corresponding spectra.

The spectrum of ρ is given by

(B.27)  σ(ρ) = \overline{M}.

To see this observe that F is real holomorphic near λ ∉ σ(ρ) and hence Im(F (λ)) = 0 in this case. Thus M ⊆ σ(ρ) and since σ(ρ) is closed we even have \overline{M} ⊆ σ(ρ). Conversely, if λ ∈ σ(ρ), then 0 < ρ((λ − ε, λ + ε)) = ρ((λ − ε, λ + ε) ∩ M) for all ε > 0 and we can find a sequence λ_{n} ∈ (λ − 1/n, λ + 1/n) ∩ M converging to λ from inside M. This is the remaining part σ(ρ) ⊆ \overline{M}.

Moreover, σ(ρ_{ac}) can be recovered from the essential closure of M_{ac}, that is,

(B.28)  σ(ρ_{ac}) = \overline{M}_{ac}^{ess},

where

(B.29)  \overline{M}_{ac}^{ess} = \{λ ∈ ℝ||(λ − ε, λ + ε) ∩ M_{ac}| > 0 for all ε > 0\}.

Note that \overline{M}_{ac}^{ess} is closed, whereas we might have M_{ac} ⊄ \overline{M}_{ac}^{ess}. To prove (B.28) we use that 0 < ρ_{ac}((λ − ε, λ + ε)) = ρ_{ac}((λ − ε, λ + ε) ∩ M_{ac}) is equivalent to |(λ − ε, λ + ε) ∩ M_{ac}| > 0. One direction follows from the definition of absolute continuity and the other from minimality of M_{ac}.

Next we define the derivative

(B.30)  Dρ(λ) = lim sup_{ε↓0} \frac{ρ((λ − ε, λ + ε))}{2ε}.

Now we can say more about how to extend a Herglotz function F to the real axis.

Theorem B.8. Let F be a Herglotz function and ρ its spectral measure. (i). For all λ ∈ ℝ, Dρ(λ) and Im(F (λ)) are either both zero, both in (0, ∞), or both infinite. Moreover, Dρ(λ) exists as ordinary limit if λ ∈ L(ρ) and

(B.31)  Dρ(λ) = \frac{1}{π} Im(F (λ)),  λ ∈ L(ρ).

Moreover, dρ_{ac}(λ) = Dρ(λ)dλ. (ii). For almost all λ_{0} ∈ ℝ with respect to Lebesgue measure, the derivative Dρ(λ_{0}) and

(B.32)  R_{F} (λ_{0}) = a + bλ_{0} + lim_{ε↓0} ∫_{ℝ\(λ_{0}−ε,λ_{0}+ε)} (\frac{1}{λ − λ_{0}} − \frac{λ}{1 + λ^{2}} ) dρ(λ)

both exist (as ordinary limit) and are finite. For those λ_{0} we have

(B.33)  F (λ_{0} + i0) = lim_{ε↓0} F (λ_{0} + iε) = R_{F} (λ_{0}) + iπDρ(λ_{0}).

(iii). If F (z) and G(z) are Herglotz functions and the boundary values coincide F (λ+i0) = G(λ+i0) for λ in a set of positive Lebesgue measure, then F (z) = G(z).

Next, we want to generalize the decomposition of ρ by taking the α-dimensional Hausdorff measure h^{α} rather than the Lebesgue measure. The splitting dρ = dρ_{ac}+ dρ_{s} will correspond to the case α = 1.

For any given Borel set B ⊆ ℝ and α ∈ [0, 1] we define

(B.34)  h_{ε}^{α}(B) = inf { ∑_{j∈ℕ} |I_{j}|^{α}| |I_{j}| < ε, B ⊆ ⋃_{j∈ℕ} I_{j}}, ε > 0,

the infimum over all countable covers by intervals I_{j} of length at most ε. Since h_{ε}^{α}(B) is increasing with respect to ε (the number of covers decreases) we can define the α-dimensional Hausdorff measure of B as

(B.35)  h^{α}(B) = lim_{ε↓0} h_{ε}^{α}(B).

Note that h^{α_{1}}(B) ≥ h^{α_{2}}(B) if α_{1} ≤ α_{2} and h^{0}, h^{1} correspond to counting, Lebesgue measure, respectively. The Hausdorff dimension α(B) of B is the unique number for which h^{α}(B) = ∞, α < α(B) and h^{α}(B) = 0, α > α(B).

Now any measure can be uniquely decomposed with respect to the Hausdorff measure h^{α}, that is,

(B.36)  dρ = dρ_{αc} + dρ_{αs},

where ρ_{αc} is absolutely continuous with respect to h^{α} (i.e., ρ_{αc}(B) = 0 for all B with h^{α}(B) = 0) and ρ_{αs} is singular with respect to h^{α} (i.e., ρ_{αs} is supported on a set B with h^{α}(B) = 0).

Next, define the α-derivative of ρ by

(B.37)  D^{α}ρ(λ) = lim sup_{ε↓0} \frac{ρ((λ − ε, λ + ε))}{(2ε)^{α}}.

Theorem B.9. Let F be a Herglotz function with associated measure dρ. (i). Let C^{α}(ρ) = {λ ∈ ℝ| D^{α}ρ(λ) < ∞}, α ∈ [0, 1], then we have dρ_{αc} = χ_{C^{α}(ρ)}dρ and dρ_{αs} = (1 − χ_{C^{α}(ρ)})dρ. (ii). Set

Q^{α}ρ(λ) = lim sup_{ε↓0} ε^{1−α}ImF (λ + iε),
(B.38)  R^{α}ρ(λ) = lim sup_{ε↓0} ε^{1−α}|F (λ + iε)|.

Then D^{α}ρ(λ) and Q^{α}ρ(λ), α ∈ [0, 1], are either both zero, both in (0, ∞), or both infinite.

If α ∈ [0, 1), Q^{α}ρ(λ) can be replaced by R^{α}ρ(λ) and for any α ∈ [0, 1] we have 2^{α−1}D^{α}ρ(λ) ≤ Q^{α}ρ(λ) ≤ R^{α}ρ(λ).

In particular,

(B.39)  C^{α}(ρ) = {λ ∈ ℝ| Q^{α}ρ(λ) < ∞} ⊇ {λ ∈ ℝ| R^{α}ρ(λ) < ∞},

where all three sets are equal if α ∈ [0, 1). For α = 1 this is not true in general, as the example F (λ) = ln(λ) (see below) shows. For α = 0 we have C^{0}(ρ) = ℝ.

Again there is a value

(B.40)  α(λ) = lim inf_{ε↓0} \frac{ln ρ((λ − ε, λ + ε))}{ln ε}

such that D^{α}ρ(λ) = 0 for α < α(λ) and D^{α}ρ(λ) = ∞ for α > α(λ).

Lemma B.10. We have

(B.41)  lim_{ε↓0} \frac{ε}{i}F (λ + iε) = ρ(\{λ\})

and hence Q^{0}ρ(λ) = R^{0}ρ(λ) = ρ({λ}). Moreover, if ρ((λ_{0}, λ_{1})) = 0, we also have

(B.42)  lim_{ε↓0}(−ε)F (λ_{0} + ε) = ρ({λ_{0}}) and lim_{ε↓0} εF (λ_{1} − ε) = ρ({λ_{1}}).

Proof. We only prove the first identity of (B.42), the remaining claims being similar. After splitting the integral in (B.11) into a part over (λ_{0} − δ, λ_{1}) and one over ℝ\(λ_{0} − δ, λ_{1}), we see that it suffices to consider only the first one (since the second one is holomorphic near λ_{0}). By our assumption ρ((λ_{0}, λ_{1})) = 0, the integral is only taken over (λ_{0} −δ, λ_{0}] and because of |\frac{−ε}{λ_{0}+ε−λ}| ≤ 1, λ ∈ (λ_{0} −δ, λ_{0}], the desired result follows from the dominated convergence theorem. □

Now, we want to consider an alternate integral representation of Herglotz functions connected to the logarithm.

Let ln(z) be defined such that

(B.43)  ln(z) = ln |z| + i arg(z), −π < arg(z) ≤ π.

Then ln(z) is holomorphic and Im( ln(z)) > 0 for z ∈ ℂ_{+}, hence ln(z) is a Herglotz function. The representation of ln(z) according to (B.11) reads

(B.44)  ln(z) = ∫_{ℝ} (\frac{1}{λ − z} − \frac{λ}{1 + λ^{2}} )χ_{(−∞,0)}(λ) dλ, z ∈ ℂ_{±},

which can be easily verified.

The sum of two Herglotz functions is again a Herglotz function, similarly the composition of two Herglotz functions is Herglotz. In particular, if F (z) is a Herglotz function, the same holds for ln (F (z)) and −\frac{1}{F (z)}. Thus, using the representation (B.11) for ln (F (z)), we get another representation for F (z). The main feature of this new representation is that, by Lemma B.7, the corresponding measure is purely absolutely continuous since the imaginary part of ln(z) is uniformly bounded.

Theorem B.11. A given function F is Herglotz if and only if it has the representation

(B.45)  F (z) = exp \{c + ∫_{ℝ} (\frac{1}{λ − z} − \frac{λ}{1 + λ^{2}} ) ξ(λ) dλ\}, z ∈ ℂ_{±},

where c = ln |F (i)| ∈ ℝ, ξ ∈ L^{1}(ℝ, (1 + λ^{2})^{−1}dλ) real-valued and ξ is not identically zero. Moreover,

(B.46)  ξ(λ) = \frac{1}{π} lim_{ε↓0} Im( ln (F (λ + iε))) = \frac{1}{π} lim_{ε↓0} arg (F (λ + iε))

for a.e. λ ∈ ℝ, and 0 ≤ ξ(λ) ≤ 1 for a.e. λ ∈ ℝ. Here −π < arg(F (λ + iε)) ≤ π according to the definition of ln(z).

Proof. F (z) is Herglotz, therefore ln(F (z)) is Herglotz with

(B.47)  |Im(ln F (z))| = | arg(F (z))| ≤ π.

Hence by Theorem B.2

(B.48)  ln(F (z)) = c + ∫_{ℝ} (\frac{1}{λ − z} − \frac{λ}{1 + λ^{2}} ) dρ(λ),  z ∈ ℂ_{±},

where ρ(λ) is (by Lemma B.7) absolutely continuous with respect to Lebesgue measure, that is, dρ(λ) = ξ(λ)dλ, ξ ∈ L_{loc}^{1}(ℝ). According to (B.31), ξ is given by (B.46) and taking the Herglotz property of F into account immediately implies 0 ≤ ξ(λ) ≤ 1 for a.e. λ ∈ ℝ. The converse is easy. □

Some additional properties are collected in the following lemma.

Lemma B.12. Let F be a Herglotz function with spectral measure ρ and exponential Herglotz measure ξ(λ)dλ. (i). The set {λ ∈ ℝ|0 < ξ(λ) < 1} is a minimal support for ρ_{ac}. Moreover, if there are constants 0 < c_{1} < c_{2} < 1 such that c_{1} ≤ ξ(λ) ≤ c_{2}, λ ∈ (λ_{1}, λ_{2}), then ρ is purely absolutely continuous in (λ_{1}, λ_{2}). (ii). Fix n ∈ ℕ and set ξ_{+}(λ) = ξ(λ), ξ_{−}(λ) = 1 − ξ(λ). Then

(B.49)  ∫_{ℝ} |λ|^{n}ξ_{±}(λ)dλ < ∞

if and only if

(B.50)  ∫_{ℝ} |λ|^{n} dρ(λ) < ∞ and lim_{z→i∞} ±F (z) = ±a ∓ ∫_{ℝ} \frac{λdρ(λ)}{1 + λ^{2}} > 0.

(iii). We have

(B.51)  F (z) = ±1 + ∫_{ℝ} \frac{dρ(λ)}{λ − z} with ∫_{ℝ} dρ(λ) < ∞

if and only if

(B.52)  F (z) = ± exp ( ± ∫_{ℝ} ξ_{±}(λ)\frac{dλ}{λ − z} ) with ξ_{±} ∈ L^{1}(ℝ)

(ξ_{±} from above). In this case

(B.53)  ∫_{ℝ} dρ(λ) = ∫_{ℝ} ξ_{±}(λ)dλ.

Observe that the set

(B.54)  M̃_{ac} = {λ ∈ ℝ|0 < ξ(λ) < 1}

is a minimal support for ρ_{ac}.

In addition to these results, we will also need some facts on matrix valued measures. Let

(B.55)  dρ = ( \begin{matrix}dρ_{0,0} & dρ_{0,1} \\ dρ_{1,0} & dρ_{1,1} \end{matrix} ) , dρ_{0,1} = dρ_{1,0},

where dρ_{i,j} are (in general) signed measures. Associated with dρ is the trace measure dρ^{tr} = dρ_{0,0} + dρ_{1,1}.

We require dρ to be positive, that is, the matrix (ρ_{i,j}(B))_{0≤i,j≤1} is nonnegative for any Borel set B. Equivalently, ρ^{tr}(B) ≥ 0 and ρ_{0,0}(B)ρ_{1,1}(B) − ρ_{0,1}(B)^{2} ≥ 0 for any Borel set. This implies that dρ_{i,i} are positive measures and that

(B.56)  |ρ_{0,1}(B)| ≤ \frac{1}{2}ρ^{tr}(B).

Hence dρ_{i,j} is absolutely continuous with respect to dρ^{tr}. Assuming ρ^{tr}(ℝ) < ∞ we can also define the Borel transform

(B.57)  F (z) = ∫_{ℝ} \frac{dρ(λ)}{z − λ}.

The matrix F (λ) satisfies

(B.58)  ± Im(F (z)) ≥ 0,  z ∈ ℂ_{±}.

Next, consider the sesquilinear form

(B.59)  ⟨\underline{f}, \underline{g}⟩ = ∫_{ℝ} ∑_{i,j=0}^{1} \overline{f_{i}}(λ)g_{j}(λ)dρ_{i,j}(λ).

Suppose \underline{f} is a simple function, that is, \underline{f}(λ) = ∑_{k=1}^{n} χ_{B_{k}}(λ)(f_{k,1}, f_{k,2}), where (f_{k,1}, f_{k,2}) ∈ ℂ^{2} and B_{k} are disjoint Borel sets. Then our assumptions ensures that ∥\underline{f}∥^{2} = ∑_{k=1}^{n} ∑_{i,j=0}^{1} \overline{f_{k,i}}f_{k,j}ρ_{i,j}(B_{k}) ≥ 0. If \underline{f} is such that f_{i} ∈ L^{2}(ℝ, dρ^{tr}) we can approximate \underline{f} by simple functions implying ∥f∥ ≥ 0. As a consequence we get a separable Hilbert space L^{2}(ℝ, ℂ^{2}, dρ) with the above scalar product. As before we have

Lemma B.13. The set σ(ρ^{tr}) is precisely the spectrum σ(H̃) of the multiplication operator H̃\underline{f}(λ) = λ\underline{f}(λ), 𝔇(H̃) = \{\underline{f} ∈ L^{2}(ℝ, ℂ^{2}, dρ)|λ\underline{f}(λ) ∈ L^{2}(ℝ, ℂ^{2}, dρ)\}.

Proof. The proof is like the one of Lemma B.5 except that ρ and f_{n} have to be replaced by ρ^{tr} and \underline{f}_{n} = ρ^{tr}((λ − \frac{1}{n}, λ + \frac{1}{n}))^{−1/2}χ_{(λ− \frac{1}{n} ,λ+ \frac{1}{n})}(1, 1), respectively. □

The trace measure can even be used to diagonalize dρ as follows. Since dρ_{i,j} is absolutely continuous with respect to dρ^{tr}, there is a symmetric (integrable) matrix R(λ) such that

(B.60)  dρ(λ) = R(λ)dρ^{tr}(λ)

by the Radon-Nikodym theorem. Moreover, the matrix R(λ) is nonnegative, R(λ) ≥ 0, and given by

(B.61)  R_{i,j}(λ) = lim_{ε↓0} \frac{Im(F_{i,j}(λ + iε))}{Im(F_{0,0}(λ + iε) + F_{1,1}(λ + iε))}.

Note also that we have tr(R(λ)) = 1.

Next, there is a measurable unitary matrix U(λ) which diagonalizes R(λ), that is,

(B.62)  R(λ) = U(λ)^{∗} ( \begin{matrix}r_{1}(λ) & 0 \\ 0 & r_{2}(λ) \end{matrix} ) U(λ),

where 0 ≤ r_{1,2}(λ) ≤ 1 are the (integrable) eigenvalues of R(λ). Note r_{1}(λ)+r_{2}(λ) = 1 by tr(R(λ)) = 1. The matrix U(λ) provides a unitary operator (B.63)

\begin{matrix}L^{2}(ℝ, ℂ^{2}, dρ) & → & L^{2}(ℝ, ℂ^{2},(\begin{matrix}r_{1} & 0 \\ 0 & r_{2} \end{matrix} )dρ^{tr}) = L^{2}(ℝ, r_{1}dρ^{tr}) ⊕ L^{2}(ℝ, r_{2}dρ^{tr}) \\ \underline{f}(λ) & ↦ & U(λ)\underline{f}(λ) \end{matrix}

which leaves H̃ invariant. This allows us to investigate the spectral multiplicity of H̃.

Lemma B.14. Define

B_{1} = {λ ∈ σ(ρ^{tr})| det R(λ) = r_{1}(λ)r_{2}(λ) = 0},
(B.64)  B_{2} = {λ ∈ σ(ρ^{tr})| det R(λ) = r_{1}(λ)r_{2}(λ) > 0}.

Then H̃ = χ_{B_{1}} H̃ ⊕ χ_{B_{2}} H̃ and the spectral multiplicity of χ_{B_{1}} H̃ is one and the spectral multiplicity of χ_{B_{2}} H̃ is two.

Proof. It is easy to see that χ_{B_{1}} H̃ is unitary equivalent to multiplication by λ in L^{2}(ℝ, χ_{B_{1}}dρ^{tr}). Moreover, since r_{i}χ_{B_{2}}dρ^{tr} and χ_{B_{2}}dρ^{tr} are mutually absolutely continuous, χ_{B_{2}} H̃ is unitary equivalent to multiplication by λ in the Hilbert space L^{2}(ℝ, ℂ^{2}, 1l_{2}χ_{B_{1}}dρ^{tr}). □

Appendix C

Jacobi Difference Equations

with Mathematica

The purpose of this chapter is to show how Mathematica can be used to make some calculations with difference equations, in particular, Jacobi difference equations. I assume that you are familiar with Mathematica, version 3.0. The calculations require the packages DiscreteMath’DiffEqs’ and DiscreteMath’JacOp’ which are available via

• ftp://ftp.mat.univie.ac.at/pub/teschl/book-jac/DiffEqs.m
• ftp://ftp.mat.univie.ac.at/pub/teschl/book-jac/JacDEqs.m

and need to be stored in the AddOns/Applications/DiscreteMath subfolder of your Mathematica folder. On multi-user systems, you can install an add-on either in the central Mathematica directory (provided you have access to it) or else in your individual user’s Mathematica directory (usually ~/.Mathematica/3.0/).

The Mathematica notebook used to make the calculations below is also available

• ftp://ftp.mat.univie.ac.at/pub/teschl/book-jac/JacDEqs.nb

C.1. The package DiffEqs and first order difference equations

We first load the package DiscreteMath’DiffEqs’.

  In[1]:= Needs[”DiscreteMath‘DiffEqs‘”]

The package DiscreteMath’DiffEqs’ provides some basic commands for dealing with differences and difference equations.

S[f[n],{n,j}] shifts the sequence f[n] by j places

 FDiff[f[n],n]  (=f[n+1]-f[n]) forward difference operator
 BDiff[f[n],n]  (=f[n-1]-f[n]) backward difference

operator

SymbSum[f[j],{j,m,n}] sum for symbolic manipulations
SymbProduct[f[j],{j,m,n}] product for symbolic manipulations

Casoratian[f_{1}, .., f_{k}, n] Casoratian of a list of sequences

SplitSum[expr] splits sums into smaller parts
SplitProduct[expr] splits products into smaller parts
SplitAll[expr] splits sums and products into smaller parts
TestDiff[f[n]==g[n],{n,m}] simple test for equality based on the

fact that f = g if equality holds at one

                             point m and FDiff[f[n] -g[n],n]=0

The main purpose of the built-in Sum command is to search for a closed form. It is, however, not suitable for manipulating sums which cannot be brought to a closed form. For example,

  In[2]:= Simplify[∑_{j=1}^{n+1}f[j] − ∑_{j=1}^{n}f[j]]

 Out[2]=  ∑_{j=1}^{n+1}f[j] − ∑_{j=1}^{n}f[j]

Moreover, the built-in Sum is zero if the upper limit is smaller than the lower limit. This can lead to, at first sight, surprising results:

  In[3]:= S_{1}[n ] := SymbSum[j, {j, 1, n}]; S_{2}[n ] := Sum[j, {j, 1, n}]; S_{3}[n ] = Sum[j, {j, 1, n}]; {{S_{1}[n], S_{1}[−3]}, {S_{2}[n], S_{2}[−3]}, {S_{3}[n], S_{3}[−3]}}//MatrixForm

 Out[3]//MatrixForm=

      ( \begin{matrix}\begin{matrix}n \\ ∑ & ∗ \\ j=1 \end{matrix} j & 3 \\ \frac{1}{2}n (n + 1) & 0 \\ \frac{1}{2}n (n + 1) & 3 \end{matrix} )

(note that Mathematica assumes n to be positive). Observe that SymbSum does not look for a closed form; which can be obtained by switching to the built-in Sum:

  In[4]:= S_{1}[n] /. SymbSum−> Sum

 Out[4]=   \frac{1}{2}n (1 + n)

Observe also the definition for situations where the upper limit is smaller than the lower limit:

  In[5]:= \{ \begin{matrix}1 \\ ∑ & ∗ \\ j=0 \end{matrix} f[j], ∑_{j=0}^{1}f[j], \begin{matrix}−3 \\ ∑ & ∗ \\ j=0 \end{matrix} f[j], ∑_{j=0}^{−3}f[j]\}

Out[5]= {f[0] + f[1], f[0] + f[1], −f[−2] − f[−1], 0}

Similarly for symbolic products

  In[6]:= \{ \begin{matrix}1 \\ ∏ & ∗ \\ j=0 \end{matrix} f[j], ∏_{j=0}^{1}f[j], \begin{matrix}−3 \\ ∏ & ∗ \\ j=0 \end{matrix} f[j], ∏_{j=0}^{−3}f[j]\}

 Out[6]= \{f[0]f[1], f[0]f[1], \frac{1}{f[−2] f[−1]}, 1\}

The main purpose of the commands SplitSum and SplitProduct is to simplify expressions involving sums and products (as pointed out earlier, Mathematica’s built-in capabilities in this respect are limited). The command SplitSum will try to break sums into parts and pull out constant factors such that the rest can be done with built-in commands. In particular, it will split off extra terms

  In[7]:= SplitSum[ \begin{matrix}n+1 \\ ∑ & ∗ \\ j=0 \end{matrix} f[j]]

 Out[7]=  f[1 + n] + \begin{matrix}n \\ ∑ & ∗ \\ j=0 \end{matrix} f[j]

expand sums and pull out constants

  In[8]:= SplitSum[ \begin{matrix}n \\ ∑ & ∗ \\ j=0 \end{matrix} (hf[j] + g[j])]

 Out[8]=  h \begin{matrix}n \\ ∑ & ∗ \\ j=0 \end{matrix} f[j] + \begin{matrix}n \\ ∑ & ∗ \\ j=0 \end{matrix} g[j]

The command SplitProduct will perform similar operations with products.

  In[9]:= SplitProduct[ \begin{matrix}n−1 \\ ∏ & ∗ \\ j=0 \end{matrix} \frac{g}{f[j]}]

 Out[9]=   \frac{g^{n}f[n]}{\begin{matrix}n \\ ∏ & ∗ \\ j=0 \end{matrix} f[j]}

Finally, the command SplitAll also handles nested expressions

  In[10]:=SplitAll[ \begin{matrix}n \\ ∑ & ∗ \\ m=0 \end{matrix} \begin{matrix}m \\ ∑ & ∗ \\ j=0 \end{matrix} (f[j] + \begin{matrix}j \\ ∏ & ∗ \\ i=0 \end{matrix} xg[i])

 Out[10]=  \begin{matrix}n \\ ∑ & ∗ \\ m=0 \end{matrix} \begin{matrix}m \\ ∑ & ∗ \\ j=0 \end{matrix} f[j] + \begin{matrix}n \\ ∑ & ∗ \\ m=0 \end{matrix} \begin{matrix}m \\ ∑ & ∗ \\ j=0 \end{matrix} x^{1+j} \begin{matrix}j \\ ∏ & ∗ \\ i=0 \end{matrix} g[i]

312 C. Jacobi Difference Equations with Mathematica

We can shift a sequence f(n) by j places using S[f[n],{n,j}]. For example,

In[11]:=S[f[g[n^{2}]], {n, 1}]
Out[11]= f[g[(1 + n)^{2}]]

or

In[12]:=S[f[n, m], {n, 3}]
Out[12]= f[3 + n, m]

Similarly, we can compute forward, backward differences using FDiff[f[n],n], BDiff[f[n],n], respectively. For example,

In[13]:={FDiff[f[n], n], BDiff[f[n], n]}
Out[13]= {−f[n] + f[1 + n], f[−1 + n] − f[n]}

A simple test for equality of sequences is based on the fact that f(n) = 0 if f(n) is constant, that is, f(n + 1) = f(n) and f(n_{0}) = 0 at one fixed point.

In[14]:=TestDiff[f[n] == 0, {n, n_{0}}]
Out[14]= −f[n] + f[1 + n] == 0&&f[n_{0}] == 0

As a first application we can verify Abel’s formula (summation by parts)

In[15]:= \begin{matrix}n \\ ∑ & ∗ \\ j=m \end{matrix} g[j]FDiff[f[j], j] == (g[n]f[n + 1] − g[m − 1] f[m] + \begin{matrix}n \\ ∑ & ∗ \\ j=m \end{matrix} BDiff[g[j], j]f[j]))// TestDiff[#, \{n, m\}]&

Out[15]= True

Note also the following product rules

In[16]:=FDiff[f[n]g[n], n] == f[n]FDiff[g[n], n] + g[n + 1]FDiff[f[n], n]// Simplify
Out[16]= True
In[17]:=BDiff[f[n]g[n], n] == f[n]BDiff[g[n], n] + g[n − 1]BDiff[f[n], n]// Simplify
Out[17]= True

Next, the product

In[18]:=P[n ] := \begin{matrix}n−1 \\ ∏ & ∗ \\ j=1 \end{matrix} f[j]

satisfies the difference equation P (n + 1) = f(n)P (n) plus the initial condition P (0) = 1

  In[19]:={P[0], SplitProduct[P[n + 1] − f[n]P[n]]}
 Out[19]= {1, 0}

Moreover, the sum

  In[20]:=F[n ] := \begin{matrix}n−1 \\ ∑ & ∗ \\ j=1 \end{matrix} f[j]

satisfies the difference equation ∂F (n) = F (n + 1) − F (n) = f(n) plus the initial condition F (0) = 0

  In[21]:={F[0], SplitSum[FDiff[F[n], n] − f[n]]}
 Out[21]= {0, 0}

Similarly, the general solution of the difference equation F (n+1) = f(n)F (n)+g(n) with initial condition F (0) = F_{0} is given by

  In[22]:=F[n ] := P[n] (F_{0} + \begin{matrix}n−1 \\ ∑ & ∗ \\ i=0 \end{matrix} \frac{g[i]}{P[i + 1]})

In fact, we can easily verify this claim:

  In[23]:={F[0], SplitAll[F[n + 1] − (f[n]F[n] + g[n])]}
 Out[23]= {F_{0}, 0}

C.2. The package JacDEqs and Jacobi difference

equations

Now we come to Jacobi difference equations. The necessary commands are provided in the package DiscreteMath’JacDEqs’ which we load first.

  In[1]:= Needs[”DiscreteMath‘JacDEqs‘”]

This will also load the package DiscreteMath’DiffEqs’ if necessary. The following commands are provided in the package JacDEqs :

JacobiDE[f[n],n] computes the Jacobi difference expression,

associated with a[n], b[n] of f[n]

JacobiDE[k,f[n],n] applies the Jacobi difference expression k times.
SolutionJacobi[u] tells Mathematica that u[z,n] solves

                          JacobiDE[u[z,n],n] = z u[z,n]

 SolutionJacobi[u,x,y,m] tells Mathematica that u[z,n] solves

                          JacobiDE[u[z,n],n] = z u[z,n] and

satisfies the initial conditions u[z,m]=x,

                          u[z,m+1]=y

   SolutionJacobi[u,x,y] tells Mathematica that u[z,n,m] solves

                          JacobiDE[u[z,n,m],n] = z u[z,n,m] and

satisfies the initial conditions u[z,m,m]=x,

                    u[z,m+1,m]=y

SolutionJacobi[f,g] tells Mathematica that f[n] satisfies

                    JacobiDE[f[n],n] = g[n]

The Jacobi difference expression τ applied to a sequence gives the following result

In[2]:= JacobiDE[f[n], n]
Out[2]=  a[−1 + n]f[−1 + n] + b[n]f[n] + a[n]f[1 + n]

This can also be written as

In[3]:= JacobiDE[f[n], n] == −BDiff[a[n]FDiff[f[n], n], n] + (a[n − 1] + a[n] + b[n])f[n]//Simplify
Out[3]=  True

or

In[4]:= JacobiDE[f[n], n] == −FDiff[a[n−1]BDiff[f[n], n], n]+(a[n−1]+a[n]+ b[n])f[n]//Simplify
Out[4]=  True

The command

In[5]:= SolutionJacobi[u]

will tell Mathematica that u(z, n) satisfies the Jacobi equation τu(z) = z u(z). Let’s see how this works:

In[6]:= JacobiDE[u[z, n], n] == zu[z, n]
Out[6]=  True

We can also define solutions u(z, n, m) satisfying the initial conditions u(z, m, m) = u_{0} and u(z, m + 1, m) = u_{1} by

In[7]:= Clear[u]; SolutionJacobi[u, u_{0}, u_{1}]

Indeed, we obtain

In[8]:= {JacobiDE[u[z, n, m], n] == zu[z, n, m], u[z, m, m], u[z, m + 1, m]}
Out[8]= {True, u_{0}, u_{1}}

The solutions c(z, n, m) and s(z, n, m) are predefined by the package

In[9]:= {JacobiDE[c[z, n, m], n] == zc[z, n, m], c[z, m, m], c[z, m + 1, m]}
Out[9]= {True, 1, 0}
In[10]:={JacobiDE[s[z, n, m], n] == zs[z, n, m], s[z, m, m], s[z, m + 1, m]}
Out[10]= {True, 0, 1}

Note that you can tell Mathematica that u(n) satisfies (τ − z)u = 0 using C.3. Simple properties of Jacobi difference equations 315

In[11]:=Clear[u]; SolutionJacobi[u, (zu[#])&]

Indeed,

In[12]:=JacobiDE[u[n], n] == zu[n]
Out[12]= True

C.3. Simple properties of Jacobi difference

equations

Let us first load our package

In[1]:= Needs[”DiscreteMath‘JacDEqs‘”]

The Wronskian of two solutions W_{n}(u(z), v(z)) is defined as a(n) times the Casoratian of two solutions C_{n}(u(z), v(z))

In[2]:= W[u , v , n ] := a[n]Casoratian[u, v, n]

Please note that the argument n tells Mathematica what to consider as index. To compute the value of the Wronskian use

In[3]:= W[u[n], v[n], n] /. n−> 0
Out[3]=  a[0](−u[1]v[0] + u[0]v[1])

and not

In[4]:= W[u[n], v[n], 0]
Out[4]=  0

Defining two solutions u(z, n) and v(z, n)

In[5]:= Clear[u, v]; SolutionJacobi[u]; SolutionJacobi[v];

we can easily verify that the Wronskian is independent of n

In[6]:= FDiff[W[u[z, n], v[z, n], n], n]//Simplify
Out[6]=  0

Similarly, we can verify Green’s formula

In[7]:= \begin{matrix}n \\ ∑ & ∗ \\ j=m \end{matrix} (f[j]JacobiDE[g[j], j]− g[j]JacobiDE[f[j], j]) == W[f[n], g[n], n]− (W[f[n], g[n], n] /. n−> m − 1)//TestDiff[#, n, m]&
Out[7]=  True

We can also define a second solution using the formula

In[8]:= Clear[v]; v[z , n ] := \begin{matrix}n−1 \\ ∑ & ∗ \\ j=0 \end{matrix} \frac{u[z, n]}{a[j]u[z, j]u[z, j + 1]};

Indeed, the sequence v(z, n) defined as above satisfies the Jacobi equation

In[9]:= SplitSum[JacobiDE[v[z, n], n] − zv[z, n]]//Simplify
Out[9]=  0

The Wronskian of u(z, n) and v(z, n) is one:

In[10]:=SplitSum[W[u[z, n], v[z, n], n]]//Simplify
Out[10]= 1

Next, recall that the equality u(n) = v(n) holds if equality holds at two consecutive points n = m, m + 1 and the Jacobi difference equations applied to both sides gives the same result, that is, τu(z) = τv(z). We can define a simple test using this fact:

In[11]:=TestJac[eqn Equal, {n , m }] := Simplify[(JacobiDE[#, n] − z#) & /@ eqn]&&(eqn /. n−> m)&& (eqn /. n−> m + 1);

As an application we verify that any solution can be written as a linear combination of the two fundamental solutions c(z, n, m) and s(z, n, m) as follows

In[12]:=SolutionJacobi[u, u_{0}, u_{1}]; u[z, n, m] == u_{0}c[z, n, m] + u_{1}s[z, n, m]//TestJac[#, {n, m}]&
Out[12]= True

The solutions c(z, n, m) and s(z, n, m) have some interesting properties.

In[13]:=s[z, n, m + 1] == −\frac{a[m + 1]}{a[m]}c[z, n, m] // TestJac[#, \{n, m\}]&
Out[13]= True
In[14]:=s[z, n, m − 1] == c[z, n, m] + \frac{z − b[m]}{a[m]}s[z, n, m] // TestJac[#, \{n, m\}]&
Out[14]= True
In[15]:=c[z, n, m + 1] == \frac{z − b[m + 1]}{a[m]}c[z, n, m] + s[z, n, m] // TestJac[#, \{n, m\}]&
Out[15]= True
In[16]:=c[z, n, m − 1] == −\frac{a[m − 1]}{a[m]}s[z, n, m] // TestJac[#, \{n, m\}]&
Out[16]= True

Unfortunately, this does not work for the following formula

In[17]:=c[z, m, n] == \frac{a[n]}{a[m]}s[z, n + 1, m] // TestJac[#, \{n, m\}]&

C.3. Simple properties of Jacobi difference equations 317

Out[17]= a[−1 + n]c[z, m, −1 + n] + (−z + b[n])c[z, m, n] + a[n] c[z, m, 1 + n] == \frac{1}{a[m]}a[−1+n]^{2}s[z, n, m]−a[n](a[n]s[z, n, m]+(−b[n]+b[1+n])s[z, 1+n, m])

and we can only check it for specific values of n:

 In[18]:=Simplify[c[z, m, n] == \frac{a[n]}{a[m]}s[z, n + 1, m] /. n−> m + 3]
Out[18]= True

Similarly for the following asymptotics w = \frac{1}{z}

 In[19]:=(( ∏_{j=m+1}^{n−1}a[j])\frac{w^{n−m−2}}{−a[m]} c[\frac{1}{w}, n, m] − (1 − w ∑_{j=m+2}^{n−1}b[j] + O[w]^{2})) /. n−> m + 3

Out[19]= O[w]^{2}
 In[20]:=(( ∏_{j=m+1}^{n−1}a[j])w^{n−m−1}s[\frac{1}{w}, n, m] − (1 − w ∑_{j=m+1}^{n−1}b[j] + O[w]^{2})) /. n−> m + 3

Out[20]= O[w]^{2}
 In[21]:=(( ∏_{j=n}^{m−1}a[j])w^{m−n}c[\frac{1}{w}, n, m] − (1 − w ∑_{j=n+1}^{m}b[j] + O[w]^{2})) /. n−> m + 3

Out[21]= O[w]^{2}
 In[22]:=(( ∏_{j=n}^{m−1}a[j])\frac{w^{m−n−1}}{−a[m]} s[\frac{1}{w}, n, m] − (1 − w ∑_{j=n+1}^{m−1}b[j] + O[w]^{2})) /. n−> m + 3

Out[22]= O[w]^{2}

Defining the Jacobi matrix

 In[23]:=Delta[i , j ] := If[Simplify[i == j], 1, 0]; J[i , j ] := a[j]Delta[i − 1, j] + a[i]Delta[i + 1, j] + b[j]Delta[i, j]; JacobiMatrix[m , n ] := Table[J[m + i, m + j], {i, n − m − 1}, {j, n − m − 1}]

In[24]:=MatrixForm[JacobiMatrix[m, m + 6]]
Out[24]//MatrixForm=

       b[1 + m] a[1 + m]  0       0       0
       a[1 + m] b[2 + m] a[2 + m]  0       0
          0     a[2 + m] b[3 + m] a[3 + m]  0
          0       0     a[3 + m] b[4 + m] a[4 + m]
          0       0       0     a[4 + m] b[5 + m]

we can check

In[25]:=(s[z, #, m] == \frac{Det[zIdentityMatrix[# − m − 1] − JacobiMatrix[m, #]]}{∏_{j=m+1}^{#−1}a[j]})&[m + 4] // Simplify

Out[25]= True

or compute traces

In[26]:=Tr[M ] := ∑_{i=1}^{Length[M]} M[[i, i]]; (∗ NotneedforMathematica4.x ∗)
In[27]:=Expand[(Tr[JacobiMatrix[m, #].JacobiMatrix[m, #]] − (2 ∑_{j=m+1}^{#−2}a[j]^{2} + ∑_{j=m+1}^{#−1}b[j]^{2}))&[m + 3]]

Out[27]= 0

Let us next investigate the inhomogeneous Jacobi operator. The command

In[28]:=SolutionJacobi[f, g];

tells Mathematica that f(n) satisfies the inhomogeneous Jacobi equation τf = g. In fact,

In[29]:=JacobiDE[f[n], n] == g[n]
Out[29]= True

Next, let us express f(n) in terms of s(z, n, m). We define the kernel

In[30]:=K[z , n , j ] := \frac{s[z, n, j]}{a[j]}

which satisfies

In[31]:=Factor[{K[z, n, n − 2], K[z, n, n − 1], K[z, n, n], K[z, n, n + 1], K[z, n, n + 2]}]

Out[31]= \{\frac{z − b[−1 + n]}{a[−2 + n]a[−1 + n]}, \frac{1}{a[−1 + n]}, 0, −\frac{1}{a[n]}, −\frac{z − b[1 + n]}{a[n]a[1 + n]}\}

and the sequence f(n)

C.4. Orthogonal Polynomials 319

In[32]:=Clear[f]; f[n ] := \begin{matrix}n \\ ∑ & ∗ \\ j=1 \end{matrix} K[z, n, j]g[j]

In[33]:=SplitSum[JacobiDE[f[n], n] − zf[n]]//Factor
Out[33]= g[n]

Finally, we consider the case with a(n) = 1/2, b(n) = 0.

In[34]:=Unprotect[a, b]; a[n ] = \frac{1}{2}; b[n ] = 0;

Then we have

In[35]:=JacobiDE[(z − (z + \sqrt{z^{2} − 1})^{n}, n] == z(z + \sqrt{z^{2} − 1})^{n} // Simplify
Out[35]= True
In[36]:=JacobiDE[(z − (z − \sqrt{z^{2} − 1})^{n}, n] == z(z − \sqrt{z^{2} − 1})^{n} // Simplify
Out[36]= True

and

In[37]:=s[z, n, 0] == \frac{(z + \sqrt{z^{2} − 1})^{n} − (z − \sqrt{z^{2} − 1})^{n}}{2\sqrt{z^{2} − 1}} // TestJac[#, \{n, 0\}]&
Out[37]= True
In[38]:=c[z, n, 0] == −s[z, n − 1, 0] // TestJac[#, {n, 0}]&
Out[38]= True

C.4. Orthogonal Polynomials

We first introduce the matrices C(k), D(k), and P (z, k). We use Cm[k] for the matrix C(k) and Cd[k] for its determinant. Similar for the other three matrices.

In[1]:= m[0] := 1; Cd[0] = 1; Cd[k ] := det Cm[k]; Cm[k ?IntegerQ] := Table[m[i + j], {i, 0, k − 1}, {j, 0, k − 1}]; Dd[0] = 0; Dd[k ] := det Dm[k]; Dm[k ?IntegerQ] := Table[If[j == k − 1, m[i + j + 1], m[i + j]], {i, 0, k − 1}, {j, 0, k − 1}]; Pd[z , 0] = 0; Pd[z , k ] := det Pm[z, k]; Pm[z , k ?IntegerQ] := Table[If[i == k − 1, z^{j}, m[i + j]], {i, 0, k − 1}, {j, 0, k − 1}];

Here is how the matrices look like.

In[2]:= Print[MatrixForm[Cm[3]], ” ”, MatrixForm[Dm[3]], ” ”, MatrixForm[Pm[z, 3]]] ( \begin{matrix}1 & m[1] & m[2] \\ m[1] & m[2] & m[3] \\ m[2] & m[3] & m[4] \end{matrix} ) ( \begin{matrix}1 & m[1] & m[3] \\ m[1] & m[2] & m[4] \\ m[2] & m[3] & m[5] \end{matrix} ) ( \begin{matrix}1 & m[1] & m[2] \\ m[1] & m[2] & m[3] \\ 1 & z & z^{2} \end{matrix} )

The determinants of C(k) and D(k) appear as the first two coefficients in the expansion of the determinant of P (z, k) for large z.

In[3]:= Simplify[(Series[Pd[1/t, k], {t, 0, −1}] == Cd[k − 1](1/t)^{k−1} − Dd[k − 1](1/t)^{k−2})/.k− > 3]
Out[3]=  O[t]^{0} == 0

The polynomials s(z, k)

In[4]:= s[z , k ] := \frac{Pd[z, k]}{\sqrt{Cd[k]Cd[k − 1]}};

are orthogonal with respect to the measure ρ whose moments are m(k) = ∫ λ^{k}dρ(λ). Moreover, defining

In[5]:= a[k ] := \frac{\sqrt{Cd[k + 1]}\sqrt{Cd[k − 1]}}{Cd[k]}; a[0] := 0; b[k ] := \frac{Dd[k]}{Cd[k]} − \frac{Dd[k − 1]}{Cd[k − 1]};

we have that s(z, k) solves the corresponding Jacobi equation.

In[6]:= Simplify[a[k]s[z, k + 1] + a[k − 1]s[z, k − 1] + (b[k] − z)s[z, k] /. k− > 3]
Out[6]=  0

Moreover, the polynomials s(z, k) are orthogonal with respect to the scalar product

In[7]:= ScalarProduct[P , Q , z ] := ∑_{i=0}^{Exponent[P,z]} ∑_{j=0}^{Exponent[Q,z]} m[i + j] Coefficient[P, z, i]Coefficient[Q, z, j]/; PolynomialQ[P, z]&&PolynomialQ[Q, z]

which can be checked as follows

In[8]:= Simplify[ScalarProduct[s[z, j], s[z, k], z] == If[j == k, 1, 0] /. {k− > 3, j− > 3}]

C.5. Recursions 321

C.5. Recursions

Now we want to look at the matrix elements g_{j}(n) = ⟨δ_{n}, H^{j}δ_{n}⟩ and h_{j}(n) = 2a(n)⟨δ_{n+1}, H^{j}δ_{n}⟩, which are important when dealing with high energy expansions for the resolvent of our Jacobi operator H.

In[1]:= Needs[”DiscreteMath‘JacDEqs‘”]

For computational purpose it is most convenient to define them using the following recursions.

In[2]:= g[0, n ] = 1; h[0, n ] = 0; g[j ?IntegerQ, n ] := g[j, n] = \frac{1}{2}(h[j − 1, n] + h[j − 1, n − 1]) + b[n]g[j − 1, n]; h[j ?IntegerQ, n ] := h[j, n] = 2a[n]^{2} ∑_{l=0}^{j−1}g[j − l − 1, n]g[l, n + 1] − \frac{1}{2} ∑_{l=0}^{j−1}h[j − l − 1, n]h[l, n];

These definitions can be used to compute arbitrary coefficients as needed.

In[3]:= Simplify[g[2, n]]
Out[3]=  a[−1 + n]^{2} + a[n]^{2} + b[n]^{2}
In[4]:= Simplify[h[2, n]]
Out[4]=  2a[n]^{2}(a[−1 + n]^{2} + a[n]^{2} + a[1 + n]^{2} + b[n]^{2} + b[n]b[1 + n] + b[1 + n]^{2})

In addition, we have the third sequence γ_{j}^{β}(n) = ⟨(δ_{n+1} + βδ_{n}), H^{j}(δ_{n+1} + βδ_{n})⟩.

In[5]:= γ[j , n ] := g[j, n + 1] + \frac{β}{a[n]}h[j, n] + β^{2}g[j, n];

But now we want to see that the recursions indeed produce the matrix elements from above. For this purpose we need a delta sequence.

In[6]:= Delta[n ] := If[n == 0, 1, 0];

Using this we have

In[7]:= g[j, n] == JacobiDE[j, Delta[n − m], m] /. j− > 3 /. m− > n//Simplify
Out[7]=  True

and

In[8]:= h[j, n] == 2a[n]JacobiDE[j, Delta[n − m], m] /. j− > 3 /. m− > n + 1//Simplify
Out[8]=  True

Hence the recursions produce indeed the desired matrix elements.

There is another relation for g_{j} and h_{j} which can be verified for given j.

In[9]:= Simplify[h[j + 1, n] − h[j + 1, n − 1] == 2(a[n]^{2}g[j, n+1]−a[n−1]^{2}g[j, n−1])+b[n](h[j, n]−h[j, n−1])/.j− > 4]

Out[9]=  True

Moreover, there is a recursion involving only g_{j}.

In[10]:=Clear[g]; g[0, n ] = 1; g[1, n ] = b[n]; g[2, n ] = a[−1 + n]^{2} + a[n]^{2} + b[n]^{2}; g[j ?IntegerQ, n ] := −a[n]^{2}g[j − 2, n + 1] + a[n − 1]^{2}g[j − 2, n − 1] − b[n]^{2}g[j − 2, n] + 2b[n]g[j − 1, n] − \frac{1}{2} ∑_{l=0}^{j−2}k[j − l − 2, n]k[l, n] + 2a[n]^{2} ∑_{l=0}^{j−2}g[j − l − 2, n + 1]g[l, n] − 2b[n] ∑_{l=0}^{j−3}g[j − l − 3, n]g[l, n + 1] + b[n]^{2} ∑_{l=0}^{j−4}g[j − l − 4, n]g[l, n + 1]; k[0, n ] = −b[n]; k[j , n ] := a[n]^{2}g[j − 1, n + 1] − a[n − 1]^{2}g[j − 1, n − 1] + b[n]^{2}g[j − 1, n] − 2b[n]g[j, n] + g[j + 1, n];

Again, it can be checked, that it is equivalent to our other one for given j.

In[11]:=g[j, n] == JacobiDE[j, Delta[n − m], m] /. j− > 4 /. m− > n//Simplify
Out[11]= True

In particular, these relations will be used in the sections on the Toda and Kac van Moerbeke lattice.

C.6. Commutation methods

C.6.1. Single commutation method. In this section we want to consider the single commutation method. As first steps we load our package

In[1]:= Needs[”DiscreteMath‘JacDEqs‘”]

and introduce a few solutions of our original operator

In[2]:= SolutionJacobi[u]; SolutionJacobi[v]; SolutionJacobi[f, g]; W[f , n ] := a[n]Casoratian[f, n];

Next, we declare the commuted operator H_{σ}

In[3]:= a_{σ}[n ] := \frac{\sqrt{−a[n]}\sqrt{−a[n + 1]}\sqrt{u[λ, n]}\sqrt{u[λ, n + 2]}}{u[λ, n + 1]}; b_{σ}[n ] := λ − a[n] (\frac{u[λ, n]}{u[λ, n + 1]} + \frac{u[λ, n + 1]}{u[λ, n]}); JacobiDE_{σ}[f , n ] := a_{σ}[n] S[f, \{n, 1\}] + a_{σ}[n − 1] S[f, \{n, −1\}] + b_{σ}[n]f; W_{σ}[f , n ] := a_{σ}[n]Casoratian[f, n];

and the operator A

C.6. Commutation methods 323

In[4]:= v_{σ}[λ , n ] := \frac{1}{−a[n]u[λ, n]u[λ, n + 1]}; A[f , n ] := v_{σ}[λ, n]W[u[λ, n], f, n];

Note that v_{σ} is a solution of H_{σ}u = λu

In[5]:= Simplify[JacobiDE_{σ}[v_{σ}[λ, n], n] == λv_{σ}[λ, n]]
Out[5]=  True

Now we can verify that Au(z) satisfies H_{σ}(Au(z)) = zAu(z)

In[6]:= Simplify[JacobiDE_{σ}[A[u[z, n], n], n] == zA[u[z, n], n]]
Out[6]=  True

and that Af satisfies H_{σ}(Af) = Ag.

In[7]:= Simplify[JacobiDE_{σ}[A[f[n], n], n] == A[g[n], n]]
Out[7]=  True

Moreover, the following relations for the Wronskians are valid:

In[8]:= Simplify[W_{σ}[A[u[z, n], n], A[v[z, n], n], n] == (z − λ)W[u[z, n], v[z, n], n]]
Out[8]=  True
In[9]:= Simplify[W_{σ}[A[u[z, n], n], A[v[zz, n], n], n] == \frac{−a[n]a[n + 1]Casoratian[u[λ, n], u[z, n], v[zz, n], n]}{u[λ, n + 1]}]
Out[9]=  True

In[10]:=Simplify[W_{σ}[A[u[z, n], n], A[v[zz, n], n], n] == (z − λ)W[u[z, n], v[zz, n], n] − \frac{(z − zz)v[zz, 1 + n]W[u[z, n], u[λ, n], n]}{u[λ, 1 + n]}]

Out[10]= True

For additional material see the section on the Kac-van Moerbeke lattice below.

C.6.2. Double commutation method. In this section we want to consider the double commutation method. Again we load our package

In[1]:= Needs[”DiscreteMath‘JacDEqs‘”]

and introduce two solutions of our original operator.

In[2]:= SolutionJacobi[u]; SolutionJacobi[v]; W[f , n ] := a[n]Casoratian[f, n];

Next, we declare the doubly commuted operator H_{γ}

In[3]:= c_{γ}[n ] = \frac{1}{γ} +\begin{matrix}n \\ ∑ & ∗ \\ j=−∞ \end{matrix} u[λ, j]^{2}; a_{γ}[n ] = a[n]\frac{\sqrt{c_{γ}[n − 1]}\sqrt{c_{γ}[n + 1]}}{c_{γ}[n]}; b_{γ}[n ] = b[n] − BDiff[\frac{a[n]u[λ, n]u[λ, n + 1]}{c_{γ}[n]}, n]; JacobiDE_{γ}[f , n ] := a_{γ}[n] S[f, \{n, 1\}] + a_{γ}[n − 1]S[f, n, −1] + b_{γ}[n]f; W_{γ}[f , n ] := a_{γ}[n]Casoratian[f, n];

As a first fact we note that u_{γ}

In[4]:= u_{γ}[n ] = \frac{u[λ, n]}{\sqrt{c_{γ}[n − 1]}\sqrt{c_{γ}[n]}};

satisfies H_{γ}u_{γ} = λu_{γ}.

In[5]:= Simplify[SplitSum[JacobiDE_{γ}[u_{γ}[n], n] == λu_{γ}[n]]]
Out[5]=  True

To obtain more solutions we introduce the operator A

In[6]:= A[f , \{z , n \}] := \begin{matrix}√ & \overline{c_{γ}[n]} \\ √ & \overline{c_{γ}[n − 1]} \end{matrix}f − \frac{1}{z − λ}u_{γ}[n]W[u[λ, n], f, n];

and set

In[7]:= u_{γ}[z , n ] = A[u[z, n], {z, n}];

The sequence u_{γ}(z) satisfies H_{γ}u_{γ}(z) = z u_{γ}(z)

In[8]:= Simplify[SplitSum[JacobiDE_{γ}[u_{γ}[z, n], n] == zu_{γ}[z, n]]]
Out[8]=  True

and for u_{γ}(z, n)^{2} we have

In[9]:= Simplify[SplitSum[u_{γ}[z, n]^{2} == u[z, n]^{2} + \frac{1}{(z − λ)^{2}}BDiff[\frac{W[u[λ, n], u[z, n], n]^{2}}{c_{γ}[n]} , n]]]
Out[9]=  True

In addition, we note two equivalent forms for u_{γ}(z)

In[10]:=Simplify[SplitSum[u_{γ}[z, n] == \sqrt{\frac{c_{γ}[n − 1]}{c_{γ}[n]}}u[z, n] − \frac{1}{z − λ}u_{γ}[n]S[W[u[λ, n], u[z, n], n], \{n, −1\}]]]

In[11]:=AA[f , n ] := \begin{matrix}√ & \overline{c_{γ}[n]} \\ √ & \overline{c_{γ}[n − 1]} \end{matrix}f − u_{γ}[n]\begin{matrix}n \\ ∑ & ∗ \\ j=−∞ \end{matrix} u[λ, j](f /. n− > j);

In[12]:=Simplify[AA[u[λ, n], n] == \frac{u_{γ}[n]}{γ}]
Out[12]= True

and the following relations for the Wronskian:

In[13]:=Simplify[SplitSum[W_{γ}[u_{γ}[n], A[u[z, n], {z, n}], n]]]

Out[13]= \frac{γa[n](u[z, 1 + n]u[λ, n] − u[z, n]u[λ, 1 + n])}{1 + γ\begin{matrix}n \\ ∑ & ∗ \\ j=−∞ \end{matrix} u[λ, j]^{2}}

In[14]:=Simplify[SplitSum[W_{γ}[A[u[z, n], \{z, n\}], A[v[zz, n], \{zz, n\}], n] == W[u[z, n], v[zz, n], n] + \frac{(z − zz)W[u[λ, n], u[z, n], n]W[u[λ, n], v[zz, n], n]}{(z − λ)(zz − λ)c_{γ}[n])}]]

Out[14]= True

Finally, we turn to the solutions c_{β} and s_{β}.

In[15]:=u[λ, 0] = −Sin[α]; u[λ, 1] = Cos[α]; SolutionJacobi[sb, −Sin[α], Cos[α], 0]; SolutionJacobi[cb, Cos[α], Sin[α], 0];

They are transformed into c_{β̂,γ} and s_{β̂,γ}

In[16]:=cn = c_{γ}[0]; cp = SplitSum[c_{γ}[n + 1]] /. n− > 0; cm = SplitSum[c_{γ}[n − 1]] /. n− > 0; β = Cot[α]; ^β = \begin{matrix}√ & \overline{cm} \\ √ & \overline{cp} \end{matrix}β; δ = \frac{a[0]}{cn(z − λ)} − \frac{β}{cp(1 + {^ \atop β^{2}})}; sb_{γ}[z , n ] = \frac{\sqrt{cm}\sqrt{cp}}{cn}A[sb[z, n], \{z, n\}]; cb_{γ}[z , n ] = \frac{cp}{cn}\frac{1 + {^ \atop β^{2}}}{1 + β^{2}} (A[cb[z, n], \{z, n\}] − δA[sb[z, n], \{z, n\}]);

as can be seen from

In[17]:=Simplify[SplitSum[\frac{sb_{γ}[z, n + 1]}{sb_{γ}[z, n]}] /. n− > 0]/ ^β
Out[17]= −1

and

In[18]:=Simplify[SplitSum[\frac{cb_{γ}[z, n + 1]}{cb_{γ}[z, n]}] ^β /. n− > 0]
Out[18]= 1

C.6.3. Dirichlet commutation method. In this section we want to consider the Dirichlet commutation method. As before we load our package

In[1]:= Needs[”DiscreteMath‘JacDEqs‘”]

and introduce two solutions of our original operator.

In[2]:= SolutionJacobi[u]; SolutionJacobi[v]; u[μ_{0}, 0] = 0; W[f , n ] := a[n]Casoratian[f, n];

Next, we introduce the operator H_{(μ,σ)}

In[3]:= W_{\{μ,σ\}}[n ] = \frac{W[u[μ_{0}, n], u[μ, n], n]}{μ − μ_{0}}; a_{\{μ,σ\}}[n ] := \frac{a[n]\sqrt{W_{\{μ,σ\}}[n − 1]}\sqrt{W_{\{μ,σ\}}[n + 1]}}{W_{\{μ,σ\}}[n]}; b_{\{μ,σ\}}[n ] := b[n] − BDiff[\frac{a[n]u[μ_{0}, n]u[μ, n + 1]}{W_{\{μ,σ\}}[n]}, n]; JacobiDE_{\{μ,σ\}}[f , n ] := a_{\{μ,σ\}}[n] S[f, \{n, 1\}] + a_{\{μ,σ\}}[n − 1]S[f, \{n, −1\}] + b_{\{μ,σ\}}[n]f; W_{\{μ,σ\}}[f , n ] := a_{\{μ,σ\}}[n]Casoratian[f, n];

We note the alternate expression for b_{(μ,σ)}

In[4]:= Simplify[b_{\{μ,σ\}}[n] == μ_{0} − \frac{a[n]u[μ, n + 1]W_{\{μ,σ\}}[n − 1]}{u[μ, n]W_{\{μ,σ\}}[n]} − \frac{a[n − 1]u[μ, n − 1]W_{\{μ,σ\}}[n]}{u[μ, n]W_{\{μ,σ\}}[n − 1]}]

Out[4]=  True

and W_{(μ,σ)}

In[5]:= Simplify[SplitSum[FDiff[ \begin{matrix}n \\ ∑ & ∗ \\ j=0 \end{matrix} u[μ_{0}, j]u[μ, j] == W_{\{μ,σ\}}[n], n] ]]
Out[5]=  True

Defining the quantities

In[6]:= u_{μ}[n ] := \frac{u[μ_{0}, n]}{\sqrt{W_{\{μ,σ\}}[n − 1]}\sqrt{W_{\{μ,σ\}}[n]}}; u_{μ_{0}}[n ] := \frac{u[μ, n]}{\sqrt{W_{\{μ,σ\}}[n − 1]}\sqrt{W_{\{μ,σ\}}[n]}}; A[f , z , n ] := \begin{matrix}√ & \overline{W_{\{μ,σ\}}[n]} \\ √ & \overline{W_{\{μ,σ\}}[n − 1]} \end{matrix}f − \frac{u_{μ_{0}}[n]W[u[μ_{0}, n], f, n]}{z − μ_{0}};

Out[6]=  True

we get several solutions of our new operator.

 In[7]:= Simplify[JacobiDE_{{μ,σ}}[u_{μ}[n], n] == μu_{μ}[n]]
Out[7]=  True
 In[8]:= Simplify[JacobiDE_{{μ,σ}}[u_{μ_{0}}[n], n] == μ_{0}u_{μ_{0}}[n]]
Out[8]=  True
 In[9]:= Simplify[JacobiDE_{{μ,σ}}[A[u[z, n], z, n], n] == zA[u[z, n], z, n]]
Out[9]=  True

In addition, we note

 In[10]:=Simplify[A[u[z, n], z, n]^{2} == u[z, n]^{2} + \frac{u[μ, n]}{(z − μ_{0})^{2}}u[μ_{0}, n]BDiff[\frac{W[u[μ_{0}, n], u[z, n], n]^{2}}{W_{\{μ,σ\}}[n]} , n]]
Out[10]= True

and the following formulas for Wronskians

In[11]:=Simplify[W_{\{μ,σ\}}[u_{μ_{0}}[n], A[u[z, n], z, n], n] == \frac{W[u[μ, n], u[z, n], n]}{W_{\{μ,σ\}}[n]}]

Out[11]= True

In[12]:=Simplify[W_{\{μ,σ\}}[u_{μ}[n], A[u[z, n], z, n], n] == \frac{z − μ}{z − μ_{0}} \frac{W[u[μ_{0}, n], u[z, n], n]}{W_{\{μ,σ\}}[n]}]

Out[12]= True

In[13]:=Simplify[W_{\{μ,σ\}}[A[u[z, n], z, n], A[v[zz, n], zz, n], n] == \frac{z − μ}{z − μ_{0}}W[u[z, n], v[zz, n], n] + \frac{z − zz}{(z − μ_{0})(zz − μ_{0})} \frac{W[u[μ, n], u[z, n], n]W[u[μ_{0}, n], v[zz, n], n]}{W_{\{μ,σ\}}[n]}]

Finally, we turn to the solutions c(z, n) and s(z, n) which transform into

In[14]:=s_{\{μ,σ\}}[z , n ] = \begin{matrix}√ & \overline{W_{\{μ,σ\}}[m + 1]} \\ √ & \overline{W_{\{μ,σ\}}[m]} \end{matrix}A[s[z, n, 0], z, n] /. m− > 0; c_{\{μ,σ\}}[z , n ] = \frac{z − μ_{0}}{z − μ} (A[c[z, n, 0], z, n] + \frac{(μ − μ_{0})u[μ, m + 1]A[s[z, n, 0], z, n]}{(z − μ_{0})}u[μ, m]) /. m− > 0;

Out[14]= True

as can be seen from

In[15]:=s_{{μ,σ}}[z, 0]
Out[15]= 0
In[16]:=Simplify[s_{{μ,σ}}[z, 1]]
Out[16]= 1
In[17]:=Simplify[c_{{μ,σ}}[z, 0]]
Out[17]= 1
In[18]:=Simplify[c_{{μ,σ}}[z, 1]]
Out[18]= 0

Note also

In[19]:=Simplify[\frac{W_{\{μ,σ\}}[n + 1]}{W_{\{μ,σ\}}[n]} /. n− > 0]

Out[19]= \frac{a[0]u[μ, 0] + (−μ + μ_{0})u[μ, 1]}{a[0]u[μ, 0]}

C.7. Toda lattice

Now we come to the Toda lattice. Our package is needed again.

In[1]:= Needs[”DiscreteMath‘JacDEqs‘”]

To define the Toda hierarchy, we recall the recursions for g_{j} and h_{j} found before. We only consider the homogeneous case for simplicity.

C.7. Toda lattice 329

In[2]:= g[0, n ] = 1; h[0, n ] = 0; g[j ?IntegerQ, n ] := g[j, n] = \frac{1}{2}(h[j − 1, n] + h[j − 1, n − 1]) + b[n]g[j − 1, n]; h[j ?IntegerQ, n ] := h[j, n] = 2a[n]^{2} ∑_{l=0}^{j−1}g[j − l − 1, n]g[l, n + 1] − \frac{1}{2} ∑_{l=0}^{j−1}h[j − l − 1, n]h[l, n];

The following command will compute the r’th (homogeneous) Toda equation for us

In[3]:= TL[r ] := Simplify[ {∂_{t}a[n, t] == a[n](g[r + 1, n + 1] − g[r + 1, n]), ∂_{t}b[n, t] == h[r + 1, n] − h[r + 1, n − 1]}] /. {a[n ]− > a[n, t], b[n ]− > b[n, t]}

For example

In[4]:= TL[1]//TableForm
Out[4]//TableForm=

a^{(0,1)}[n, t] == −a[n, t](a[−1 + n, t]^{2} − a[1 + n, t]^{2} + b[n, t]^{2} − b[1 + n, t]^{2})

b^{(0,1)}[n, t] == −2a[−1 + n, t]^{2}(b[−1 + n, t] + b[n, t])
+ 2a[n, t]^{2}(b[n, t] + b[1 + n, t])

Next, we introduce the second operator in the Lax pair which will be denoted by

TodaP.
In[5]:= TodaP[r , f , n ] := −JacobiDE[r + 1, f, n] + ∑_{j=0}^{r}2(a[n]g[j, n]S[JacobiDE[r − j, f, n], {n, 1}] − h[j, n]JacobiDE[r − j, f, n]) + g[r + 1, n]f;

Here is the first one applied to a sequence f(n)

In[6]:= TodaP[0, f[n], n]
Out[6]= −a[−1 + n]f[−1 + n] + a[n]f[1 + n]

or, alternatively, in matrix form (cut off after 2 terms in both directions).

In[7]:= m = 2; Table[Coefficient[TodaP[0, f[n], n], f[n + i − j]], {i, −m, m}, {j, −m, m}]//MatrixForm m = .;

Out[7]//MatrixForm=

( \begin{matrix}0 & −a[−1 + n] & 0 & 0 & 0 \\ a[n] & 0 & −a[−1 + n] & 0 & 0 \\ 0 & a[n] & 0 & −a[−1 + n] & 0 \\ 0 & 0 & a[n] & 0 & −a[−1 + n] \\ 0 & 0 & 0 & a[n] & 0 \end{matrix} )

Similarly, the second one applied to a sequence f(n)

In[9]:= TodaP[1, f[n], n]//Simplify

Out[9]= −a[−2 + n]a[−1 + n]f[−2 + n] − a[−1 + n](b[−1 + n] + b[n])f[−1 + n] + a[n](b[n]f[1 + n] + b[1 + n]f[1 + n] + a[1 + n]f[2 + n])

The following command will display the coefficients in a handy format.

In[10]:=CoeffTodaP[k ] := Block[{dummy, i, n}, dummy = TodaP[k, f[n], n]; TableForm[ Table[{f[n + i], ” : ”, Simplify[Coefficient[dummy, f[n + i]]]}, {i, −k − 2, k + 2}]] ];

For example we obtain for r = 0

In[11]:=CoeffTodaP[0]
Out[11]//TableForm=

\begin{matrix}f[−2 + n] : & 0 \\ f[−1 + n] : & −a[−1 + n] \\ f[n] : & 0 \\ f[1 + n] : & a[n] \\ f[2 + n] : & 0 \end{matrix}

and r = 1

In[12]:=CoeffTodaP[1]
Out[12]//TableForm=

\begin{matrix}f[−3 + n] : & 0 \\ f[−2 + n] : & −a[−2 + n]a[−1 + n] \\ f[−1 + n] : & −a[−1 + n](b[−1 + n] + b[n]) \\ f[n] : & 0 \\ f[1 + n] : & a[n](b[n] + b[1 + n]) \\ f[2 + n] : & a[n]a[1 + n] \\ f[3 + n] : & 0 \end{matrix}

We can also compute the commutator of H and P_{2r+2}

In[13]:=Commutator[r , f , n ] := JacobiDE[TodaP[r, f, n], n] − TodaP[r, JacobiDE[f, n], n];
In[14]:=CoeffCommutator[r ] := Block[{dummy, i, n}, dummy = Commutator[r, f[n], n]; TableForm[Table[{f[n + i], ” : ”, Simplify[ Coefficient[dummy, f[n + i]]]}, {i, −2, 2}]] ];

and display its matrix coefficients

In[15]:=CoeffCommutator[0]
Out[15]//TableForm=

C.8. Kac-van Moerbeke lattice 331

\begin{matrix}f[−2 + n] : & 0 \\ f[−1 + n] : & a[−1 + n](b[−1 + n] − b[n]) \\ f[n] : & 2(a[−1 + n]^{2} − a[n]^{2}) \\ f[1 + n] : & a[n](b[n] − b[1 + n]) \\ f[2 + n] : & 0 \end{matrix}

C.8. Kac-van Moerbeke lattice

Finally we come to the Kac van Moerbeke lattice.

In[1]:= Needs[”DiscreteMath‘JacDEqs‘”]

Again we start with the recursions for G_{j} and H_{j} and only consider the homogeneous case for simplicity.

In[2]:= G[0, n ] = 1; H[0, n ] = 0; G[j ?IntegerQ, n ] := G[j, n] = \frac{1}{2}(H[j − 1, n] + H[j − 1, n − 2]) + (ρ[n]^{2} + ρ[n − 1]^{2})G[j − 1, n]; H[j ?IntegerQ, n ] := H[j, n] = 2(ρ[n]ρ[n+1])^{2} ∑_{l=0}^{j−1}G[j−l−1, n]G[l, n+1]− \frac{1}{2} ∑_{l=0}^{j−1}H[j−l−1, n]H[l, n];

We can easily compute the first few

In[3]:= Table[Simplify[{G[j, n], H[j, n]}], {j, 0, 1}]//TableForm
Out[3]//TableForm=

\begin{matrix}1 & 0 \\ ρ[−1 + n]^{2} + ρ[n]^{2} & 2ρ[n]^{2}ρ[1 + n]^{2} \end{matrix}

This command will display the r’th Kac-van Moerbeke equation

In[4]:= KM[r ] := Simplify[ ∂_{t}ρ[n, t] == ρ[n](G[r + 1, n + 1] − G[r + 1, n])] /. ρ[n ]− > ρ[n, t];

For example

In[5]:= KM[1]

Out[5]= ρ^{(0,1)}[n, t] == −ρ[n, t](ρ[−2 + n, t]^{2}ρ[−1 + n, t]^{2} + ρ[−1 + n, t]^{4} +ρ[−1+n, t]^{2}ρ[n, t]^{2} −ρ[1+n, t]^{2}(ρ[n, t]^{2} +ρ[1+n, t]^{2} +ρ[2+n, t]^{2})

Next, we need two pairs of sequences a_{1}, b_{1} and a_{2}, b_{2}

In[6]:= a_{1}[n ] := ρ_{e}[n]ρ_{o}[n]; b_{1}[n ] := ρ_{e}[n]^{2} + ρ_{o}[n − 1]^{2}; a_{2}[n ] := ρ_{e}[n + 1]ρ_{o}[n]; b_{2}[n ] := ρ_{e}[n]^{2} + ρ_{o}[n]^{2};

plus the corresponding quantities g_{k,j}, h_{k,j}, k = 1, 2.

332 C. Jacobi Difference Equations with Mathematica

In[7]:= g_{k} [0, n ] = 1; h_{k} [0, n ] = 0; g_{k} [j ?IntegerQ, n ] := g_{k}[j, n] = \frac{1}{2}(h_{k}[j − 1, n] + h_{k}[j − 1, n − 1]) + b_{k}[n]g_{k}[j − 1, n]; h_{k} [j ?IntegerQ, n ] := h_{k}[j, n] = 2a_{k}[n]^{2} ∑_{l=0}j − 1g_{k}[j − l − 1, n]g_{k}[l, n + 1] − \frac{1}{2} ∑_{l=0}j − 1h_{k}[j − l − 1, n]h_{k}[l, n];

They satisfy several relations which can be checked for given j:

In[8]:= Simplify[(g_{2}[j + 1, n] == ρ_{o}[n]^{2}g_{1}[j, n + 1] + ρ_{e}[n]^{2}g_{1}[j, n] + h_{1}[j, n]) /. j− > 5]
Out[8]=  True
In[9]:= Simplify[(h_{2}[j + 1, n] − h_{1}[j + 1, n] == ρ_{o}[n]^{2}(2(ρ_{e}[n + 1]^{2}g_{1}[j, n + 1] − ρ_{e}[n]^{2}g_{1}[j, n]) + (h_{1}[j, n + 1] − h_{1}[j, n]))) /. j− > 5]
Out[9]=  True
In[10]:=Simplify[(g_{1}[j + 1, n] == ρ_{e}[n]^{2}g_{2}[j, n] + ρ_{o}[n − 1]^{2}g_{2}[j, n − 1] + h_{2}[j, n − 1]) /. j− > 5]
Out[10]= True
In[11]:=Simplify[(h_{2}[j + 1, n − 1] − h_{1}[j + 1, n] == ρ_{e}[n]^{2}(−2(ρ_{o}[n]^{2}g_{2}[j, n] − ρ_{o}[n − 1]^{2}g_{2}[j, n − 1]) − (h_{2}[j, n] − h_{2}[j, n − 1]))) /. j− > 5]
Out[11]= True
In[12]:=Simplify[(h_{2}[j, n − 1] − h_{1}[j, n] == 2ρ_{e}[n]^{2}(g_{1}[j, n] − g_{2}[j, n])) /. j− > 5]
Out[12]= True
In[13]:=Simplify[(h_{1}[j + 1, n] − h_{2}[j + 1, n − 1] == 2(a_{1}[n]^{2}g_{1}[j, n]−a_{2}[n−1]^{2}g_{2}[j, n−1])−ρ_{o}[n]^{2}(h_{2}[j, n−1]−h_{1}[j, n])+ ρ_{e}[n]^{2}(h_{2}[j, n] − h_{2}[j, n − 1])) /. j− > 5]
Out[13]= True

Now we introduce the corresponding operators A, A^{∗}, H_{k}, P_{k,2r+2}, D, Q_{2r+2}, k = 1, 2.

C.8. Kac-van Moerbeke lattice 333

In[14]:=A[f , n ] := ρ_{o}[n]S[f, {n, 1}] + ρ_{e}[n]f; A^{∗}[f , n ] := ρ_{o}[n − 1]S[f, {n, −1}] + ρ_{e}[n]f; JacobiDE_{k} [f , n ] := a_{k}[n]S[f, {n, 1}] + S[a_{k}[n]f, {n, −1}] + b_{k}[n]f; JacobiDE_{k} [r ?IntegerQ, f , n ] := Nest[JacobiDE_{k}[#, n]&, f, r]; DiracDE[{f , g }, n ] := A[g, n], A^{∗}[f, n]; TodaP_{k} [r , f , n ] := −JacobiDE_{k}[r + 1, f, n] + ∑_{j=0}r(2a_{k}[n]g_{k}[j, n]S[JacobiDE_{k}[r − j, f, n], n, 1] − h_{k}[j, n]JacobiDE_{k}[r − j, f, n]) + g_{k}[r + 1, n]f; KMQ[r , {f , g }, n ] := {TodaP_{1}[r, f, n], TodaP_{2}[r, g, n]};

Again we can use Mathematica to check some simple formulas. For example A^{∗}A = H_{1}

In[15]:=Simplify[A^{∗}[A[f[n], n], n] == JacobiDE_{1}[f[n], n]]
Out[15]= True

or AA^{∗} = H_{2}

In[16]:=Simplify[A[A^{∗}[f[n], n], n] == JacobiDE_{2}[f[n], n]]
Out[16]= True

Moreover, we have H_{2}^{r}A = AH_{1}^{r}

In[17]:=Simplify[(JacobiDE_{2}[r, A[f[n], n], n] == A[JacobiDE_{1}[r, f[n], n], n]) /. r− > 5]
Out[17]= True

respectively A^{∗}H_{2}^{r} = H_{1}^{r}A^{∗}

In[18]:=Simplify[(A^{∗}[JacobiDE_{2}[r, f[n], n], n] == JacobiDE_{1}[r, A^{∗}[f[n], n], n]) /. r− > 5]
Out[18]= True

and P_{1,2r+2}A − AP_{2,2r+2} = ρ_{o}(g_{1,r+1}^{+} − g_{2,r+1})S^{+} + ρ_{e}(g_{2,r+1} − g_{1,r+1})

In[19]:=Simplify[(TodaP_{2}[r, A[f[n], n], n] − A[TodaP_{1}[r, f[n], n], n] == ρ_{o}[n](g_{1}[r + 1, n + 1] − g_{2}[r + 1, n])f[n + 1] + ρ_{e}[n](g_{2}[r + 1, n] − g_{1}[r + 1, n])f[n]) /. r− > 5]
Out[19]= True

respectively P_{1,2r+2}A^{∗} − A^{∗}P_{2,2r+2} = −ρ_{o}^{−}(g_{2,r+1}^{−} − g_{1,r+1})S^{−} + ρ_{e}(g_{2,r+1} − g_{1,r+1})

In[20]:=Simplify[(TodaP_{1}[r, A^{∗}[f[n], n], n] − A^{∗}[TodaP_{2}[r, f[n], n], n] == − ρ_{o}[n − 1](g_{2}[r + 1, n − 1] − g_{1}[r + 1, n])f[n − 1] + ρ_{e}[n](g_{2}[r + 1, n] − g_{1}[r + 1, n])f[n]) /. r− > 5]

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Glossary of notations

a_{R}(n)  = a(2n_{0} − n − 1) reflected coefficient, 15
\underline{A}_{p_{0}}  … Abel map, 283
\underline{α}_{p_{0}}  … Abel map for divisors, 284
arg(.)  … argument of a complex number, arg(z) ∈ (−π, π]
b_{R}(n)  = b(2n_{0} − n) reflected coefficient, 15
b^{(ℓ)}(n)  … expansion coefficient of ln g(z, n), 112
b^{β,(ℓ)}(n)  … expansion coefficient of ln γ^{β}(z, n), 113
B_{±}(τ)  … 69
B_{±,f}(τ)  … 69
c(z, n)  = c(z, n, 0)
c(z, n, n_{0})  … fundamental solution of τu = zu vanishing at n_{0} + 1, 6
c_{β}(z, n)  = c_{β}(z, n, 0)

c_{β}(z, n, n_{0}) … fundamental solution of τu = zu corresponding to general

boundary conditions at n_{0}, 37

ℂ  … the set of complex numbers
ℂ_{±}  = {z ∈ ℂ| ± Im(z) > 0}
C^{α}(ρ)  … support of the α-continuous part of ρ, 304
C(M_{1}, M_{2}) … continuous functions from M_{1} to M_{2}
C^{∞}(M_{1}, M_{2}) … smooth functions from M_{1} to M_{2}
χ_{Λ}(.)  … characteristic function of the set Λ
deg  … degree of a divisor (or of a polynomial), 279
δ_{n}  … δ_{n}(m) = δ_{n,m}
δ_{n,m}  … Kronecker delta δ_{n,m} = 0, if n ≠ m and δ_{n,n} = 1
dim  … dimension of a linear space
dist(A, B)  … distance between two sets A, B
D^{α}  … α-derivative, 304
D  … divisor on a Riemann surface, 279
Div(M)  … ring of divisors on M, 279
𝔇(.)  … domain of an operator

e  … exponential function, e^{z} = exp(z)
𝔼(.)  … expectation of a random variable, 87
f_{±}(k, n)  … Jost solutions, 167
g(z, n)  = G(z, n, n)
G_{r}(z, n)  … polynomial, 144
G(z, n, m) … Green function of H, 16
γ^{±}(z)  … Lyapunov exponent, 7
γ^{β}(z, n)  = g(z, n + 1) + \frac{β}{a(n)}h(z, n) + β^{2}g(z, n), 18
h^{α}  … Hausdorff measure, 304
h(z, n)  = 2a(n)G(z, n, n + 1) − 1
H  … Jacobi operator, 13
H_{±,n_{0}}^{β}  … Jacobi operator on ℓ^{2}(n_{0}, ±∞), 16
H_{n_{0}}^{β}  = H_{−,n_{0}}^{β} ⊕ H_{−,n_{0}}^{β}
H_{±,n_{0}}  = H_{±,n_{0}}^{∞}, 16
H_{±}  = H_{±,0}
H_{n_{1},n_{2}}^{β_{1},β_{2}}  … Jacobi operator on ℓ^{2}(n_{1}, n_{2}), 16
H_{n_{1},n_{2}}  = H_{n_{1},n_{2}}^{∞,∞}
H_{R}  … reflected Jacobi operator, 15
Ĥ  … Helmholtz operator, 20
ℌ  … (separable) Hilbert space
H_{r+1}(z, n) … polynomial, 145
hull(c)  … hull of a sequence, 100
Hyp_{±}(Φ) … set of hyperbolic numbers, 7
i  … complex unity, i^{2} = −1
i(D)  = dim L^{1}(D) index of specialty, 281
Im(.)  … imaginary part of a complex number
Iso_{R}(Σ)  … set of reflectionless Jacobi operators with spectrum Σ, 141
J_{n_{1},n_{2}}  … Jacobi matrix, 12
J(M)  … Jacobian variety of the Riemann surface M, 283
KM_{r}(ρ)  … Kac-van Moerbeke hierarchy, 260
K̃M_{r}(ρ)  … homogeneous Kac-van Moerbeke hierarchy, 261
Ker  … kernel of an operator
K_{r+1}^{β}(z, n) … polynomial, 150
ℓ(I, M)  … set of M-valued sequences with index set I ⊆ ℤ, 3
ℓ(I)  = ℓ(I, ℂ)
ℓ^{p}(I, M) … sequences in ℓ(I, M) being p summable, 3
ℓ^{p}(I)  = ℓ^{p}(I, ℂ)
ℓ^{∞}(I, M) … sequences in ℓ(I, M) being bounded, 3
ℓ^{∞}(I)  = ℓ^{∞}(I, ℂ)

ℓ_{0}(I, M) … sequences in ℓ(I, M) with only finitely many nonzero values,

3 ℓ_{0}(I)  = ℓ_{0}(I, ℂ)

ℓ_{±}^{p}(ℤ, M) … sequences in ℓ(I, M) being ℓ^{p} near ±∞, 3
ℓ_{±}(ℤ)  = ℓ_{±}(ℤ, ℂ)
ℓ^{2}(ℤ; w)  = weighted ℓ^{2} space, 20

L_{loc}^{1}(x_{0}, x_{1}) … space of (equivalence classes of) locally integrable functions

over the interval (x_{0}, x_{1})

L^{2}(x_{0}, x_{1}) … Hilbert space of (equivalence classes of) square integrable

functions over the interval (x_{0}, x_{1})

L(D)  = {f ∈ M(M)|f = 0 or (f) ≥ D}, 281
L^{1}(D)  = {ω ∈ M^{1}(M)|ω = 0 or (ω) ≥ D}, 281
λ  … a real number

λ_{j}^{β}(n) … eigenvalues associated with general boundary conditions,

139

m^{±}(z)  … Floquet multipliers, 116
m_{±}(z, n)  … Weyl m-functions, 27
m_{±}(z)  = m_{±}(z, 0)
m̃_{±}(z, n)  … Weyl m̃-functions, 29
m̃_{±}(z)  = m̃_{±}(z, 0)

m_{±}^{β}(z) … Weyl m-functions associated with general boundary condi-

tions, 29

m̃_{±}^{β}(z) … Weyl m̃-functions associated with general boundary condi-

tions, 39

M  … manifold or Riemann surface
M(z)  … Weyl matrix, 46

M^{β}(z) … Weyl matrix corresponding to general boundary conditions,

46

M(M)  … meromorphic functions on M
M^{1}(M)  … meromorphic forms on M
μ_{j}(n)  … Dirichlet eigenvalue, 135
ℕ  … the set of positive integers
ℕ_{0}  = ℕ ∪ {0}
N_{±}(τ)  … set of λ for which no subordinate solution exists, 68
p  … point on M
P_{Λ}(.)  … family of spectral projections of an operator
Φ(z, n, n_{0}) … fundamental matrix, 7
Φ(z, n)  = Φ(z, n, 0)
ℚ  … the set of rational numbers
q(z)  … Floquet momentum, 116
Q^{α}  … 304
r(D)  = dim L(D), 281
Re(.)  … real part of a complex number
ℝ  … the set of real numbers
R_{2r+2}^{1/2}(z)  … 143
R_{j}(n)  … residue associated with the Dirichlet eigenvalue μ_{j}(n), 136
R_{±}(k)  … reflection coefficient(constant background), 173
R_{±}(λ)  … reflection coefficient, 129
Ran  … range of an operator
R^{α}  … 304
ρ  … a Borel measure

s(z, n)  = s(z, n, 0)
s(z, n, n_{0}) … fundamental solution of τu = zu vanishing at n_{0}, 6
s_{β}(z, n)  = s_{β}(z, n, 0)

s_{β}(z, n, n_{0}) … fundamental solution of τu = zu corresponding to general

boundary conditions at n_{0}, 37

S(k)  … scattering matrix (constant background), 173
S(λ)  … scattering matrix, 129
S^{±}  … shift expressions, 4
S_{±}(H)  … scattering data of H, 174
S(τ)  … 61
S_{f}(τ)  … 61
S_{±}(τ)  … 60
S_{f,±}(τ)  … 60
σ(H)  … spectrum of an operator H, 15
σ_{ac}(H)  … absolutely continuous spectrum of H, 41, 46
σ_{sc}(H)  … singular continuous spectrum of H, 41, 46
σ_{pp}(H)  … pure point spectrum of H, 41, 46
σ_{s}(H)  = σ_{sc}(H) ∪ σ_{pp}(H)
σ_{p}(H)  … point spectrum (set of eigenvalues) of H, 42
σ_{ess}(H)  … essential spectrum of H, 62
σ_{d}(H)  … discrete spectrum of H
σ_{αc}(H)  … α-continuous spectrum of H, 42
σ_{αs}(H)  … α-singular spectrum of H, 42
σ_{j}(n)  … sign associated with the Dirichlet eigenvalue μ_{j}(n), 136
σ_{j}^{β}(n)  … sign associated with the eigenvalue λ_{j}^{β}(n), 151

sgn(.) … sign of a real number, sgn(λ) = −1 if λ < 0, sgn(0) = 0,

and sgn(λ) = 1 if λ > 0

T (k)  … transmission coefficient(constant background), 173
T (λ)  … transmission coefficient, 129
τ  … second order, symmetric difference expression, 5
\underline{τ}  = (τ_{i,j})_{i,j=1}^{g}, matrix of b periods of ζ_{j}, 278
θ(\underline{z})  … Riemann theta function, 286
TL_{r}(a, b) … Toda hierarchy, 229
T̃L_{r}(a, b) … homogeneous Toda hierarchy, 230
tr  … trace of an operator
u(z, n)  … solution of τu = zu
u_{±}(z, .)  … solution of τu = zu being square summable near ±∞, 30
U(t, s)  … unitary propagator, 230
W_{n}(., ..)  … Wronskian of two sequences, 6
ξ(λ, n)  … xi function for Dirichlet boundary conditions, 112
ξ^{β}(λ, n)  … xi function for general boundary conditions, 113
\underline{Ξ}_{p_{0}}  … vector of Riemann constants, 287
ℤ  … the set of integers
z  … a complex number

1l  … identity operator
\sqrt{z}  … standard branch of the square root, 19
\overline{z}  … complex conjugation
∥.∥_{p}  … norm in the Banach space ℓ^{p}, 3
∥.∥  … norm in the Hilbert space ℓ^{2}, 13
⟨., ..⟩_{ℌ}  … scalar product in ℌ, 4
⟨., ..⟩  … scalar product in ℓ^{2}(I), 13
⊕  … direct sum of linear spaces or operators
∂  … forward difference, 5
∂^{∗}  … backward difference, 5
∑ ∗  … indefinite sum, 9
∏ ∗  … indefinite product, 10
⊤  … transpose of a matrix
#(u)  … total number of nodes of u, 77
#_{(m,n)}(u)  … number of nodes of u between m and n, 77
#W (u_{1}, u_{2})  … total number of nodes of W_{.}(u_{1}, u_{2}), 83
#_{(m,n)}W (u_{1}, u_{2}) … number of nodes of W_{.}(u_{1}, u_{2}) between m and n, 83
[.]_{n}  … Weyl bracket, 31
[.]_{±}  … upper/lower triangular part of a matrix, 5
[.]_{0}  … diagonal part of a matrix, 5
[[x]]  = sup{n ∈ ℤ | n < x}, 76
\overline{M}^{ess}  … essential closure of a set, 303
(λ_{1}, λ_{2})  = {λ ∈ ℝ | λ_{1} < λ < λ_{2}}, open interval
[λ_{1}, λ_{2}]  = {λ ∈ ℝ | λ_{1} ≤ λ ≤ λ_{2}}, closed interval

Index

Abel transform, 5 Abel’s map, 281 Abel’s theorem, 282 Abelian differentials, 275 Additive process, 91 Almost Mathieu operator, 101 Almost periodic sequence, 100 Anderson model, 88 Antisymmetric part, 15 Asymptotic expansion, 106

Bäcklund transformation, 259 Baker-Akhiezer function, 147 Banach space, 3 Bessel function, 25 Birkhoff’s ergodic theorem, 91 Birman-Schwinger type bound, 71 Borel

measure, 295

set, 295

transform, 296 Boundary condition, 16, 18 Branch number, 274 Burchnall-Chaundy polynomial, 234

Canonical divisor, see divisor Canonical form, 40 Canonical homology basis, 274 Casoratian, 189 Cauchy

transform, 296 Characteristic polynomial, 12 Closure

essential, 299

operator, 47 Commutation method, 185

Dirichlet deformation method, 207 double, 198 single, 187

Commutator, 224 Conservation law, 252 Continued fraction, 10 Crystal

exponential interaction, 221

linear interaction (harmonic), 22 C^{∗}-algebra, 227 Cyclic vector, 40

Deficiency indices, 48

Degree, 274

Degree of a divisor, see divisor Density of states, 95 Deterministic sequences, 100 Difference expression, 4 commuting, 231 matrix representation, 4 order, 4 second order, symmetric, 5 skew-symmetric, 4 symmetric, 4 triangular parts, 5 Dirac operator, 262 Dirichlet boundary condition, 16 Dirichlet eigenvalues, 134 Discriminant, 116 Dispersion, 25 relation, 25 Divisor, 277

associated with eigenvalues, 154 canonical, 278 degree, 277 index of specialty, 279 nonspecial, 279 positive, 278 principal, 278 relatively prime, 278 Divisor class group, 278 Divisor of poles, 278 Divisor of zeros, 278 Double commutation method, 198 Dynamical system, 87

ergodic, 88

Eigenfunction expansion, 41, 45 Elliptic functions, 163 Elliptic integral, 163 Ergodic, 88 Essential spectrum, 62 Finite lattices, 231 Floquet

discriminant, 116 functions, 118 momentum, 117 monodromy matrix, 115 multipliers, 117 solutions, 117 theory, 115

Fourier transform, 20, 296 Fundamental matrix, 7 Fundamental polygon, 273 Fundamental solutions, 6

Gel’fand-Levitan-Marchenko equation, 176 Genus, 273 Green function, 16 Green’s formula, 6 Gronwall-type inequality, 177

Haar measure, 100 Hamburger moment problem, 42 Hamiltonian system, 22, 222 Hausdorff dimension, 300 Hausdorff measure, 300 Helmholtz operator, 20 Herglotz function, 296 Hilbert space, 4, 13

weighted, 20 Hilbert-Schmidt operator, 51 Holomorphic differentials, 274 Holomorphic structure, 273 Homology basis, 274 Hull, 100 Hyperbolic numbers, set of, 8 Hyperelliptic Riemann surface, 281 Hypergeometric function (generalized), 25

Index of specialty, see divisor Initial conditions, 6 Initial value problem

Kac-van Moerbeke system, 258 Toda system, 230

Intersection matrix, 293 Inverse scattering transform, 251 Isospectral operators, 140

Jacobi equation, 6

inhomogeneous, 9 Jacobi inversion problem, 288 Jacobi matrix, 12, 119

trace, 12 Jacobi operator, 13

almost periodic, 100 critical, 36 free, 19 maximal, 47 minimal, 47 periodic, 115 quasi-periodic, finite-gap, 153 random, 87 reflectionless, 133

finite-gap, 142 subcritical, 36 supercritical, 36 unbounded, 47

Jacobian variety, 281 Jensen’s inequality, 93, 102 Jordan canonical form, 116 Jost solutions, 167 Kac-van Moerbeke hierarchy, 258

homogeneous, 259 Krein spectral shift, 109

Langmuir lattice, 262 Lax equation, 224 Lax operator, 227

homogeneous, 225 Lebesgue decomposition, 41, 45 Limit circle, 39, 48 Limit point, 39, 48 Liouville number, 103 Lusin’s theorem, 295 Lyapunov exponent, 7

Matrix valued measure, 44 Measure

Baire, 100

Borel, 295

ergodic, 88

growth point, 297

invariant, 88

Jacobi, 52

probability, 87

stationary, 88 Meromorphic differentials, 274 Meromorphic functions, 274 m-function, 27, 51 m̃-function, 29, 51 Minimal positive solution, 35 Miura-type transformation, 259 Möbius transformation, 38 Moment problem, 42, 52 monic, 230 Monodromy matrix, 115

Neumann expansion, 105 Nevanlinna-Pick function, 296 Newton’s interpolation formula, 124 Node

sequence, 77

Wronskian, 83 Noether gap, 281 Nonspecial divisor, see divisor Norming constants, 174

oliton solutions, 265 Order

difference expression, 4 function, 274

Orthogonal polynomials, 40 Oscillatory, 78

Periodic coefficients, 115 Perturbation determinant, 111, 172 Phase space, 23 Picard group, 278 Pick function, 296 Plücker identity, 50 Poisson-Jensen formula, 174 Positive divisor, see divisor Principal divisor, see divisor Principal solutions, 35 Probability space, 87 Propagator, 228 Prüfer variables, 75

Quantum mechanics

supersymmetric, 15 Quasi-periodic, 153 Radon-Nikodym Theorem, 298 RAGE Theorem, 90 Random

Jacobi operator, 87

variable, 87 Recessive solutions, 35 Reflection, 14 Reflection coefficient, 174 Reflectionless, see Jacobi operator Relatively prime, see divisor Residue, 274 Resolvent, 16

Neumann expansion, 105 set, 16

Riccati equation (discrete), 10 Riemann inequality, 280 Riemann surface, 273 Riemann theta function, 283 Riemann vanishing theorem, 288 Riemann-Hilbert problem, 272 Riemann-Hurwitz relation, 274 Riemann-Roch Theorem, 280 Rotation number, 95

Scalar product, 4, 13 Scattering data, 174 Scattering matrix, 173 Scattering theory, 128, 167 Shift expressions, 4 Single commutation method, 187 Solution

weak, 234 Spectral measure

Herglotz function, 297

Jacobi operator, 28, 44 Spectral projections, 28 Spectral resolution, 28 Spectral shift function, 111 Spectrum, 15

absolutely continuous, 41, 46 point, 42, 61 pure point, 41, 46 singular continuous, 41, 46

Stable manifold, 8 Stieltjes inversion formula, 297 Stone-Weierstrass Theorem, 296 Sturm’s separation theorem, 77 Sturm-Liouville expression, 5 Subadditive ergodic theorem, 91 Subadditive process, 91 Subharmonic, 92 Submean, 92 Subordinacy, 67 Subordinacy, principle of, 70 Subordinate solution, 67 Summation by parts, 5 Supersymmetric, 15, 262 Support, 298

essential, 298

minimal, 298 Symmetric part, 15 Symmetric power of a manifold, 286 Symplectic form, 23 Symplectic transform, 23

Theorem Birkhoff’s ergodic, 91 Lusin, 295 Radon-Nikodym, 298 RAGE, 90 Riemann-Roch, 280 Stone-Weierstrass, 296 Sturm, 77 subadditive ergodic, 91 Torelli, 283

Thouless formula, 97

Tight binding approximation, 87

Toda hierarchy, 227 homogeneous, 228 stationary, 231 Topological group, 100 Torelli’s theorem, 283 Total branching number, 274 Trace measure, 45 Trace relations, 120 Transfer matrix, 7 Transformation operator, 169, 188, 211 Transmission coefficient, 174

Unstable manifold, 8 Uppersemicontinuous, 92

Variation of constants formula, 9 Vector of Riemann constants, 285 Vector valued polynomial, 44 Volterra sum equation, 126

Weak solution, 234 Weierstrass gap, 281 Weierstrass point, 281 Weyl

bracket, 31 circle, 38 matrix, 46 m-function, 27, 51 m̃-function, 29, 51

Wronskian, 6

Xi function, 111

Young’s inequality, 179

Zeros of the Riemann theta function, 286