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%%     date="12.12.95",
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%%     journal="PhD Thesis, University of Missouri, 1995", 
%%     copyright="Copyright (C) G.Teschl".
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\author{Gerald Teschl}
\title{Spectral Theory for Jacobi Operators}

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%\keywords{Jacobi operators, spectral theory, oscillation theory, commutation methods}
%\subjclass{Primary 36A10, 39A70; Secondary 34B24, 34L05}

\begin{document}

\newtheorem{thm}{Theorem}[chapter]
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\begin{titlepage}
\begin{center}
\vspace*{2.3cm}
{\Large SPECTRAL THEORY FOR JACOBI OPERATORS}\\[12mm]
\rule{10cm}{.01in}\\[5mm]
A Dissertation\\[1mm]
presented to\\[1mm]
the Faculty of the Graduate School\\[1mm]
University of Missouri-Columbia\\[5mm]
\rule{10cm}{.01in}\\[5mm]
In Partial Fulfillment\\[1mm]
of the Requirements for the Degree\\[1mm]
Doctor of Philosophy\\[5mm]
\rule{10cm}{.01in}\\[3mm]
by \\
GERALD TESCHL\\[3mm]
Professor Fritz Gesztesy, Dissertation Supervisor \\[5mm]
DECEMBER 1995
\end{center}
\end{titlepage}

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\addcontentsline{toc}{section}{Acknowledgements}

\vspace*{0.5in}
\begin{center}
ACKNOWLEDGEMENTS
\end{center}

I would like to thank my advisor Fritz Gesztesy
for two fruitful years of collaboration and for his excellent supervision.

Special thanks are also due to Karl Unterkofler and all students of Fritz:
Ronnie Dickson, Miroslaw Mystkowski, Ray Nowell, Ratnam Ratnaseelan, Walter Renger,
Don Steiger, Wilhelm Sticka, Susanne Timischl, and Mehmet Unal. Moreover, I would
like to acknowledge the hospitality of the Department of Theoretical Physics,
Technical University of Graz, Austria during the summer months of 1994 and 1995.

Last but not least I am indebted to my parents Edeltrude Teschl and Karl
Prenner and particularly to Susanne Timischl for
always being there when I
needed someone.


\newpage
\thispagestyle{plain}
\addcontentsline{toc}{section}{Abstract}
\vspace*{0.5in}
\begin{center}
{\large SPECTRAL THEORY FOR JACOBI OPERATORS}\\[3mm]
Gerald Teschl\\
Professor Fritz Gesztesy, Dissertation Supervisor\\[2mm]
\large ABSTRACT
\end{center}

%\begin{abstract}
The present thesis discusses various aspects of spectral theory for Jacobi
operators.

The first chapter reviews Weyl-Titchmarsh theory for these operators and provides
the necessary background for the following chapters.

In the second chapter we provide a comprehensive treatment of oscillation theory 
for Jacobi operators with  separated boundary conditions. Moreover, we present a
reformulation of oscillation theory in terms of Wronskians of solutions, thereby
extending the range of applicability for this theory. Furthermore, these results
are applied to establish the finiteness of the number of eigenvalues in essential
spectral gaps of perturbed periodic Jacobi operators.

In the third chapter we offer two methods of inserting eigenvalues into spectral
gaps of a given background Jacobi operator: The single commutation method which
introduces eigenvalues into the lowest spectral gap of a given semi-bounded
background Jacobi operator and the double commutation method which inserts
eigenvalues into arbitrary spectral gaps. Moreover, we prove unitary equivalence of
the commuted operators, restricted to the orthogonal complement of the eigenspace
corresponding to the newly inserted eigenvalues, with the original background
operator. Finally, we show how to iterate the above methods. Concrete applications
include an explicit realization of the isospectral torus for algebro-geometric
finite-gap Jacobi operators and the $N$-soliton solutions  of the Toda and Kac-van
Moerbeke lattice equations with respect to arbitrary background solutions.
%\end{abstract}



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\chapter[Weyl-Titchmarsh Theory]{Weyl-Titchmarsh Theory for Jacobi Operators}



\section{General Background}


First of all we need to fix some notation. For $I \subseteq \Z$ we denote by
$\ell(I)$ the set of $\C$-valued sequences $\{ f(n) \}_{n \in I}$. For $M,N \in \Z
\cup \{ \pm\infty \}$ we abbreviate $\ell(M,N) = \ell(\{n\in \Z | M < n < N\})$
(sometimes we will also write $\ell(N,-\infty)$ instead of $\ell(-\infty,N)$).
$\ell^2(I)$ is the Hilbert space of all square summable sequences with scalar
product and norm defined as 
\begin{equation}
\spr{f}{g} = \sum_{n \in I} \ol{f(n)} g(n), \quad \| f \| =
\sqrt{\spr{f}{f}}, \quad f,g \in \ell^2(I).
\end{equation}
Furthermore, $\ell_0(I)$ denotes the set of sequences with only
finitely-many values being nonzero, $\ell^1(I)$ the set of summable sequences,
$\ell^\infty(I)$ the set of bounded sequences, and $\ell^2_\pm(\Z)$ denotes
the set of sequences in $\ell(\Z)$ which are $\ell^2$ near $\pm\infty$. For
brevity we focus in the following on the case $I=\Z$.

To set the stage, we shall consider operators on $\lz$ associated with the
difference expression
\begin{equation}
(\tau f)(n) = a(n) f(n+1) + a(n-1) f(n-1) -b(n) f(n),
\end{equation}
where $a,b \in \ell(\Z)$ satisfy the following hypothesis.

\bh \label{hypoab}
Suppose $a,b \in \ell(\Z)$ satisfy
\begin{equation}
a(n) \in \R \mzer, \quad b(n) \in \R, \quad n \in \Z.
\end{equation}
\eh

If $\tau$ is limit point ($l.p.$) at both $\pm\infty$ (cf., e.g., \cite{at},
\cite{be}) then $\tau$ gives rise to a unique self-adjoint operator $H$ when
defined maximally. Otherwise we need to fix a boundary condition at each endpoint
where $\tau$ is limit circle ($l.c.$). Throughout this thesis we denote by
$u_\pm(z,.)$, $z \in \C$ nontrivial solutions of $\tau u = z u$ which satisfy
the boundary condition at $\pm\infty$ (if any) and $u_\pm(z,.) \in
\ell^2_\pm(\Z)$, respectively. $u_\pm(z,.)$ might not exist for $z \in \R$
(cf.\ Lemma~\ref{upmholz}) but if it exists it is unique up to a constant
multiple.

Picking $z_0 \in \C \bs \R$ we can characterize $H$ by
\begin{equation}
\ba{lccl} H :& \db(H) & \to & \lz \\ & f &\mapsto& \tau f \ea ,
\end{equation}
where the domain of $H$ is explicitly given by
\begin{equation}
\db(H) = \{ f \in \lz | \ba[t]{l} \tau f \in \lz, \: \lim_{n \to +\infty}
W_n(u_+(z_0),f) = 0, \\ \lim_{n \to -\infty}
W_n(u_-(z_0),f) = 0 \} \ea
\end{equation}
and
\begin{equation} \label{wr}
W_n(f,g) = a(n) \Big( f(n)g(n+1) - f(n+1)g(n) \Big)
\end{equation}
denotes the (modified) Wronskian. By $\sig(.)$, $\sig_p(.)$, and
$\sig_{ess}(.)$ we denote the spectrum, point spectrum (i.e., the set of
eigenvalues), and essential spectrum of an operator, respectively.

A simple calculation yields Green's formula for 
$f,g \in \lf$
\begin{equation} \label{gf}
\sum_{j=m}^n \Big( f (\tau g) - 
g \tau f \Big)(j) = W_n(f,g) - W_{m-1}(f,g).
\end{equation}
A glance at (\ref{gf}) shows that the modified Wronskian of 
two solutions of
\begin{equation} \label{gl}
\tau u = z u
\end{equation}
is constant and
nonzero if and only if they are linearly independent. 
If we choose $f=u(z)$, $g=\ol{u(z)}$ in
(\ref{gf}), where $u(z)$ is a solution of (\ref{gl}) 
with $z\in \C \bs \R$, we
obtain
\bea \label{weylkl}
[u(z)]_n = [u(z)]_{m-1} - \sum_{j=m}^n |u(z,j)|^2,
\eea
where $[.]_n$ denotes the Weyl bracket
\bea
[u(z)]_n = \frac{W_n(u(z),\ol{u(z)})}{2 \I \im(z)} = 
a(n) \frac{\im (u(z,n)
\ol{u(z,n+1)})}{\im(z)}, \quad n \in \Z.
\eea
Taking limits in (\ref{gf}) shows that 
$W_{\pm\infty}(f,g) = \lim_{n \to
\pm\infty} W_n(f,g)$ exists if $f,g,\tau f$, and 
$\tau g$ are square summable
near $\pm\infty$.

The following sections generalize some well-known facts  
about Sturm--Liouville operators (to be found, e.g.,
in \cite{cole},\cite{com},\cite{9},\cite{wd})
to Jacobi operators. The presented material is 
essentially taken from \cite{ak},\cite{at},\cite{be},\cite{cl}.



\section{Weyl $m$-functions}
\label{secwm}

Let $c_\alpha(z,.),s_\alpha(z,.)$ be 
the fundamental system of (\ref{gl})
corresponding to the initial conditions
\bea \label{funsy}
\ba{l@{\qquad}l} s_\alpha(z,0) = -\sin(\alpha), 
& s_\alpha(z,1) =
\cos(\alpha), \\ c_\alpha(z,0) = 
\D \frac{\cos(\alpha)}{a(0)}, &
c_\alpha(z,1) = 
\D \frac{\sin(\alpha)}{a(0)} \ea
\eea
such that
\bea
W(c_\alpha(z),s_\alpha(z)) =  1.
\eea
Next pick $\lam_1 \in \R$ and define the following 
rational function with respect
to $z$,
\bea \label{mfun}
m_N(z,\alpha) = \frac{W_N(s_\alpha(\lam_1),
c_\alpha(z))}{W_N(s_\alpha(\lam_1), 
s_\alpha(z))}, \qquad N \in 
\Z \bs \{ 0 \},
\eea
which has poles at the zeros $\lam_j(N) \in \R$, 
$\lam_1(N) \equiv \lam_1$ of $W_N(s_\alpha(\lam_1),
s_\alpha(.)) = 0$. The fact that one can rewrite $m_N(z,\alpha)$ 
with $\lam_1$ replaced by $\lam_j(N)$ together with
\bea
\lim_{z \to \lam_j(N)} W_N(s_\alpha(\lam_j(N)), 
c_\alpha(z)) &=& -1, \\
\label{limw}
\lim_{z \to \lam_j(N)} 
\frac{W_N(s_\alpha(\lam_j(N)), s_\alpha(z))
}{z-\lam_j(N)} &=& W_N(s_\alpha(\lam_j(N)), 
\frac{d}{dz} 
s_\alpha(\lam_j(N)) )
\eea
imply that all poles of $m_N(z,\alpha)$ are simple.
Using (\ref{gf}) to evaluate
(\ref{limw}) one infers that $\mp 1$ times the residue at 
$\lam_j(N)$ is given by
\bea
\gam_j(\alpha,N) = 
\Big( \sum_{n={1 \atop N+1}}^{N \atop 0}
s_\alpha(\lam_j(N),n)^2 \Big)^{-1}, 
\quad N \gele 0.
\eea
The $\gam_j(\alpha,N)$ are called norming constants. 
Hence one gets
\bea
m_N(z,\alpha) = 
\sum_j \frac{\mp\gam_j(\alpha,N)}{z - \lam_j(N)} + \left\{
\ba{l} \frac{\pm\tan(\alpha)^{\pm1}}{a(0)}, \: \alpha \in
{[0,\pi) \atop (0,\pi]}  \\ \frac{\pm z - b({1 \atop 0})}{a(0)^2}, \:
\alpha = {\pi \atop 0}\ea \right. ,
\quad N \gele 0.
\eea
(We note that $\lam_j(N)$ depend on 
$\alpha$ for $j>1$.) Furthermore, the
function
\bea
u_N(z,n) = c_\alpha(z,n) - 
m_N(z,\alpha) s_\alpha(z,n)
\eea
satisfies
\bea
\sum_{n={1 \atop N+1}}^{N \atop 0} 
|u_N(z,n)|^2 = \pm
\frac{\im(m_N(z,\alpha))}{\im(z)}, 
\quad N \gele 0,
\eea
i.e., $\pm m_N(z,\alpha)$ are Herglotz functions 
for $N \gele 0$.

Next we want to investigate the limits 
$N \to \pm\infty$. Fix $z \in \C \bs \R$.
Then, as in the Sturm-Liouville case, the 
function $m_N(z,\alpha)$ (for different values of
$\lam_1 \in \R$) lies on a  circle given by
\bea
\{ m \in \C | [c_\alpha(z) - 
m s_\alpha(z)]_N =0 \}.
\eea
Since $[.]_N$ is decreasing in $N$ for $N>0$, the circle 
corresponding to $N+1$
lies inside the circle corresponding to $N$.
Similarly for $N<0$. Hence 
these circles either tend to
a limit point or a limit circle, depending on 
whether
\bea
\sum^{\pm\infty} |s_\alpha(z,n)|^2 = \infty, 
\quad\mb{or}\quad
\sum^{\pm\infty} |s_\alpha(z,n)|^2 < \infty.
\eea
Accordingly, one says that $\tau$ is limit 
point ($l.p.$) respectively limit
circle ($l.c.$) at $\pm\infty$. One can show that this definition is
independent of $z \in \C \bs \R$. Thus the pointwise convergence of 
$m_N(z,\alpha)$ is clear 
in the $l.p.$ case. In the $l.c.$ case both Wronskians 
in (\ref{mfun}) converge
and we may set
\bea
\ti{m}_\pm(z,\alpha) = \lim_{N \to \pm\infty} m_N(z,\alpha).
\eea

\begin{rem}
(i).\ $\ti{m}_\pm(z,0)$ are not the usual Weyl $m$-functions
defined in the literature. For a connection with the standard
Weyl $m$-functions $m_\pm(z)$ see (\ref{mpl}), (\ref{mmi}).
We have chosen to introduce $\ti{m}_\pm(z,\alpha)$ in order to simplify
our notation in various places.\\
(ii).\ This explicit construction of converging 
sequences, even in the
$l.c.$ case, also works for Sturm-Liouville 
operators and
seems to be novel to the best of our knowledge. 
Previously one usually
proved the existence of such sequences using Helly's 
selection theorem (cf.,
e.g., \cite{cole}).
\end{rem}

Moreover, the above sequences are locally bounded in 
$z$ (fix an $N$ 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 (\cite{tit}, p.\ 168) they 
converge uniformly on every
compact set in $\C_\pm =
\{z \in \C | \pm\im(z)>0 \}$, implying that
$\pm\ti{m}_\pm(z,\alpha)$ are again Herglotz functions.

Upon setting
\bea
u_\pm(z,n) = c_\alpha(z,n) - 
\ti{m}_\pm(z,\alpha) s_\alpha(z,n) 
\eea
we get a function which is square summable near 
$\pm\infty$
\bea
\sum_{n= {1 \atop -\infty}}^{\infty \atop 0} 
|u_\pm(z,n)|^2 = \pm
\frac{\im(\ti{m}_\pm(z,\alpha))}{\im(z)}, 
\quad N \gele 0.
\eea
In addition,
\bea
W_{\pm\infty}(s_\alpha(\lam_1),u_\pm(z))=0,
\eea
if $\tau$ is $l.c.$ at $\pm\infty$. We remark 
that (independently of the $l.c.$
and $l.p.$ case at $\pm\infty$)
\bea
\ti{m}_\pm(z) = \ti{m}_\pm(z,0) =
\frac{-u_\pm(z,1)}{a(0)u_\pm(z,0)}
\eea
and that $\ti{m}_\pm(z,\alpha)$ can be expressed in
terms of $\ti{m}_\pm(z,\beta)$ (use that $u_\pm$ is unique 
up to a constant) by
\bea
\hspace*{1cm} \ti{m}_\pm(z,\alpha) = 
\frac{1}{a(0)} \frac{a(0) \cos(\beta-\alpha)
\ti{m}_\pm(z,\beta) - \sin(\beta-\alpha)}{a(0) \sin(\beta-\alpha)
\ti{m}_\pm(z,\beta)  + \cos(\beta-\alpha)}.
\eea




\section[Weyl-Titchmarsh Theory on $\N$]{Weyl-Titchmarsh Theory on {\Symb \N}}
\label{secwtn}

Let $H_+$ be a given self-adjoint operator 
associated with $\tau$ on $\N$
and a Dirichlet boundary condition at $n=0$. 
Abbreviate $s(z,n)=s_0(z,n)$
and let $u_+(z,n)$, $z \in \C \backslash \sig(H_+)$ be
a solution of (\ref{gl}) which  is square summable near
$\infty$ and fulfills the boundary 
condition at $\infty$ (if any). The resolvent of 
$H_+$ then reads
\bea
((H_+ - z)^{-1} f)(n) &=& \sum_{m \in \N}
G_+(z,m,n) f(m), \quad z \in \C \backslash \sig(H_+),
\eea
where
\bea
G_+(z,m,n) = \frac{1}{W(s(z),
u_+(z))} \left\{ \ba{l@{,\quad}l} s(z,n)
u_+(z,m) & m \ge n \\ s(z,m) 
u_+(z,n) & m \le n \ea
\right. .
\eea

Since $s(z,n)$ is a polynomial in $z$ we 
infer by induction 
\bea \label{relhl}
s(H_+,n) \delta_{1} = \delta_{n}, \qquad \delta_n(k) = \left\{
\ba{c@{\quad}l} 1, & k=n \\ 0, & k \ne n \ea \right. , 
\eea
implying that $\delta_{1}$ is a cyclic vector for 
$H_+$. If $E_+(.)$ denotes the
family of spectral projections corresponding to 
$H_+$ we introduce
the measure
\bea
d\rho_+(.) = d\spr{\delta_1}{E_+(.) \delta_1}.
\eea
Equation (\ref{relhl}) now shows that the 
polynomials $s(z,n), \: n \in \N$
are orthogonal with respect to this 
measure, i.e.,
\bea
\spr{s(j)}{s(k)} = 
\int\limits_{-\infty}^\infty s(\lam,j)
s(\lam,k) \, d\rho_+(\lam) = \delta_j(k) ,
\eea
implying
\bea \label{polab}
a(n) = \spr{s(n+1)}{\lam s(n)}, 
\quad b(n)= -\spr{s(n)}{\lam s(n)},
\:\: n \in \N.
\eea

Now consider the following transformation $U$ 
from the set $\ell_0(\N)$
onto the set of polynomials
\bea
(Uf)(\lam) &=& \sum_{n=1}^\infty f(n) s(\lam,n), 
\\ \label{unitary2} (U^{-1} F)(n)
 &=& \int_\R s(\lam,n) F(\lam) d\rho_+(\lam).
\eea
A simple calculation for $F(\lam) = (Uf)(\lam)$ 
shows that
\bea
\sum_{n=1}^\infty |f(n)|^2 = 
\int_\R |F(\lam)|^2 d\rho_+(\lam).
\eea
Thus $U$ extends to a unitary transformation
\bea
\tilde{U}: \ell^2(\N) \to L^2(\R,d\rho_+)
\eea
(since the set of polynomials is dense in 
$L^2(\R,d\rho_+)$, \cite{be},
Theorem VII.1.7) which maps the operator
$H_+$ to the multiplication operator by $\lam$,
\bea
\tilde{U} H_+ \tilde{U}^{-1} = \tilde{H},
\eea
where
\bea
\tilde{H} F(\lam) = \lam F(\lam), 
\quad \db(\tilde{H})= \{F \in L^2(\R,d\rho_+) |
\lam F(\lam) \in L^2(\R,d\rho_+)\}.
\eea

This is easily verified for $f \in \ell_0(\N)$. 
If $\tau$
is $l.p.$ at $\infty$ note that $\ell_0(\N)$ 
is a
core for $H_+$ and if $\tau$ is $l.c.$\ at 
$\infty$ note that
$d\rho_+$ is a pure point measure and that 
eigenfunctions are mapped onto
eigenfunctions (all finite linear combinations
 of eigenfunctions form again a
core).

This implies that the spectrum of $H_+$ can be 
characterized as follows.
Let the Lebesgue decomposition of $d\rho_+$ be 
given by
\bea
d\rho_+ = d\rho_{+,p} + d\rho_{+,ac} + 
d\rho_{+,sc},
\eea
then we have ($\rho_+(\lam) = 
\int_{(-\infty,\lam]} d\rho_+$)
\bea
\sig(H_+) &=& \{\lam \in \R |\mb{$\lam$ is a 
growth point of $\rho_+$}\},\\
\sig_{p}(H_+) &=& \{\lam \in \R |\mb{$\lam$ is 
a growth point of 
$\rho_{+,p}$}\},\\
\sig_{ac}(H_+) &=& \{\lam \in \R |\mb{$\lam$ is 
a growth
point of $\rho_{+,ac}$}\},\\
\sig_{sc}(H_+) &=& \{\lam \in \R |\mb{$\lam$ is 
a growth
point of $\rho_{+,sc}$}\}.
\eea

The Stieltjes transform of the spectral 
function $\rho_+$ is called the Weyl
$m$-function
\bea
m_+(z) = \int_\R  \frac{d\rho_+(\lam)}{z-\lam}, 
\qquad z
\in \C \bs \R.
\eea
Conversely, the spectral function $\rho_+$ can be 
recovered from $m_+(z)$ by the Stieltjes inversion formula
\bea
\rho_+(\lam) = 
\frac{-1}{\pi} \lim_{\delta \downarrow 0} 
\lim_{\eps \downarrow 0} 
\int\limits_{-\infty}^{\lam+\delta} \im(m_+(\nu + 
\I \eps)) d\nu.
\eea
We have normalized $\rho_+$ such that it is right 
continuous and satisfies
$\lim\limits_{\lam \to -\infty} \rho_+(\lam) = 0$. 
One infers
\bea \label{mpl}
m_+(z) = G_+(z,1,1) = \frac{-u_+(1)}{a(0) u_+(0)} = \ti{m}_+(z), 
\eea
and we remark that the local compact convergence of 
$m_N(z,0)$ to $\ti{m}_+(z) = m_+(z)$ implies the
convergence of the associated  spectral functions 
at every point of continuity (\cite{ad}, p.\ 332). The
second Weyl $m$-function is usually defined as
\bea \label{mmi}
m_-(z) = G_-(z,-1,-1) = \frac{-u_-(-1)}{a(-1) u_-(0)} = -\frac{
z+b(0) + a(0)^2 \ti{m}_-(z)}{a(-1)^2}.
\eea
$m_\pm(z)$, like $\pm\ti{m}_\pm(z)$, are Herglotz functions.




\section[Weyl--Titchmarsh Theory on $\Z$]{Weyl--Titchmarsh Theory on {\Symb \Z}}
\label{secwtz}

In Section~\ref{secwtn} we have dealt with the half-line $\N$. 
In this section we extend these results to all of $\Z$.

Let $H$ be a given self-adjoint operator associated 
with $\tau$. Let $u_\pm(z,n)$
be a solution of (\ref{gl}) which is square summable 
near $\pm\infty$ (provided
such a solution exists) and fulfills the boundary 
condition at $\pm\infty$ if
any. The resolvent of $H$ then reads
\bea
((H - z)^{-1} f)(n) 
&=& \sum_{m \in \Z} G(z,m,n) f(m), \quad z
\in \rho(H),
\eea
where
\bea
G(z,m,n) = \frac{1}{W(u_-(z),u_+(z))} 
\left\{ \ba{l@{\quad}l} u_-(z,n) u_+(z,m),
& m \ge n \\ u_-(z,m) u_+(z,n), & m \le n \ea
\right. .
\eea

Consider the vector-valued polynomials
\bea
\ul{S}(z,n) = 
\Big( s(z,n), c(z,n) \Big),
\eea
where $s(z,n)$, $c(z,n)$ are solutions of 
(\ref{gl}) satisfying the
initial conditions
\bea
\ba{c@{\qquad}c} s(z,0) = 0, & s(z,1) = 
1,\\ c(z,0)
= 1, & c(z,1) = 0. \ea
\eea
The analog of (\ref{relhl}) reads
\bea \label{relwl}
s(H,n) \delta_1 + c(H,n) \delta_0 = 
\delta_n.
\eea
This is obvious for $n=0,1$ and the rest follows 
from induction upon applying
$H$ to (\ref{relwl}). If $E(.)$ denotes the 
spectral resolution
of the identity corresponding to $H$ we introduce 
the measures
\bea
d\rho_{j,k}(.) = d\spr{\delta_j}{E(.) \delta_k},
\eea
and the (hermitian) matrix-valued measure
\bea
d\rho = \left( \ba{cc} d\rho_{1,1} 
& d\rho_{1,2} \\ d\rho_{2,1} & d\rho_{2,2}
\ea \right).
\eea
By (\ref{relwl}) the vector-valued polynomials are 
orthogonal with respect to
$d\rho$
\bea \nn
\spr{\ul{S}(m)}{\ul{S}(n)} 
&=& \sum_{j,k=1}^2 \int_\R S_j(\lam,m)
\; S_k(\lam,n) d\rho_{j,k}(\lam)\\ 
&\equiv& \int_\R \ul{S}(\lam,m)
d\rho(\lam)
\; \ul{S}(\lam,n) = \delta_n(m).
\eea
The analogous formulas to (\ref{polab}) 
then read
\bea
a(n) = \spr{\ul{S}(n+1)}{\lam \ul{S}(n)}, 
\quad b(n) = \spr{\ul{S}(n)}
{\lam \ul{S}(n)}, \:\: n \in \Z.
\eea

Next we consider the following transformation 
$U$ from the set $\ell_0(\Z)$
onto the set of vector-valued polynomials
\bea
(Uf)(\lam) 
&=& \sum_{n\in \Z} f(n) \ul{S}(\lam,n), \\
(U^{-1} \ul{F})(n) 
&=& \int_\R \ul{S}(\lam,n) d\rho(\lam) \ul{F}(\lam).
\eea
Again a simple calculation for 
$\ul{F}(\lam) = (Uf)(\lam)$ shows that
\bea
\sum_{n \in \Z} |f(n)|^2 = 
\int_\R \ol{\ul{F}(\lam)} d\rho(z) \ul{F}(\lam).
\eea
Thus $U$ extends to a unitary transformation
\bea
\tilde{U}: \ell^2(\Z) \to L^2(\R,d\rho)
\eea
which maps the operator $H$ to the 
multiplication operator by $\lam$,
\bea
\tilde{U} H \tilde{U}^{-1} = \tilde{H},
\eea
where
\bea
\tilde{H} \ul{F}(\lam) = z \ul{F}(\lam), 
\quad \db(\tilde{H})= \{\ul{F} \in
L^2(\R,d\rho) | \lam \ul{F}(\lam) \in L^2(\R,d\rho)\}.
\eea

In order to characterize the spectrum of 
$H$ one only needs to consider the 
trace $d\rho^t$ of $d\rho$
\bea
d\rho^t = d\rho_{1,1} + d\rho_{2,2}.
\eea

Let the Lebesgue decomposition of $d\rho^t$ be 
given by
\bea
d\rho^t = d\rho^t_{p} + d\rho^t_{ac} + d\rho^t_{sc},
\eea
then we have ($\rho^t(\lam) 
=\int_{(-\infty,\lam]} d\rho^t$, etc.)
\bea
\sig(H) &=& \{\lam \in \R |\mb{$\lam$ is a 
growth point of $\rho^t$}\},\\
\sig_{p}(H) &=& \{\lam \in \R |\mb{$\lam$ is a 
growth point of 
$\rho^t_{p}$}\},\\
\sig_{ac}(H) &=& \{\lam \in \R |\mb{$\lam$ 
is a growth
point of $\rho^t_{ac}$}\},\\
\sig_{sc}(H) &=& \{\lam \in \R |\mb{$\lam$ 
is a growth
point of $\rho^t_{sc}$}\}.
\eea

The Weyl-matrix $M(z)$ is defined as
\bea
M(z) = \int\limits_{-\infty}^\infty  
\frac{d\rho(\lam)}{z-\lam}, \qquad z
\in \C \bs \R.
\eea
Explicit evaluation yields
\bea \nn
&& M(z) = \left( \ba{cc} G(z,0,0) &
G(z,1,0) \\ 
G(z,0,1) & G(z,1,1) \ea \right)\\
&& = \frac{a(0)^{-2}}{\ti{m}_-(z) - \ti{m}_+(z)} \left( \ba{cc} 1 &
-a(0) \ti{m}_+(z) \\ -a(0) \ti{m}_+(z) & a(0)^2 \ti{m}_+(z) \ti{m}_-(z) \ea
\right).
\eea
Finally, assuming $\rho$ to be right 
continuous and normalizing
$\rho(-\infty)=0$ one obtains
\bea
\rho_{j,k}(\lam) = \frac{-1}{\pi} 
\lim_{\delta \downarrow 0} \lim_{\eps
\downarrow 0} \int\limits_{-\infty}^{\lam+\delta} 
\im(M_{j,k}(\nu + \I \eps))
d\nu, \quad 1 \le j,k \le 2.
\eea




\section{Some Useful Lemmas}


This section provides some useful results needed later on. Denote by $s(z,n)$ and
$c(z,n)$ the solutions of $\tau u = z u$ corresponding to the initial conditions
$s(z,0)=c(z,1)=0$, $s(z,1)=c(z,0)=1$.

\bl \label{upmholz}
Let $\lam_0<\lam_1$ be such that $[\lam_0,\lam_1] \cap \sig_{ess}(H_+) =
\emptyset$. Then there exists a solution $u_+(z,.)\in\ell^2_+(\Z)$ of $\tau u =z
u$ satisfying the boundary condition of $H$ at $+\infty$ (if any) which is
holomorphic with respect to $z$ for $z \in \C \bs ((-\infty,\lam_0] \cup
[\lam_1,\infty))$.  In addition, we can assume $u_+(z,.) \not\equiv 0$ and
$\ol{u_+(z,.)} = u_+(\ol{z},.)$.

Similarly,  $[\lam_0,\lam_1] \cap \sig_{ess}(H_-) = \emptyset$ implies the
existence of a solution $u_-(z,.)\in\ell_-(\Z)$ fulfilling the boundary
condition of $H$ at $-\infty$ (if any) and, as a function of $z$, satisfies
the same conditions as $u_+(z,.)$.
\el

\bpf
Explicitly, we can set
\begin{equation}
u_\pm(z,n) = \Big(\prod_{\mu \in \sig(H_+) \cap [\lam_0,\lam_1]}
\hspace*{-3mm} (z-\mu) \Big) \Big( a(0)^{-1}c(z,n) - \ti{m}_\pm(z) s(z,n) \Big).
\end{equation}
\epf


\bl \label{upmpos}
Suppose $a(n)<0$ and let $\lam < \inf\sig(H)$. Then we can assume
\begin{equation}
u_\pm(\lam,n) >0, \quad n \in \Z,
\end{equation}
\begin{equation}
n \, s(\lam,n) >0, \quad n \in \Z \mzer.
\end{equation}
They solutions $u_\pm(\lam,.)$ are called principal 
solutions of $(H-\lam)u=0$ near $\pm\infty$ in \cite{har}.
\el

\bpf
From $(H-\lam)>0$ one infers $(H_{+,n} - \lam)>0$ and hence
\begin{equation} \label{posauou}
0 < \spr{\delta_{n+1}}{(H_{+,n} -\lam)^{-1}\delta_{n+1}}
= \frac{u_+(\lam,n+1)}{-a(n) u_+(\lam,n)}
\end{equation}
showing that $u_+(\lam)$ can be chosen to be positive. Furthermore,
for $n>0$ we obtain
\begin{equation}
0 < \spr{\delta_n}{(H_+ -\lam)^{-1}\delta_n}
= \frac{u_+(\lam,n)s(\lam,n)}{-a(0) u_+(\lam,0)}
\end{equation}
implying $s(\lam,n)>0$ for $n>0$. Similarly one proves the remaining results.
\epf

Let $u_\pm(z,n)$ are solutions of $\tau u =z u$
as in Lemma \ref{upmholz}. Then Green's formula (\ref{gf}) implies
\begin{equation}
W_n(u_+(z),u_+(\ti{z})) = (z-\ti{z}) \sum_{j=n+1}^\infty u_+(z,j) u_+(\ti{z},j)
\end{equation}
and furthermore,
\bea \nn
W_n(u_+(z),\dot{u}_+(z)) &=& \lim_{\ti{z} \to z}
W_n(u_+(z),\frac{u_+(z)-u_+(\ti{z})}{z-\ti{z}})\\
&=& \sum_{j=n+1}^\infty u_+(z,j)^2.
\eea
Here the dot denotes the derivative with respect to $z$. An analogous
result holds for $u_-(z,n)$. Interchanging limit and summation can be 
justified using (cf.~Remark~\ref{rembc})
\begin{equation}
u_+(\ti{z},j) = const(\ti{z}) (H^\beta_{+,n-1} - \ti{z})^{-1} \delta_n(j)
\quad\mb{ for } j \le n
\end{equation}
(with $\beta$ such that $z \not\in \sig(H^\beta_{+,n-1})$) and the first resolvent
identity. Summarizing (compare \cite{at}, Theorem 4.2.2):

\bl \label{wuudot}
Let $u_\pm(z,n)$ be solutions of $\tau u =z u$ as in Lemma \ref{upmholz}. Then
we have
\begin{equation}
W_n(u_\pm(z),\dot{u}_\pm(z)) = \left\{ \ba{l} -\sum\limits_{j=n+1}^\infty
u_+(z,j)^2 \\ \sum\limits_{j=-\infty}^n u_-(z,j)^2 \ea \right. .
\end{equation}
\el





\chapter{Oscillation Theory}




\section{Introduction}

In 1836 Sturm originated the investigations of oscillation properties of
solutions of second-order differential and difference equations
\cite{stu}. Since then numerous extensions have been made. Especially, around
1948, Hartman and  others have shown the following in a series of papers
(\cite{har1}, \cite{har2}, \cite{har3}). For a given Sturm--Liouville operator
$H$ on $L^2(0,\infty)$, the dimension of the spectral projection $P_{(-\infty,
\lam)}(H)$ equals the number of zeros of certain solutions of $H u = \lam u$.
Moreover, the dimension of $P_{(\lam_1, \lam_2)}(H)$ can be obtained by
considering the difference of the number of zeros inside a finite interval
$(0,x)$ of two solutions corresponding to their respective spectral parameters
$\lam_1$ and $\lam_2$, and performing a limit $x \to \infty$. Only
recently it was shown in \cite{gst} by F.~Gesztesy, B.~Simon, and myself that
these limits can be avoided by using a renormalized version of oscillation
theory, that is, counting zeros of Wronskians of solutions instead.

This naturally raises the question whether similar results hold for
second-order difference equations. Despite a variety of literature on this
subject (cf., e.g., \cite{at}, \cite{bo}, \cite{do}, \cite{ft}, \cite{crit},
\cite{glz}, Sections 14 and 37, \cite{har}, \cite{hl}, \cite{hp}, \cite{hkp},
\cite{kp}, \cite{pat1}, \cite{pat2} and the references therein) only a few things
concerning the connections between oscillation properties of solutions and spectra
of the corresponding operators appear to be known. In particular, the analogs of the
aforementioned theorems seem to be unknown. Moreover, even the analog of the
well-known fact that the $n$-th eigenfunction of a Sturm-Liouville operator
(below the essential spectrum) has $n-1$ nodes is only known in the special
case of finite Jacobi operators (i.e., finite tri-diagonal matrices)
\cite{at}, Theorem~4.3.5, \cite{ft}. The present thesis aims at filling these
gaps and provides a complete solution to these problems.

Now, we want to give the reader an intuitive idea of how oscillation theory
works. In the sequel a solution of $\tau u = \lam u$, $\lam \in \R$ will
always mean a real-valued, non-zero solution. We first need to define what
we mean by a node of a real-valued sequence $u \in \ell(\Z)$. A point $n \in
\Z$, is called a node of $u$ if either
\begin{equation}
u(n) =0 \quad \mbox{or} \quad a(n)u(n)u(n+1)>0.
\end{equation}
In the special case $a(n)<0$, $n \in \Z$ a node of $u$ is precisely a sign flip
of $u$ as one would expect. In the general case, however, one has to take the sign
of $a(n)$ into account.

For simplicity we shall assume $a(n)<0$ (cf.\ Lemma~\ref{lemaeps}) and $a,b$
bounded (implying $H$ bounded) for the remainder of this section.

By Lemma~\ref{upmholz} $u_-(\lam,.)$ can be assumed to be continuous with
respect to $\lam$ as long as $\lam$ is below the essential spectrum of $H$. In
addition, $u_-(\lam,.)$ can be assumed positive for $\lam$ below the spectrum
of $H$ and hence has no nodes in this case. Increasing $\lam$ one needs to
observe three things: (i) Nodes of $u_-(\lam)$ move to the right (by
(\ref{thetaminc})) without colliding; (ii) $u_-(\lam)$ cannot pick up nodes
locally (by (\ref{sigch})); (iii) $u_-(\lam)$ cannot lose nodes at $-\infty$.
By (i) and (ii) we infer that $u_-(\lam)$ can only pick up nodes at $+\infty$.
Intuitively this happens if $u_-(\lam) \in \lz$ (or equivalently, if $\lam$ an
eigenvalue of $H$) and hence $\lim_{n\to\infty} u_-(\lam,n) =0$. Summarizing,
$u_-(\lam)$ has no nodes below the spectrum of $H$ and picks up one
additional node whenever $\lam$ is an eigenvalue of $H$. Since no nodes get
lost we are lead to (cf.\ Theorem~\ref{thmbelesssph})
\begin{equation}
\dim\Ran\, P_{(-\infty,\lam)}(H) = \#(u_-(\lam)),
\end{equation}
where $\#(u)$ denotes the total number of nodes of $u$ and $P_\Omega(H)$ is the
spectral projection of $H$ corresponding to the Borel set $\Omega \subseteq
\R$. As a corollary we conclude, as already anticipated, that the
$n$-th eigenfunction (below the essential spectrum) has $n-1$ nodes.

To obtain the number of eigenvalues between two given values $\lam_1$ and
$\lam_2$ it seems natural to consider $\#(u_-(\lam_2)) - \#(u_-(\lam_1))$.
This gives nothing new below the essential spectrum and otherwise we have
$\#(u)=\infty$ for any solution of $\tau u = \lam u$ with $\lam$ above the
infimum of the essential spectrum. Hence, a naive use of oscillation theory in
the latter case yields $\infty-\infty$. There are two ways to overcome this
problem. The first, due to \cite{har2} in the case of differential operators,
uses a limiting procedure which only works for half-line operators and can be
found in Theorem~\ref{thmhartman}. The second, due to \cite{gst} in the case
of differential operators, uses the fact that the nodes of the Wronskian of
two solutions $u_1, u_2$ corresponding to $\lam_1, \lam_2$, respectively,
essentially counts the additional nodes of $u_2$ with respect to $u_1$ (cf.\
Corollary~\ref{cornodw}). In this sense the Wronskian comes with a built-in
renormalization. Moreover, the nodes of Wronskians behave similar to the
nodes of solutions and satisfy the above properties (i), (ii), and (iii) as
well. Hence, similar techniques apply.

To give rigorous proofs for the indicated results we first introduce and
investigate Pr\"ufer variables in Section~\ref{secpruf}. They will be our
main tool in Section~\ref{secosc} and Section~\ref{secrnosc} where our major
theorems are derived. Section~\ref{secapp} uses the results of
Section~\ref{secosc} and \ref{secrnosc} to investigate the spectra of
short-range perturbations of periodic Jacobi operators.




\section{Pr\"ufer Variables}
\label{secpruf}


For the rest of this chapter we assume for convenience

\bh \label{hypoaposb}
Suppose
\begin{equation}
a,b \in \ell(\Z), \qquad a(n)< 0, b(n) \in \R.
\end{equation}
\eh

We remark that the case $a(n) \ne 0$ can be reduced 
to the case $a(n)>0$ or $a(n)<0$ (cf., e.g., 
\cite{eil}, p.\ 141). In fact one
has

\begin{lem} \label{lemaeps}
Assume (H.\ref{hypoab}) and let $H$ be a Jacobi operator associated with the
difference expression (\ref{diffex}). Introduce $a_\eps$ by
\bea
a_\eps(n) = \eps(n)a(n), \; \eps(n) \in\{+1, -1\},
\quad n\in\Z
\eea
and the unitary operator $U_\eps$ by
\bea
U_\eps =\{ \tilde{\eps}(n) \delta_{m,n} \}_{m,n\in\Z}, 
\quad \tilde{\eps}(n)
\in\{+1,-1\}, \; 
\tilde{\eps}(n) \tilde{\eps}(n+1) = \eps(n).
\eea
Then $H_\eps$ defined as
\bea
H_\eps = U_\eps^{-1} H U_\eps,
\eea
is associated with the difference expression
\bea
(\tau_\eps f)(n) = a_\eps(n) f(n+1) + 
a_\eps(n-1) f(n-1) - b_\eps(n) f(n).	
\eea
In particular, $H_\eps$ is unitarily equivalent 
to $H$. 
\end{lem}

In addition, by a solution of $\tau u = \lam u$, $\lam \in \R$ we will
always mean a real-valued solution not vanishing identically.

Given a solution $u(\lam,.)$ of $\tau u =\lam u$, $\lam\in\R$ we introduce
Pr\"ufer variables
$\rho_u(\lam,.)$, $\theta_u(\lam,.)$ via
\bea \label{pruef1}
u(\lam,n) &=& \rho_u(\lam,n) \sin \theta_u(\lam,n), \\ \label{pruef2}
u(\lam,n+1) &=& \rho_u(\lam,n) \cos \theta_u(\lam,n).
\eea
Notice that the Pr\"ufer angle $\theta_u(\lam,.n)$ is only defined up to an
additive integer multiple of $2\pi$ (which depends on $n$).

Inserting (\ref{pruef1}), (\ref{pruef2}) into $(\tau - \lam) u=0$ yields
\begin{equation} \label{ricphi}
a(n) \cot \theta_u(\lam,n) + a(n-1) \tan \theta_u(\lam,n-1) = b(n) + \lam,
\end{equation}
\begin{equation} \label{eqrho}
\rho_u(\lam,n) \sin \theta_u(\lam,n) = \rho_u(\lam,n-1) \cos
\theta_u(\lam,n-1).
\end{equation}
Equation (\ref{ricphi}) is a discrete Riccati equation (cf.\ \cite{hp}) for
$\cot\theta_u(n)$ and (\ref{eqrho}) can be solved if $\theta_u(n)$ is known
provided it is replaced by
\begin{equation}
a(n) \rho_u(\lam,n) = a(n-1) \rho_u(\lam,n-1) =0
\end{equation}
if $\sin \theta_u(\lam,n) = \cos \theta_u(\lam,n-1) =0$ (use $\tau u =\lam u$
and (\ref{sigch}) below). The Wronskian of two solutions
$u_{1,2}(\lam_{1,2},n)$ reads
\begin{equation}
W_n(u_1(\lam_1), u_2(\lam_2)) = a(n) \rho_{u_1}(\lam_1,n) \rho_{u_2}(\lam_2,n)
\sin(\theta_{u_1}(\lam_1,n) - \theta_{u_2}(\lam_2,n)).
\end{equation}

The next lemma considers nodes of solutions and their Wronskians more
closely.

\bl
Let $u_{1,2}$ be solutions of $\tau u_{1,2} =\lam_{1,2} u_{1,2}$
corresponding to $\lam_1 \ne \lam_2$, respectively. Then
\begin{equation} \label{sigch}
u_1(n) =0 \quad\Rightarrow\quad u_1(n-1)u_1(n+1)<0.
\end{equation}
Moreover, suppose $W_n(u_1,u_2) = 0$ but $W_{n-1}(u_1,u_2) W_{n+1}(u_1,u_2) \ne
0$, then
\begin{equation}
W_{n-1}(u_1,u_2) W_{n+1}(u_1,u_2) < 0.
\end{equation}
Otherwise, if $W_n(u_1,u_2) = W_{n+1}(u_1,u_2) =0$, then necessarily
\begin{equation}
u_1(n+1)=u_2(n+1)=0, \quad\mbox{and}\quad W_{n-1}(u_1,u_2) W_{n+2}(u_1,u_2)< 0.
\end{equation}
\el

\bpf
The fact $u(n)=0$ implies $u_1(n-1)u_1(n+1) \ne 0$ (otherwise $u_1$ vanishes
identically) and $a(n) u_1(n+1) = -a(n-1) u_1(n-1)$ (from $\tau u =\lam u$) shows
$u_1(n-1) u_1(n+1)<0$.

Next, $W_n(u_1,u_2)=0$ is equivalent to $u_1(n) = c
u_2(n)$, $u_1(n+1) = c u_2(n+1)$ for some $c \ne 0$ and from (\ref{gf}) we
infer
\begin{equation}
W_{n+1}(u_1,u_2) - W_n(u_1,u_2) = (\lam_2-\lam_1) u_1(n+1) u_2(n+1).
\end{equation}
Applying the above formula gives
\begin{equation}
W_{n-1}(u_1,u_2) W_{n+1}(u_1,u_2) = - c^2
(\lam_2-\lam_1)^2 u_1(n)^2 u_1(n+1)^2
\end{equation}
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 $W_{n-1}(u_1,u_2) W_{n+1}(u_1,u_2) =
- (\lam_2-\lam_1)^2 u_1(n-1) u_1(n+2) u_2(n-1) u_2(n+2)$. Hence the claim follows
from the first part.
\epf

We can make the Pr\"ufer angel $\theta_u(\lam,.)$ unique by fixing, for
instance, $\theta_u(\lam,0)$ and requiring
\begin{equation} \label{normalth}
\intp{\theta_u(\lam,n)/ \pi} \le \intp{\theta_u(\lam,n+1)/ \pi} \le
\intp{\theta_u(\lam,n)/ \pi} +1,
\end{equation}
where
\begin{equation}
\intp{x} = \sup \{n \in \Z \,|\, n<x \}.
\end{equation}

\bl
Let $\Omega \subseteq \R$ be an interval. Suppose $u(\lam,n)$ is continuous 
with respect to $\lam \in \Omega$ and (\ref{normalth}) holds for one $\lam_0
\in \Omega$. Then it holds for all $\lam \in \Omega$ if we require
$\theta_u(.,n) \in C(\Omega)$.
\el

\bpf
Fix $n$ and set
\begin{equation}
\theta_u(\lam,n) = k \pi + \delta(\lam), \quad \theta_u(\lam,n+1) = k \pi +
\Delta(\lam), \quad k\in\Z,
\end{equation}
where $\delta(\lam) \in (0,\pi]$, $\Delta(\lam) \in (0,2\pi]$. If
(\ref{normalth}) should break down then by continuity we must have one of the
following cases for some $\lam_1 \in \Omega$. (i) $\delta(\lam_1)=0$ and
$\Delta(\lam_1) \in (\pi,2\pi)$, (ii) $\delta(\lam_1)=\pi$ and
$\Delta(\lam_1) \in (0,\pi)$, (iii) $\Delta(\lam_1)=0$ and
$\delta(\lam_1) \in (0,\pi)$, (iv) $\Delta(\lam_1)=2\pi$ and
$\delta(\lam_1) \in (0,\pi)$. Abbreviate $R = \rho(\lam_1,n)
\rho(\lam_1,n+1)$. Case (i) implies $0> \sin(\Delta(\lam_1))  =
\cos(k\pi)\sin(k\pi + \Delta(\lam_1)) = R^{-1} u(\lam_1,n+1)^2>0$,
contradicting (i). Case (ii) is similar. Case (iii) implies
$\delta(\lam_1)=\pi/2$ and hence $1=\sin(k\pi+\pi/2)\cos(k\pi) = R^{-1}
u(\lam_1,n) u(\lam_1,n+2)$ contradicting (\ref{sigch}). Again, case (iv) is
similar.
\epf

Let us call a point $n \in \Z$ a node of a solution $u$ if either
$u(n)=0$ or $a(n)u(n)u(n+1)>0$. Then, $\intp{\theta_u(n)/ \pi} =
\intp{\theta_u(n+1)/ \pi}$ implies no node at $n$. Conversely, if
$\intp{\theta_u(n+1)/ \pi} = \intp{\theta_u(n)/ \pi} +1$, then $n$ is a node by
(\ref{sigch}). 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 shall 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 $u(m) \ne 0$. Hence we conclude

\bl \label{nbofu}
Let $m<n$. Then we have for any solution $u$
\begin{equation} \label{nbunm}
\#_{(m,n)}(u) = \intp{\theta_u(n)/ \pi} - \lim_{\eps\downarrow 0}
\intp{\theta_u(m)/ \pi +\eps}
\end{equation}
and
\begin{equation}
\#(u) = \lim_{n \to \infty} \Big( \intp{\theta_u(n)/ \pi} - \intp{\theta_u(-n)/
\pi} \Big).
\end{equation}
\el

Next, we recall the well-known analog of Sturm's theorem for differential
equations and include a proof for the sake of completeness
(cf.,~\cite{at},\cite{pat2}).

\bl \label{sturm}
Let $u_{1,2}$ be solutions of $\tau u =\lam u$ corresponding to $\lam_1 \le
\lam_2$. Suppose $m<n$ are two consecutive points which are either nodes of
$u_1$ or zeros of $W_{.}(u_1,u_2)$ (the cases $m=-\infty$ or $n=+\infty$ are
allowed if $u_1$ and $u_2$ are both in $\ell_\pm^2(\Z)$ and
$W_{\pm\infty}(u_1,u_2) = 0$ 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 $m_1< \cdots < m_k$ 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
\begin{equation} \label{diffnb}
\#_{(m,n)}(u_2) \ge \#_{(m,n)}(u_1) -1.
\end{equation}
\el

\bpf
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 $m<j<n$, $u_1(n) \ge 0$, and
$u_2(j)>0$ for $m\le j\le n$. Moreover, $u_1(m) \le 0$, $u_1(n+1)<0$ and $u_2(n+1)
\ge 0$ provided $m,n$ are finite. By Green's formula (\ref{gf})
\begin{equation}
0 \le (\lam_2 - \lam_1) \sum_{j=m+1}^n u_1(j) u_2(j) = W_n(u_1,u_2)
- W_m(u_1,u_2).
\end{equation}
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
$u_2$ lower or equal than $m_{k+1}$ by $n_k > n_{k-1} > \cdots$. If $n_k >
m_k$ we are done since there are $k-1$ nodes $n$ such that $m_1 \le n \le
m_k$ by induction hypothesis. Otherwise we can find $k_0$, $0 \le k_0 \le k$
such that
$m_j=n_j$ for $1+k_0 \le j \le k$. If $k_0=0$ we are clearly done and we can
suppose $k_0 \ge 1$. By induction hypothesis it suffices to show that there are
$k-k_0$ nodes $n$ of $u_2$ with $m_{k_0} \le n \le m_{k+1}$. By assumption 
$m_j=n_j$, $1+k_0 \le j \le k$ are the only nodes $n$ of $u_2$ such that
$m_{k_0} \le n \le m_{k+1}$. 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 $0<\sum_{j=m+1}^n u_1(j) u_2(j)$ and we can proceed as
in the first part to obtain a contradiction.
\epf

We call $\tau$ oscillatory if one solution of $\tau u =0$ has an infinite
number of nodes. In addition, we call $\tau$ oscillatory at $\pm\infty$ if one
solution of $\tau u =0$ has an infinite number of nodes near $\pm\infty$.
We remark that if one solution of $(\tau - \lam)u =0$ has infinitely many nodes
so has any other (corresponding to the same $\lam$) by (\ref{diffnb}).
Furthermore, $\tau - \lam_1$ oscillatory implies $\tau - \lam_2$ oscillatory
for all $\lam_2>\lam_1$ (again by (\ref{diffnb})).

Now we turn to the special solution $s(\lam,n)$ characterized via the initial
conditions $s(\lam,0)=0$, $s(\lam,1)=1$. As in Lemma~\ref{wuudot} we infer
\bea
W_n(s(\lam),\dot{s}(\lam)) &=& \sum_{j=n+1}^0 s(\lam,j)^2, \quad n<-1, \\
W_n(s(\lam),\dot{s}(\lam)) &=& \sum_{j=1}^n s(\lam,j)^2, \quad n\ge 1.
\eea
Here the dot denotes the derivative with respect to $\lam$.
Notice also $W_{-1}(s(\lam),\dot{s}(\lam)) = W_0(s(\lam),\dot{s}(\lam)) =0$.
Evaluating the above equation using Pr\"ufer variables shows
\bea \label{thetaspos}
\dot{\theta}_s(\lam,n) &=& \frac{\sum_{j=1}^n s(\lam,j)^2}{
-a(n) \rho_s(\lam,n)^2} > 0, \quad n \ge 1,\\
\dot{\theta}_s(\lam,n) &=& \frac{\sum_{j=n+1}^0 s(\lam,j)^2}{
a(n) \rho_s(\lam,n)^2} < 0, \quad n<-1.
\eea
Notice, again that $\dot{\theta}_s(\lam,-1)=\dot{\theta}_s(\lam,0)=0$.
Equation (\ref{thetaspos}) implies that nodes of $s(\lam,n)$ for $n \in \N$
move monotonically to the left without colliding (cf., \cite{at}
Theorem~4.3.4). In addition, since $s(\lam,n)$ cannot pick up nodes locally by
(\ref{sigch}), all nodes must enter at $\infty$ and since
$\dot{\theta}_s(\lam,0) =0$ they are trapped inside
$(0,\infty)$.

We shall normalize $\theta_s(\lam,0)=0$ and hence
$\theta_s(\lam,-1)=-\pi/2$. Since $s(\lam,n)$ is a polynomial in $\lam$ we easily
infer $s(\lam,n) \gele 0$ for fixed $n\gele 0$ and $\lam$ sufficiently small. This
implies
\begin{equation} \label{thetassmlam}
-\pi < \theta_s(\lam,n) < -\pi/2, \: n < -1, \quad 0 < \theta_s(\lam,n) < \pi,
\: n \ge 1,
\end{equation}
for fixed $n$ and $\lam$ sufficiently small. Moreover, dividing (\ref{ricphi}) by
$\lam$ and letting $\lam \to -\infty$ using (\ref{thetassmlam}) shows
\begin{equation}
\lim_{\lam \to \pm\infty} \frac{\cot(\theta_s(\lam,n))^{\pm1}}{\lam} =
\frac{1}{a(n)}, \quad n \, {\ge +1 \atop < -1 }
\end{equation}
and hence
\begin{equation} \label{limthetas}
\theta_s(\lam,n) = -\frac{\pi}{2}-\frac{a(n)}{\lam} + o(\frac{1}{\lam}), \:
n<-1, \quad \theta_s(\lam,n) = \frac{a(n)}{\lam} + o(\frac{1}{\lam}), \: n
\ge 1,
\end{equation}
as $\lam \to -\infty$.

Analogously, let $u_\pm(\lam,n)$ be solutions of $\tau u =\lam u$ as in Lemma
\ref{upmholz}. Then Lemma \ref{wuudot} implies
\bea
\dot{\theta}_+(\lam,n) &=& \frac{\sum_{j=n+1}^\infty u_+(\lam,j)^2}{
a(n)\rho_+(\lam,n)^2} < 0,\\ \label{thetaminc}
\dot{\theta}_-(\lam,n) &=& \frac{\sum_{j=-\infty}^n u_-(\lam,j)^2}{
-a(n)\rho_-(\lam,n)^2} > 0,
\eea
where we have abbreviated $\rho_{u_\pm}=\rho_\pm$, $\theta_{u_\pm}=\theta_\pm$.

If $H$ is bounded from below we can normalize
\begin{equation}
0 < \theta_\mp(\lam,n) < \pi/2, \quad n \in \Z, \quad\lam < \inf\sig(H)
\end{equation}
and we get as before
\begin{equation} \label{limthetapm}
\theta_-(\lam,n) = \frac{a(n)}{\lam} + o(\frac{1}{\lam}),  \quad
\theta_+(\lam,n) = \frac{\pi}{2} - \frac{a(n)}{\lam} + o(\frac{1}{\lam}), \quad
n \in \Z
\end{equation}
as $\lam \to -\infty$.



\section{Standard Oscillation Theory}
\label{secosc}


First of all we recall (\cite{gst}, Lemma 5.2).

\bl \label{lemsrc}
Let $H,H_n$ be self-adjoint operators and $H_n \to H$ in strong resolvent sense
as $n \to \infty$. Then
\begin{equation}
\dim\Ran\,P_{(\lam_1,\lam_2)}(H)\le
\liminf_{n \to \infty} \dim\Ran\,
P_{(\lam_1,\lam_2)}(H_n).
\end{equation}
\el
 
Our first theorem considers half-line operators $H_\pm$ associated with a
Dirichlet boundary condition at $n=0$, that is,  the following restrictions of $H$
to the subspaces
$\ell^2(\pm\N)$,
\begin{equation}
\ba{llcl} H_\pm :& \db(H_\pm) & \to & \ell^2(\pm\N) \\ &
f(n) &\mapsto& \left\{ \ba{l}
a({+1 \atop -2})f(\pm 2) - b(\pm1) f(\pm 1), \:
n=\pm1 \\ (\tau f)(n), \quad n \gele \pm1 \ea
\right.
\ea ,
\end{equation}
with
\begin{equation}
\db(H_\pm) = \{f \in \ell^2(\pm\N) | \tau f \in \ell^2(\pm\N), \: \lim_{n \to
\pm\infty} W_n(u_\pm(z_0),f) =0 \}.
\end{equation}
Similarly one defines finite restriction $H_{n_1,n_2}$ to the
subspaces $\ell^2(n_1,n_2)$ with Dirichlet boundary conditions at $n=n_1$ and
$n=n_2$.

\br \label{rembc}
We only consider the case of a Dirichlet boundary condition at $n=0$ since the
operators $H_{\pm,n_0}^\beta$ on $\ell^2(n_0,\pm\infty)$ associated with the
general boundary condition
\begin{equation} \label{boundcon}
f(n_0+1) + \beta f(n_0)=0, \qquad \beta \in \R \cup \{ \infty \}
\end{equation}
at $n=n_0$ can be reduced to this case by a simple shift and altering the sequence
$b$ at one point. More precisely, we have
\begin{equation} \label{hbetann}
H^0_{+,n_0} = H_{+,n_0+1}, \quad
H^\beta_{+,n_0} = H_{+,n_0} - a(n_0) \beta^{-1}
\spr{\delta_{n_0+1}}{.}\delta_{n_0+1}, \quad \beta \ne 0,
\end{equation}
and
\begin{equation}
H^\infty_{-,n_0} = H_{-,n_0}, \quad
H^\beta_{-,n_0} = H_{-,n_0+1} - a(n_0) \beta
\spr{\delta_{n_0}}{.}\delta_{n_0}, \quad \beta \ne \infty,
\end{equation}
where $\delta_{n_0}(n) = 1$ if $n=n_0$ and $\delta_{n_0}(n) = 0$ otherwise.
Hence all one has to do is alter the definition of $b(n_0)$ or $b(n_0+1)$.
Analogously one defines the corresponding finite operators
$H_{n_1,n_2}^{\beta_1,\beta_2}$ which will be used in the next section.
\er

\bth \label{nbhpm}
Let $\lam \in \R$. Suppose $\tau$ is $l.p.$ at $+\infty$ or $\lam \in
\sig_p(H_+)$. Then
\begin{equation}
\dim\Ran\, P_{(-\infty,\lam)}(H_+) = \#_{(0,+\infty)}(s(\lam)).
\end{equation}
The same theorem holds if $+$ is replaced by $-$.
\eth

\bpf
We only carry out the proof for the plus sign (the other part following from
reflection). By virtue of (\ref{thetaspos}), (\ref{limthetas}), and
Lemma~\ref{nbofu} we infer
\begin{equation} \label{dimhhzn}
\dim\Ran\, P_{(-\infty,\lam)}(H_{0,n}) = \intp{\theta_s(\lam,n)/\pi}
=\#_{(0,n)}(s(\lam)), \quad n>1,
\end{equation}
since $\lam \in \sig(H_{0,n})$ if and only if $\theta_s(\lam,n) = 0 \mod\pi$.
Let $k = \#(s(\lam))$ if $\#(s(\lam))<\infty$, otherwise the following
argument works for arbitrary $k \in \N$. If we pick $n$ so large that $k$
nodes of $s(\lam)$ are to the left of $n$ we have $k$ eigenvalues
$\hat{\lam}_1 < \cdots < \hat{\lam}_k <\lam$ of $H_{0,n}$. Taking an arbitrary
linear combination $\eta(m) = \sum_{j=1}^k c_j s(\hat{\lam}_j,m)$, $c_j \in
\C$ for $m<n$ and $\eta(m)=0$ for $m \ge n$ a straightforward calculation
(using orthogonality of $s(\hat{\lam}_j)$) yields
\begin{equation}
\spr{\eta}{H_+ \eta} < \lam \| \eta \|^2.
\end{equation}
Invoking the spectral theorem shows
\begin{equation}
\dim\Ran\, P_{(-\infty,\lam)}(H_\pm) \ge k.
\end{equation}
For the reversed inequality we can assume $k = \#(s(\lam)) < \infty$.

We first suppose $\tau$ is $l.p.$ at $+\infty$. Consider
$\ti{H}_{0,n} = H_{0,n} \oplus \lam \id$ on $\ell^2(0,n) \oplus
\ell^2(n-1,\infty)$. Then Theorem~9.16.(i)  in \cite{wd} (take $\ell_0(\Z)$
as a core) implies strong resolvent convergence of $\ti{H}_{0,n}$ to $H_+$ as
$n\to \infty$ and by Lemma~\ref{lemsrc} we have
\begin{equation}
\dim\Ran\, P_{(-\infty,\lam)}(H_+) \le \lim_{n \to \infty}\dim\Ran\,
P_{(-\infty,\lam)}(H_{0,n}) = k
\end{equation}
completing the proof if $\tau$ is $l.p.$ at $+\infty$.

Otherwise, that is, if $\tau$ is $l.c.$ at $+\infty$ (implying that the
spectrum of $H_+$ is purely discrete), $\lam$ is an eigenvalue by hypothesis.
We first suppose $H$ bounded from below. Hence it suffices to show that the
$n$-th eigenvalue $\lam_n$, $n\in\N$ has at least $n-1$ nodes. This is trivial
for $n=1$. Suppose this is true for $\lam_n$ and let $m$ be the largest node
of $s(\lam_n)$. By $\theta_s(\lam_{n+1},m)>\theta_s(\lam_n,m)$ we infer that
$\theta_s(\lam_{n+1},m)$ has either more nodes between $0$ and $m$ or there
is at least one additional node of $\theta_s(\lam_{n+1},m)$ larger than $m$
by Lemma~\ref{sturm}. In the case where $H$ is not bounded from below 
we can label the eigenvalues $\lam_n$, $n\in\Z$. The same argument as before
shows that the eigenfunction corresponding to $\lam_m$ has $|m-n|$ nodes
more than the one corresponding to $\lam_n$. Letting $m \to -\infty$ shows
that the eigenfunction corresponding to $\lam_n$ has infinitely many nodes.
This completes the proof.   
\epf

\br \label{rembcch}
(i) The $l.p.$ / $\lam\in\sig_p(H_+)$ assumption is crucial since we need
some information about the boundary condition at $+\infty$.\\
(ii) The previous remark implies the following.
Let $\lam \in \R$. Suppose $\tau$ is $l.p.$ at $+\infty$ or $\lam \in
\sig_p(H^\beta_{+,n_0})$ and $\beta \ne 0$. Then
\begin{equation}
\dim\Ran\, P_{(-\infty,\lam)}(H^\beta_{+,n_0}) =
\#_{(0,+\infty)}(s_\beta(\lam,.,n_0)),
\end{equation}
where $s_\beta(\lam,.,n_0)$ is a sequence satisfying $\tau s = \lam s$ and
the boundary condition (\ref{boundcon}). Similar modifications apply to
Theorems~\ref{thmhartman}, \ref{thmwrfin}, and \ref{thmzerwronshl}.
\er

As a consequence of Theorem~\ref{nbhpm} we infer

\bk \label{corosc}
We have
\begin{equation}
\dim\Ran\, P_{(-\infty,\lam)}(H_\pm) < \infty
\end{equation}
if and only if $\tau-\lam$ is non-oscillatory near $\pm\infty$, respectively,
and hence
\begin{equation}
\inf\sig_{ess}(H_\pm) = \inf \{ \lam \in \R \,|\, (\tau - \lam) \mbox{ is
oscillatory at }\pm\infty \}.
\end{equation}
Moreover, let $H_\pm$ be bounded from below and $\lam_1 < \cdots < \lam_k <
\cdots$ be the eigenvalues of $H_\pm$ below the essential spectrum of $H_\pm$.
Then the eigenfunction corresponding to $\lam_k$ has precisely $k-1$ nodes
inside
$(0,\pm\infty)$.
\ek
We remark that the first part of Corollary~\ref{corosc} can be found in
\cite{glz}, Theorem 32 (see also \cite{hl}).

\br
Consider the following example
\begin{equation}
a(n) = -\frac{1}{2}, \: n \in\N, \quad b(1) =1, b(2) =b_2, b(3)
=\frac{1}{2}, b(n)=0, \: n \ge 4.
\end{equation}
The essential spectrum of $H_+$ is given by $\sig_{ess}(H_+) = [-1,1]$ and
one might expect that $H_+$ has no eigenvalues below the essential spectrum
if $b_2 \to -\infty$. However, since we have
\begin{equation}
s(-1,0)=0, s(-1,1)=1, s(-1,2)=0, s(-1,n)=-1, \: n \ge 3,
\end{equation}
Theorem~\ref{nbhpm} shows that, independent of $b_2 \in \R$, there is always
precisely one eigenvalue below the essential spectrum.
\er

In a similar way we obtain

\bth \label{thmbelesssph}
Let $\lam <\inf\sig_{ess}(H)$. Suppose $\tau$ is $l.p.$ at $-\infty$
or $\lam \in \sig_p(H)$. Then
\begin{equation}
\dim\Ran\, P_{(-\infty,\lam)}(H) = \#(u_+(\lam)).
\end{equation}
The same theorem holds if $l.p.$ at $-\infty$ and $u_+(\lam)$ is replaced by
$l.p.$ at $+\infty$ and $u_-(\lam)$.
\eth

\bpf
Again it suffices to prove the minus case. If $H$ is not bounded from below the
same is true for $H_- \oplus H_+$ (which can be embedded into $\lz$ and
considered as a finite rank perturbation of $H$). Hence $H_-$ or $H_+$ (or both) is
not bounded from below implying $\tau-\lam$ oscillatory near $-\infty$ or
$+\infty$ by Corollary~\ref{corosc} and we can suppose $H$ bounded from below.

By virtue of (\ref{thetaminc}) and
(\ref{limthetapm}) we infer
\begin{equation}
\dim\Ran\, P_{(-\infty,\lam)}(H_{-,n}) = \intp{\theta_-(\lam,n)/\pi},
\quad n\in\Z.
\end{equation}
We first want to show $\intp{\theta_-(\lam,n)/\pi} =
\#_{(-\infty,n)}(u_-(\lam))$ or equivalently
\begin{equation}
\lim_{n \to \infty} \intp{\theta_-(\lam,n)/\pi} =0.
\end{equation}
Suppose $\lim_{n \to \infty}
\intp{\theta_-(\lam_1,n)/\pi} =k\ge 1$ for some $\lam_1 \in \R$ (saying that
$u_-(.,n)$ loses at least one node at $-\infty$). In this case we can find $n$
such that $\theta_-(\lam_1,n) > k\pi$ for $m \ge n$. Now pick $\lam_0$ such that
$\theta_-(\lam_0,n) = k\pi$. Then $u_-(\lam_0,.)$ has a node at $n$ but no node
between $-\infty$ and $n$ (by Lemma \ref{nbofu}). Now apply Lemma \ref{sturm} to
$u_-(\lam_0,.)$, $u_-(\lam_1,.)$ to obtain a contradiction. The rest follows as
in the proof of Theorem~\ref{nbhpm}.
\epf

As before we obtain

\bk \label{coresssp}
We have
\begin{equation}
\dim\Ran\, P_{(-\infty,\lam)}(H) < \infty
\end{equation}
if and only if $\tau-\lam$ is non-oscillatory and hence
\begin{equation}
\inf\sig_{ess}(H) = \inf\{ \lam \in \R \,|\, (\tau - \lam) \mbox{ is
oscillatory} \}.
\end{equation}
Furthermore, let $H$ be bounded from below and $\lam_1 < \cdots < \lam_k <
\dots$ be the eigenvalues of $H$ below the essential spectrum of $H$. Then the
eigenfunction corresponding to $\lam_k$ has precisely $k-1$ nodes.
\ek

\br \label{remabovesp}
Corresponding results for the projection $P_{(\lam,\infty)}(H)$ can be obtained 
from $P_{(\lam,\infty)}(H) = P_{(-\infty,-\lam)}(-H)$. 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 $P_{(-\infty,\lam)}(H)$ to $P_{(\lam,\infty)}(H)$ in
all results of this section.
\er

Now we turn to the analog of \cite{har2}, Theorem I.

\bth \label{thmhartman}
Let $\lam_1<\lam_2$. Suppose $\tau-\lam_2$ is oscillatory near $+\infty$ and
$\tau$ is $l.p.$ at $+\infty$. Then
\begin{equation}
\dim\Ran\, P_{(\lam_1,\lam_2)}(H_+) = \liminf_{n \to
+\infty}\Big( \#_{(0,n)}(s(\lam_2)) - \#_{(0,n)}(s(\lam_1)) \Big).
\end{equation}
The same theorem holds if $+$ is replaced by $-$.
\eth

\bpf
As before we only carry out the proof for the plus sign. Abbreviate $\Delta(n) =
\intp{\theta_s(\lam_2,n)/\pi} -\intp{\theta_s(\lam_1,n)/\pi} =
\#_{(0,n)}(s(\lam_2)) - \#_{(0,n)}(s(\lam_1))$. By (\ref{dimhhzn}) we infer
\begin{equation}
\dim\Ran\, P_{[\lam_1,\lam_2)}(H_{0,n}) = \Delta(n), \quad n>2.
\end{equation}
Let $k=\liminf \Delta(n)$ if $\limsup \Delta(n)<\infty$ and $k \in \N$
otherwise. We contend that there exists a $n \in \N$ such that
\begin{equation}
\dim\Ran\, P_{(\lam_1,\lam_2)}(H_{0,n}) \ge k.
\end{equation}
In fact, if $k=\limsup \Delta(n)<\infty$ it follows that $\Delta(n)$ is
eventually equal to $k$ and since $\lam_1 \not\in \sig(H_{0,m})
\cap \sig(H_{0,m+1})$, $m \in\N$ we are done in this case. Otherwise we
can pick $n$ such that $\dim\Ran\, P_{[\lam_1,\lam_2)}(H_{0,n}) \ge k+1$.
Hence $H_{0,n}$ has at least $k$ eigenvalues $\hat{\lam}_j$ with $\lam_1 <
\hat{\lam}_1 < \dots < \hat{\lam}_k < \lam_2$. Again let $\eta(m) = 
\sum_{j=1}^k c_j s(\hat{\lam}_j,n)$, $c_j \in \C$ for $m<n$ and $\eta(m)=0$
for $n \ge m$ be an arbitrary linear combination. Then
\begin{equation}
\| (H_+ - \frac{\lam_2+\lam_1}{2}) \eta \| < \frac{\lam_2-\lam_1}{2} \| \eta \|
\end{equation}
together with the spectral theorem implies
\begin{equation}
\dim\Ran\, P_{(\lam_1,\lam_2)}(H_+) \ge k.
\end{equation}
To prove the second inequality we use that $\ti{H}_{0,n} = H_{0,n}\oplus\lam_2
\id$ converges to $H_+$ in strong resolvent sense as $n \to \infty$ and proceed as
before 
\begin{equation}
\dim\Ran\, P_{(\lam_1,\lam_2)}(H_+) \le
\liminf_{n\to\infty} P_{[\lam_1,\lam_2)}(\ti{H}_{0,n}) = k
\end{equation}
since $P_{[\lam_1,\lam_2)}(\ti{H}_{0,n}) = P_{[\lam_1,\lam_2)}(H_{0,n})$.
\epf



\section{Renormalized Oscillation Theory}
\label{secrnosc}


The objective of this section is to look at the nodes of the Wronskian of two
solutions $u_{1,2}$ corresponding to $\lam_{1,2}$, respectively. We call $n\in
\Z$ a node of the Wronskian if $W_n(u_1,u_2)=0$ and $W_{n+1}(u_1,u_2)\ne 0$
or if $W_n(u_1,u_2) W_{n+1}(u_1,u_2) < 0$. Again we shall 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 $W_{n_0}(u_1,u_2) \ne 0$.  We abbreviate
\begin{equation}
\Delta_{u_1,u_2}(n) = (\theta_{u_2}(n) - \theta_{u_1}(n)) \mod 2\pi.
\end{equation}
and require
\begin{equation} \label{normwr}
\intp{\Delta_{u_1,u_2}(n)/ \pi} \le \intp{\Delta_{u_1,u_2}(n+1)/ \pi}
\le\intp{\Delta_{u_1,u_2}(n)/ \pi} +1.
\end{equation}
We shall fix $\lam_1 \in \R$ and a
corresponding solution $u_1$ and choose a second solution $u(\lam,n)$ with
$\lam \in [\lam_1,\lam_2]$. Now let us consider
\begin{equation}
W_n(u_1,u(\lam)) = -a(n) \rho_{u_1}(n) \rho_u(\lam,n)
\sin(\Delta_{u_1,u}(\lam,n))
\end{equation}
as a function of $\lam \in[\lam_1,\lam_2]$.

\bl \label{lempropwueuz}
Suppose $\Delta_{u_1,u}(\lam_1,.)$ satisfies (\ref{normwr}) then we have
\begin{equation} \label{deluoul}
\Delta_{u_1,u}(\lam,n) = \theta_{u}(\lam,n) - \theta_{u_1}(n)
\end{equation}
where $\theta_{u}(\lam,.)$, $\theta_{u_1}(.)$ both satisfy (\ref{normalth}).
That is,  $\Delta_{u_1,u}(.,n) \in C[\lam_1,\lam_2]$ and (\ref{normwr})
holds for all $\Delta_{u_1,u}(\lam,.)$ with $\lam \in [\lam_1,\lam_2]$.
In particular, the second inequality in (\ref{normalth}) is attained if and
only if $n$ is a node of $W_{.}(u_1,u(\lam))$. 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
\begin{equation} \label{eqpropwueuz}
\#_{(m,n)}W(u_1,u_2) = \intp{\Delta_{u_1,u_2}(n)/ \pi} - \lim_{\eps\downarrow
0} \intp{\Delta_{u_1,u_2}(m)/ \pi +\eps}
\end{equation}
and
\begin{equation}
\#W(u_1,u_2) = \#_{(-\infty,\infty)}W(u_1,u_2) = \lim_{n \to \infty} \Big(
\intp{\Delta_{u_1,u_2}(n)/ \pi} -
\intp{\Delta_{u_1,u_2}(-n)/ \pi}\Big).
\end{equation}
\el

\bpf
We fix $n$ and set
\begin{equation}
\Delta_{u_1,u}(\lam,n) = k \pi + \delta(\lam), \quad \Delta_{u_1,u}(\lam,n+1)
= k \pi + \Delta(\lam),
\end{equation}
where $k \in \Z, \delta(\lam_1) \in (0,\pi]$ and $\Delta(\lam_1) \in (0,2\pi]$.
Clearly (\ref{deluoul}) holds for $\lam=\lam_1$ since $W_{.}(u_1,u(\lam_1))$
is constant. If (\ref{normwr}) should break down we must have one of the
following cases for some $\lam_0 \ge \lam_1$. (i) $\delta(\lam_0) = 0$,
$\Delta(\lam_0) \in (\pi,2\pi]$, or (ii) $\delta(\lam_0) = \pi$,
$\Delta(\lam_0) \in (0,\pi]$, or (iii) $\Delta(\lam_0) = 2\pi$,
$\delta(\lam_0) \in (\pi,\pi]$, or (iv) $\Delta(\lam_0) = 0$, $\delta(\lam_0)
\in (\pi,\pi]$. For notational convenience let us set $\delta=\delta(\lam_0),
\Delta=\Delta(\lam_0)$ and $\theta_{u_1}(n) = \theta_1(n), \theta_u(\lam_0,n) =
\theta_2(n)$. Furthermore, we can assume $\theta_{1,2}(n) = k_{1,2} \pi +
\delta_{1,2}$, $\theta_{1,2}(n+1) = k_{1,2} \pi +
\Delta_{1,2}$ with $k_{1,2} \in \Z, \delta_{1,2} \in (0,\pi]$ and $\Delta_{1,2}
\in (0,2\pi]$. 

Suppose (i). Then
\begin{equation} \label{wueulam}
W_{n+1}(u_1,u(\lam_0)) = (\lam_0 - \lam_1) u_1(n+1) u(\lam_0,n+1).
\end{equation}
Inserting Pr\"ufer variables shows
\begin{equation}
\sin(\Delta_2 - \Delta_1) = \rho \cos^2(\delta_1) \ge 0
\end{equation}
for some $\rho>0$ since $\delta=0$ implies $\delta_1=\delta_2$. Moreover, $k =
(k_2 - k_1) \mod 2$ and $k\pi + \Delta = (k_2-k_1)\pi + \Delta_2 - \Delta_1$
implies $\Delta = (\Delta_2 -
\Delta_1) \mod 2\pi$. Hence we have
$\sin\Delta \ge 0$ and $\Delta \in (\pi,2\pi]$ implies $\Delta = 2\pi$. But
this says $\delta_1=\delta_2=\pi/2$ and $\Delta_1=\Delta_2 = \pi$. Since we
have at least $\delta(\lam_2-\eps)>0$ and hence $\delta_2(\lam_2-\eps)>\pi/2$,
$\Delta_2(\lam_2-\eps)>\pi$ for $\eps>0$ sufficiently small. Thus from
$\Delta(\lam_2-\eps) \in (\pi,2\pi)$ we get
\begin{equation}
0> \sin\Delta(\lam_2-\eps) = \sin(\Delta_2(\lam_2-\eps) -\pi) >0,
\end{equation}
contradicting (i).

Suppose (ii). Again by (\ref{wueulam}) we have $\sin(\Delta_2-\Delta_1)\ge 0$
since $\delta_1 =\delta_2$. But now $(k + 1) = (k_1-k_2) \mod 2$.
Furthermore, $\sin(\Delta_2-\Delta_1) = - \sin(\Delta) \ge 0$ says $\Delta =
\pi$ since $\Delta \in (0,\pi]$. Again this implies $\delta_1=\delta_2=\pi/2$
and $\Delta_1=\Delta_2 = \pi$. But since $\delta(\lam)$
increases/decreases precisely if $\Delta(\lam)$ increases/decreases for $\lam$
near $\lam_0$ (\ref{normwr}) stays valid.

Suppose (iii) or (iv). Then
\begin{equation}
W_n(u_1,u(\lam_0)) = -(\lam_0 - \lam_1) u_1(n+1) u(\lam_0,n+1).
\end{equation}
Inserting Pr\"ufer variables gives
\begin{equation} \label{sindelzmdele}
\sin(\delta_2 - \delta_1) = -\rho \sin(\Delta_1) \sin(\Delta_2)
\end{equation}
for some $\rho>0$. We first assume $\delta_2>\delta_1$. In this case we infer
$k=(k_2-k_1) \mod 2$ implying $\Delta_2-\Delta_1 = 0 \mod 2\pi$ contradicting
(\ref{sindelzmdele}). Next assume $\delta_2 \le \delta_1$. Then we obtain
$(k+1)=(k_2-k_1) \mod 2$ implying $\Delta_2-\Delta_1 = \pi \mod 2\pi$ and hence
$\sin(\delta_2 - \delta_1) \ge 0$ from (\ref{sindelzmdele}). Thus we get
$\delta_1=\delta_2=\pi/2$ $\Delta_1=\Delta_2 =
\pi$, and hence $\Delta_2-\Delta_1 = 0 \mod 2\pi$ contradicting (iii), (iv).
This settles (\ref{deluoul}).

Furthermore, if $\Delta(\lam) \in (0,\pi]$ we have no node at $n$ since
$\delta(\lam)=\pi$ implies $\Delta(\lam) =\pi$ by (ii). Conversely, if
$\Delta(\lam) \in (\pi,2\pi]$ we have a node at $n$ since $\Delta(\lam)=2\pi$
is impossible by (iii). The rest being straightforward.
\epf

Equations (\ref{nbunm}), (\ref{deluoul}), and (\ref{eqpropwueuz}) imply

\bk \label{cornodw}
Let $\lam_1 \le \lam_2$ and suppose $u_{1,2}$ satisfy $\tau u_{1,2} =
\lam_{1,2} u_{1,2}$, respectively. Then we have
\begin{equation}
| \#_{(n,m)}W(u_1,u_2) - (\#_{(n,m)}(u_2) - \#_{(n,m)}(u_1))| \le 2
\end{equation}
\ek

Now we come to a renormalized version of Theorem \ref{thmhartman}. We first
need the result for a finite interval.

\bth \label{thmwrfin}
Fix $n_1<n_2$ and $\lam_1<\lam_2$. Then
\begin{equation}
\dim\Ran\, P_{(\lam_1,\lam_2)}(H_{n_1,n_2}) =
\#_{(n_1,n_2)}W(s(\lam_1,.,n_1),s(\lam_2,.,n_2)).
\end{equation}
\eth

\bpf
We abbreviate
\begin{equation}
\Delta(\lam,n) = \Delta_{s(\lam_1,.,n_1),s(\lam,.,n_2)}(n)
\end{equation}
and normalize (perhaps after flipping the sign of $s(\lam_1,.,n_1)$)
$\Delta(\lam_1,n) \in (0,\pi]$. From (\ref{thetaspos}) we infer
\begin{equation}
\dim\Ran\, P_{(\lam_1,\lam_2)}(H_{n_1,n_2}) =
- \lim_{\eps\downarrow 0}\intp{\Delta(\lam_2,n_1)/\pi + \eps}
\end{equation}
since $\lam \in \sig(H_{n_1,n_2})$ is equivalent to
$\Delta(\lam,n_1) = 0 \mod\pi$. Using (\ref{eqpropwueuz}) completes
the proof.
\epf

\bth \label{thmzerwronshl}
Fix $\lam_1<\lam_2$ and suppose $\tau$ is in the $l.p.$ case near $+\infty$ or
$\lam_2\in\sig_p(H_+)$. Then
\begin{equation}
\dim\Ran\, P_{(\lam_1,\lam_2)}(H_+) =
\#_{(0,+\infty)}W(s(\lam_1),s(\lam_2)).
\end{equation}
The same theorem holds if $+$ is replaced by $-$.
\eth

\bpf
As usual we only prove the result for $H_+$ and set $k = \#_{(0,\infty)}
W(s(\lam_1),${}$s(\lam_2))$ provided this number is finite and $k\in\N$
otherwise. We abbreviate
\begin{equation}
\Delta(\lam,n) = \Delta_{s(\lam_1),s(\lam)}(n)
\end{equation}
and normalize $\Delta(\lam_1,n) =0$ implying $\Delta(\lam,n) >0$ for
$\lam>\lam_1$. Hence if we chose $n$ so large that all $k$ nodes are to the
left of $n$ we have
\begin{equation}
\Delta(\lam,n) > k \pi.
\end{equation}
Thus we can find $\lam_1 < \hat{\lam}_1 < \cdots < \hat{\lam}_k < \lam_2$ with
$\Delta(\hat{\lam}_j,n) = j \pi$. Now define
\begin{equation}
\eta_j(m) = \left\{ \ba{ll} s(\hat{\lam}_j,m) - \rho_j s(\lam_1,m) & m \le n \\
0 & m \ge n \ea\right. ,
\end{equation}
where $\rho_j \ne 0$ is chosen such that $s(\hat{\lam}_j,m) = \rho_j
s(\lam_1,m)$ for $m=n,n+1$. Furthermore observe that
\begin{equation}
\tau \eta_j(m) = \left\{ \ba{ll} \hat{\lam}_j s(\hat{\lam}_j,m) -
\lam_1 \rho_1 s(\lam_1,m)& m
\le n \\ 0 & m \ge n \ea\right.
\end{equation}
and that $s(\lam_1,m)$, $s(\hat{\lam}_j,.)$, $1\le j \le k$ are orthogonal on
$1,\dots,n$. Next, let $\eta = \sum_{j=1}^k c_j \eta_j$, $c_j$ be an arbitrary
linear combination, then a short calculation verifies
\begin{equation}
\| (H_+ - \frac{\lam_2+\lam_1}{2}) \eta \| < \frac{\lam_2-\lam_1}{2} \| \eta \|.
\end{equation}
And invoking the spectral theorem gives
\begin{equation} \label{ineqwsso}
\dim\Ran\, P_{(\lam_1,\lam_2)}(H_+) \ge k.
\end{equation}
To prove the reversed inequality is only necessary if $\#_{(0,\infty)}
W(s(\lam_1),${}$s(\lam_2))<\infty$. In this case we look at
$H_{0,n}^{\infty,\beta}$ with $\beta = s(\lam_2,n+1)/s(\lam_2,n)$. By
Theorem~\ref{thmwrfin} and Remark~\ref{rembcch} (ii) we have
\begin{equation}
\dim\Ran\, P_{(\lam_1,\lam_2)}(\ti{H}_{0,n}^{\infty,\beta}) = \#_{(0,n)}
W(s(\lam_1),s(\lam_2)).
\end{equation}
Now use strong resolvent convergence of $\ti{H}_{0,n}^{\infty,\beta} =
H_{0,n}^{\infty,\beta} \oplus
\lam_1\id$ to $H_+$ (due to our $l.p.$ / $\lam_2 \in \sig_p(H_+)$
assumption) as $n\to\infty$ to obtain
\begin{equation} \label{ineqwsst}
\dim\Ran\, P_{(\lam_1,\lam_2)}(H_+) \le \liminf_{n\to\infty}
\dim\Ran\, P_{(\lam_1,\lam_2)}(\ti{H}_{0,n}^{\infty,\beta}) = k
\end{equation}
completing the proof.
\epf

As a consequence we infer.

\bk \label{coresslamot}
Let $u_{1,2}$ satisfy $\tau u_{1,2} = \lam_{1,2} u_{1,2}$. Then
\begin{equation}
\#_{(0,\pm\infty)}W(u_1,u_2) < \infty \quad\Leftrightarrow\quad \dim\Ran\,
P_{(\lam_1,\lam_2)}(H_\pm) < \infty.
\end{equation}
\ek

\bpf
By Corollary~\ref{cornodw} the  result does not depend on the choice of
$u_{1,2}$. Since the proof of (\ref{ineqwsso}) does not use the $l.p.$ /
$\lam_2 \in\sig_p(H_+)$ assumption the first direction follows. Conversely,
we can replace the sequence $\beta$ in (\ref{ineqwsst}) by a sequence
$\hat{\beta}$ such that $\ti{H}_{0,n}^{\infty,\hat{\beta}}$ converges to
$H_+$. Since we have $|\dim\Ran\,\ti{H}_{0,n}^{\infty,\hat{\beta}} -
\dim\Ran\,\ti{H}_{0,n}^{\infty,\beta}| \le 1$ the corollary is proven.
\epf

Finally we turn to  our main result for Jacobi operators $H$ on $\Z$. We
emphasize that to date, Theorem~\ref{thmzerwon} appears to be the only
oscillation theoretic result concerning the number of eigenvalues in
essential spectral gaps of Jacobi operators on $\Z$.

\bth \label{thmzerwon}
Fix $\lam_1<\lam_2$ and suppose $[\lam_1,\lam_2] \cap \sig_{ess}(H) =
\emptyset$. Then
\begin{equation}
\dim\Ran\, P_{(\lam_1,\lam_2)}(H) = \#W(u_\mp(\lam_1),u_\pm(\lam_2)).
\end{equation}
In addition, if $\tau$ is $l.p.$ at $+\infty$ we even have
\begin{equation}
\dim\Ran\, P_{(\lam_1,\lam_2)}(H) = \#W(u_+(\lam_1),u_+(\lam_2)).
\end{equation}
The same result holds if $+$ is replaced by $-$.
\eth

\bpf
Since the proof is similar to the proof of Theorem \ref{thmzerwronshl} we
shall only outline the first part. Let $k = \#W(u_+(\lam_1),u_-(\lam_2))$ if
this number is finite and $k \in \N$ else. Pick $n>0$ so large that all zeros
of the Wronskian are between $-n$ and $n$. We abbreviate
\begin{equation}
\Delta(\lam,n) = \Delta_{u_+(\lam_1),u_-(\lam)}(n)
\end{equation}
and normalize $\Delta(\lam_1,n) \in [0,\pi)$ implying $\Delta(\lam,n) >0$
for $\lam>\lam_1$. Hence if we chose $n \in \N$ so large that all $k$ nodes
are between $-n$ and $n$ we can assume
\begin{equation}
\Delta(\lam,n) > k \pi.
\end{equation}
Thus we can find $\lam_1 < \hat{\lam}_1 < \cdots < \hat{\lam}_k < \lam_2$ with
$\Delta(\hat{\lam}_j,n) = 0 \mod\pi$. Now define
\begin{equation}
\eta_j(m) = \left\{ \ba{ll} u_-(\hat{\lam}_j,m) & m \le n \\
\rho_j u_+(\lam_1,m) & m \ge n \ea\right. ,
\end{equation}
where $\rho_j \ne 0$ is chosen such that $u_-(\hat{\lam}_j,m) = \rho_j
u_+(\lam_1,m)$ for $m=n,n+1$. Now proceed as in the previous theorems.
\epf

Again, we infer as a consequence.

\bk \label{coresslamote}
Let $u_{1,2}$ satisfy $\tau u_{1,2} = \lam_{1,2} u_{1,2}$. Then
\begin{equation}
\#W(u_1,u_2) < \infty \quad\Leftrightarrow\quad \dim\Ran\,
P_{(\lam_1,\lam_2)}(H) < \infty.
\end{equation}
\ek

\bpf
Follows from  Corollaries~\ref{cornodw}, \ref{coresslamot}, and $\dim\Ran\,
P_{(\lam_1,\lam_2)}(H)<\infty$ if and only if $(\dim\Ran$
$P_{(\lam_1,\lam_2)}(H_-) + \dim\Ran\, P_{(\lam_1,\lam_2)}(H_+))<\infty$.
\epf



\br
The most general three-term recurrence relation
\begin{equation}
\ti{\tau} f(n) = \ti{a}(n) f(n+1) - \ti{b}(n) f(n) + \ti{c}(n) f(n-1),
\end{equation}
with $\ti{a}(n)\ti{c}(n)>0$, can be transformed to a Jacobi recurrence relation
as follows. First we symmetrise $\ti{\tau}$ via
\begin{equation}
\ti{\tau} f(n) = \frac{1}{w(n)} \Big( c(n) f(n+1) + c(n-1) f(n-1) - d(n) f(n)
\Big),
\end{equation}
where
\bea
&w(n) = \left\{ \ba{c@{\quad\mb{ for }}l}
\prod\limits_{j=n_0}^{n-1} \frac{\ti{a}(j)}{\ti{c}(j+1)} & n > n_0 \\ 1 & 
n=n_0\\ \prod\limits_{j=n}^{n_0-1} \frac{\ti{c}(j+1)}{\ti{a}(j)} & n < n_0 \ea
\right. >0,&\\ &c(n) = w(n) \ti{a}(n) = w(n+1) \ti{c}(n+1), \quad d(n) = w(n)
\ti{b}(n).&
\eea
The natural Hilbert space for $\ti{\tau}$ is the weighted space $\ell^2(\Z,w)$
with scalar product
\begin{equation}
\spr{f}{g} = \sum_{n\in \Z} w(n) \ol{f(n)} g(n), \qquad f,g \in \ell^2(\Z,w).
\end{equation}
Let $\ti{H}$ be a self-adjoint operator associated with $\ti{\tau}$ in
$\ell^2(\Z,w)$. Then the unitary operator
\begin{equation} \label{unopgentojop}
\ba{clcl} U: & \ell^2(\Z,w) & \to & \lz \\ & u(n) & \mapsto & \sqrt{w(n)} 
u(n) \ea
\end{equation}
transforms $\ti{H}$ into a Jacobi operator $H$ in $\lz$ associated with the
sequences
\bea
a(n) &=& \frac{c(n)}{\sqrt{w(n)w(n+1)}} = \sgn(\ti{a}(n))
\sqrt{\ti{a}(n) \ti{c}(n+1)}, \\ b(n) &=&
\frac{d(n)}{w(n)} = \ti{b}(n).
\eea
In addition we infer
\bea \nn
\lefteqn{c(n) \Big( f(n) g(n+1) - f(n+1) g(n) \Big) =}\\
&& a(n) \Big( (U f)(n) (U g)(n+1) - (U f)(n+1) (U g)(n)
\Big).
\eea
Hence all the results derived for Jacobi operator thus far apply to generalized
Jacobi operators of the type $\ti{H}$ as well.
\er

\section{Applications}
\label{secapp}


One important class of Jacobi operators are periodic ones
(cf., e.g., \cite{bght}, Appendix B, \cite{kr1}, \cite{vm}). Instead of
periodic operators themselves we are interested in short-range perturbations
of these operators. In fact, we are going to prove the analog of the Theorem
by Rofe-Beketov (\cite{rof}, see also \cite{gs}, \cite{glz}, Section~67,
\cite{kl}) about the finiteness of the number of eigenvalues in essential
spectral gaps of the perturbed Hill operator. Since constant coefficients
$a,b$ are a special case of periodic ones our results contain results from
scattering theory (cf., e.g.,
\cite{dinv1}, \cite{gu}).

To set the stage, we first recall some basic facts from the theory of
periodic operators. Let $H_p$ be a Jacobi operator associated with
periodic sequences $a_p<0,b_p$, that is,
\begin{equation}
a_p(n+N) =a_p(n), \qquad b_p(n+N) = b_p(n),
\end{equation}  
for some fixed $N \in\N$. The spectrum of $H_p$ is purely absolutely
continuous and consists of a finite number of gaps, that is,
\begin{equation}
\sig(H_p) = \bigcup_{j=0}^g [E_{2j},E_{2j+1}], \qquad g \in\N_0,
\end{equation}
with $E_0 < E_1 < \cdots < E_{2g+1}$ and $g \le N-1$. Moreover, Floquet
theory implies the existence of solutions $u_{p,\pm}(z,.)$ of $\tau_p u = z
u$, $z\in\C$ ($\tau_p$ the difference expression corresponding to $H_p$)
satisfying
\begin{equation}
u_{p,\pm}(z,n+N) = m^\pm(z) u_{p,\pm}(z,n),
\end{equation}
where $m^\pm(z) \in \C$ are called Floquet multipliers. $m^\pm(z)$ satisfy
$m^+(z) m^-(z)=1$, $m^\pm(z)^2 =1$ for $z \in \{ E_j \}_{j=0}^{2g+1}$,
$|m^\pm(z)|=1$ for $z \in \sig(H_p)$, and $|m^+(z)|<1$ for $z \in
\C\bs\sig(H_p)$. (This says in particular, that $u_{p,\pm}(z,.)$ are
bounded for $z\in\sig(H_p)$ and linearly independent for $z \in \C\bs\{ E_j
\}_{j=0}^{2g+1}$.)

We are going to study perturbations $H$ of $H_p$ associated with sequences
$a,b$ satisfying $a(n) \to a_p(n)$ and $b(n) \to b_p(n)$ as $|n| \to
\infty$. Clearly, $H$ and $H_p$ are both bounded and hence defined on the
whole of $\lz$. In fact, we have
\begin{equation}
\sig(H) \subseteq [\ul{c},\ol{c}],
\end{equation}
where $\ul{c} = \inf_{n\in\Z} (b(n)+a(n-1)+a(n))$ and $\ol{c} = \sup_{n\in\Z}
(b(n)-a(n-1)-a(n))$. Using this notation our theorem reads:

\bth \label{thmappl}
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
\begin{equation} 
\sum_{n\in\Z} |n(a(n) - a_p(n))| < \infty, \quad
\sum_{n\in\Z} |n(b(n) - b_p(n))| < \infty.
\end{equation}
Then we have $\sig_{ess}(H)=\sig(H_p)$, the point spectrum of $H$ is finite
and confined to the spectral gaps of $H_p$, that is, $\sig_p(H) \subset
\R\bs\sig(H_p)$. Furthermore, the essential spectrum of $H_p$ is purely
absolutely continuous.
\eth

For the proof we will need the following lemma the proof of which is
elementary.

\bl \label{voltse}
The Volterra sum equation
\begin{equation}
f(n) = g(n) + \sum_{m=n+1}^\infty K(n,m) f(m),
\end{equation}
with
\begin{equation}
|K(n,m)| \le \hat{K}(n,m), \quad \hat{K}(n+1,m) \le \hat{K}(n,m), \quad
\hat{K}(n,.) \in \ell^1(0,\infty),
\end{equation}
has for $g \in \ell^\infty(0,\infty)$ a unique solution $f \in
\ell^\infty(0,\infty)$, fulfilling the estimate
\begin{equation}
|f(n)| \le \Big(\sup_{m>n}|g(m)|\Big) \exp \Big(  \sum_{m=n+1}^\infty 
\hat{K}(n,m) \Big).
\end{equation}
\el

\bpf (of Theorem \ref{thmappl})
The fact that $H-H_p$ is compact implies $\sig_{ess}(H)=\sig_{ess}(H_p)$.
To prove the remaining claims it suffices to show the existence of solutions
$u_\pm(\lam,.)$ of $\tau u = \lam u$ for $\lam \in \sig(H_p)$ satisfying
\begin{equation} \label{asupm}
\lim_{|n|\to \infty} |u_\pm(\lam,n) - u_{p,\pm}(\lam,n)| =0.
\end{equation}
In fact, since $u_\pm(\lam,.)$, $\lam \in \sig(H_p)$ are bounded and do not
vanish near $\pm\infty$, there are no eigenvalues in the essential spectrum of
$H$ and invoking the principal of subordinacy (cf., \cite{simac}, \cite{st})
shows that the essential spectrum of $H$ is purely absolutely continuous. Moreover,
(\ref{asupm}) with $\lam=E_0$ implies that $H -E_0$ is non-oscillatory since
we can assume (perhaps after flipping signs) $u_{p,\pm}(E_0,n) \ge \eps >0$,
$n\in\Z$ and by Corollary~\ref{coresssp} there are only finitely many
eigenvalues below $E_0$. Similarly, (using Remark~\ref{remabovesp}) there are
only finitely many eigenvalues above $E_{2g+1}$. Applying
Corollary~\ref{coresslamote} in each gap $(E_{2j-1},E_{2j})$, $1 \le j \le g$
shows that the number of eigenvalues in each gap is finite as well.

It remains to show (\ref{asupm}). Suppose $u_+(\lam,.)$, $\lam \in\sig(H_p)$
satisfies (disregarding summability for a moment)
\bea \nn
\lefteqn{u_+(\lam,n) = \frac{a_p(n)}{a(n)} u_{p,+}(\lam,n)}\\
\label{voltseq} && {}+\sum_{m=n+1}^\infty \frac{a_p(n)}{a(n)}
K(\lam,n,m) u_+(\lam,m),
\eea
with
\bea \nn
\lefteqn{K(\lam,n,m)= \frac{s_p(\lam,n,m-1)}{a_p(m-1)}(a(m-1) - a_p(m-1))}\\
&& {}+\frac{s_p(\lam,n,m+1)}{a_p(m+1)}(a(m) - a_p(m))
- \frac{s_p(\lam,n,m)}{a_p(m)}(b(m) - b_p(m)),
\eea
where $s_p(\lam,.,m)$ is the solution of of $\tau_p u = z u$
satisfying the initial conditions $s_p(z,m,m)=0$ and $s_p(z,m+1,m)=1$.
Then $u_+(\lam,.)$ fulfills $\tau u = \lam u$ and (\ref{asupm}). Hence if we
can apply Lemma~\ref{voltse} we are done. To do this we need an estimate for
$K(\lam,n,m)$ which again follows from Floquet theory
\begin{equation}
|s_p(\lam,n,m)| \le M |n-m|, \qquad \lam \in \sig(H_p),
\end{equation}
for some suitable constant $M>0$.
\epf

As anticipated, specializing to the case $a_p(n)=-1/2$, $b_p(n)=0$ we obtain
a corresponding result for the scattering case.

\bk \label{corscat}
(\cite{gu}) Suppose
\begin{equation} \label{decay}
\sum_{n\in\Z} |n(1+2a(n))| < \infty, \quad
\sum_{n\in\Z} |n\,b(n)| < \infty.
\end{equation}
Then we have
\begin{equation}
\sig_{ess}(H)=[-1,1], \quad \sig_p(H) \subseteq [\ul{c},-1) \cup
(1,\ol{c}].
\end{equation}
Moreover, the essential spectrum of $H$ is purely absolutely continuous and
the point spectrum of $H$ is finite.
\ek

Corollary~\ref{corscat} is stated in \cite{gu}
(for the case $a_p(n)=1$ -- but Lemma~\ref{lemaeps} plus a scaling transform
takes care of that). Similar results can be obtained using the Birman-Schwinger
principle.




\chapter{Spectral Deformations}



\section{Introduction}

For a variety of reasons, techniques to insert and remove 
eigenvalues in 
spectral gaps of a given one-dimensional second-order 
differential (i.e.,
Sturm-Liouville) respectively difference (i.e., Jacobi) 
operator have recently
attracted great interest. In fact, these techniques are 
vital in diverse fields
such as the inverse scattering approach used by Deift and 
Trubowitz \cite{dt}, 
supersymmetric quantum mechanics (cf.\ the literature 
cited, e.g., in
\cite{gss}), level comparison theorems (see, e.g., \cite{ba}), 
in the
construction of soliton solutions of the 
Korteweg-de Vries (KdV) and
Toda hierarchies relative to general KdV and Toda background 
solutions
(see, e.g$.$, \cite{bs}, \cite{bght}, \cite{deift}, 
\cite{dt}, \cite{eak}, Ch. 4,
\cite{fi}, \cite{gg}, \cite{gsv}, \cite{gss}, 
\cite{km}-\cite{kum}, \cite{lv},
Sect. 6.6, \cite{rs}--\cite{sh2}), and in connection 
with B\"acklund
transformations for the KdV and Toda hierarchies 
(cf$.$, e.g$.$,
\cite{bght}, \cite{ek}, \cite{ef}, \cite{fm}, \cite{gsv}, 
\cite{gw}, \cite{gss},
\cite{mk1}, \cite{mk2},
\cite{wa}).

Historically, methods of inserting eigenvalues in the 
case of differential
operators go back to Jacobi \cite{ja}, Darboux \cite{da}, 
Crum \cite{cr}, Gel'fand
and Levitan \cite{gl}, Schmincke \cite{sc}, and especially 
Deift \cite{deift}.
Two particular such methods, the so called single commutation 
or Crum-Darboux
method and the double commutation method, shortly to be 
described below, turned
out to be of particular importance. The operator theoretic 
approach developed in
\cite{deift} applies to the single commutation method and 
has been used in
\cite{deift} to give a complete spectral characterization in 
the differential
operator case. The double commutation method on the other hand 
required entirely
different methods and was only recently solved in the 
differential operator
case. A solution based on ODE techniques was given in 
\cite{com} and most
recently, a more general and at the same time greatly 
simplifying operator
theoretic approach to a spectral characterization of the 
double commutation
method appeared in \cite{gt}.

Surprisingly, a complete spectral characterization of both 
the single and double
commutation methods in the difference operator context is 
lacking in the
literature thus far. Although special cases of the single 
commutation method
with constant or algebro-geometric backgrounds have been 
discussed in
\cite{bght}, \cite{dkv},
\cite{vdo}, no treatment of general backgrounds is known to 
us. 
Moreover, with the exception of reference \cite{vdo}, where an
eigenvalue is  inserted into the spectral gap of a two-band
periodic Jacobi operator with period 2, no general
formulation of the double commutation method for finite difference
operators seems to be available in the literature. The present 
chapter fills these gaps
and provides a complete spectral characterization of the 
single commutation
method (based on Deift's operator theoretic approach) in 
Sections~\ref{secsc} and \ref{secscit} and
develops the corresponding results for the double commutation 
method in Sections~\ref{secsc}-\ref{secdcit}. Section~\ref{secappc}
gives three applications of our results.  The
discrete analog of the FIT formula for the isospectral 
torus of periodic
Schr\"odinger operators, thereby deriving an explicit 
realization of the
isospectral torus of all algebro-geometric quasi-periodic 
finite-gap Jacobi
operators, and the $N$-soliton solutions of the Toda and 
Kac-van
Moerbeke equations on an arbitrary background solution 
using the single
and double commutation methods.

In the remainder of this introduction we provide an 
informal discussion of
commutation methods and restrict ourselves to the case  of the
whole line and bounded Jacobi operators (so we don't have 
to bother with domain considerations). 

We first review the single commutation method 
\cite{TKvM}: Let $a,b \in
\ell(\Z)$ be two bounded, real-valued sequences 
satisfying
\begin{equation}
a(n)<0, \qquad b(n) \in \R,
\end{equation}
and introduce the corresponding Jacobi operator 
$H$ in $\lz$
\begin{equation}
(H f)(n) = a(n) f(n+1) + a(n-1) f(n-1) - b(n) f(n), 
\quad u \in
\ell^2(\Z).
\end{equation}
Next (cf.\ Lemma \ref{lemaeps}), assume the existence 
of two weak positive
solutions $u_\pm(\lam_1,n)$ of
\begin{equation} \label{possol}
H u_\pm = \lam_1 u_\pm, \qquad u_\pm(\lam_1,n)>0, 
\quad u_\pm(\lam_1,n) \in
\ell^2(\pm\N)
\end{equation}
(implying $b(n)+\lam_1<0$, i.e., $H-\lam_1 \ge 0$). 
$u_\pm$ are the principal
solutions as used, e.g., in \cite{crit}. Any positive 
solution can then be written
as
\begin{equation}
u_{\sig_1}(\lam_1,n) = \frac{1+\sig_1}{2} u_+(\lam_1,n) + 
\frac{1-\sig_1}{2}
u_-(\lam_1,n), \quad \sig_1 \in [-1,1].
\end{equation}
Now define the operator $A_{\sig_1}$
in $\lz$ by
\begin{equation}
(A_{\sig_1} f)(n) = \rho_{o,\sig_1}(n) f(n+1) + 
\rho_{e,\sig_1}(n) f(n),  
\quad f \in \lz,
\end{equation}
where
\begin{equation}
\rho_{o,\sig_1}(n) = - 
\sqrt{-\frac{a(n) u_{\sig_1}(\lam_1,n)
}{u_{\sig_1}(\lam_1,n+1)}},\quad \rho_{e,\sig_1}(n) = 
\sqrt{-\frac{a(n)
u_{\sig_1}(\lam_1,n+1)}{u_{\sig_1}(\lam_1,n)}}.
\end{equation}
We will always take the positive branch of all square 
roots involved. We note
that $\rho_{o,\sig_1}$ and $\rho_{e,\sig_1}$ are bounded 
sequences as can be seen
from
\begin{equation}
|\frac{a(n)u_{\sig_1}(\lam_1,n+1)}
{u_{\sig_1}^1(\lam_1,n)}| +
|\frac{a(n-1)u_{\sig_1}(\lam_1,n-1)}
{u_{\sig_1}(\lam_1,n)}| = |b(n)+\lam_1|.
\end{equation}
The adjoint operator $A_{\sig_1}^*$ of $A_{\sig_1}$ 
is given by
\begin{equation}
(A_{\sig_1}^* f)(n) = \rho_{o,\sig_1}(n-1) f(n-1) + 
\rho_{e,\sig_1}(n) f(n),
\quad f \in \lz,
\end{equation}
and for the (positive self-adjoint) operator 
$A_{\sig_1}^* A_{\sig_1}$ one infers 
\begin{equation}
A_{\sig_1}^* A_{\sig_1} = H -\lam_1.
\end{equation}
This shows that $(H-\lam_1) \ge 0$ is a necessary 
condition for the
existence of a positive solution of (\ref{possol}). We 
remark that this
condition is also sufficient (see, e.g., \cite{crit}, 
Theorem 2.8). Commuting
$A_{\sig_1}^*$ and $A_{\sig_1}$ (observing 
$(A_{\sig_1}^*)^*=A_{\sig_1}$) yields a
second positive self-adjoint bounded operator 
$A_{\sig_1} A_{\sig_1}^*$ and
further the commuted operator
\bea
H_{\sig_1} = A_{\sig_1} A_{\sig_1}^* + \lam_1.
\eea
A straightforward calculation shows
\bea
(H_{\sig_1} f)(n) = a_{\sig_1}(n) f(n+1) + 
a_{\sig_1}(n-1) f(n-1) - b_{\sig_1}(n)
f(n),
\eea
with
\bea
a_{\sig_1}(n) &=& 
-\frac{\sqrt{a(n) a(n+1) u_{\sig_1}(\lam_1,n)
u_{\sig_1}(\lam_1,n+2)}}{u_{\sig_1}(\lam_1,n+1)},
\\ b_{\sig_1}(n) &=& a(n) \Big(
\frac{u_{\sig_1}(\lam_1,n)}{u_{\sig_1}(\lam_1,n+1)} +
\frac{u_{\sig_1}(\lam_1,n+1)}
{u_{\sig_1}(\lam_1,n)} \Big) - \lam_1.
\eea
As proven by Deift \cite{deift}, the operators 
$H-\lam_1$ and $H_{\sig_1}-\lam_1$,
restricted to the orthogonal complements of their 
respective null-spaces, are
unitarily equivalent. Specifically, we have
\begin{equation} \label{sigsc}
\ba{rcl@{\quad}rcl}
\sigma(H_{\sig_1}) \!&=&\! \left\{ \ba{cl} 
\sigma(H) \cup \{ \lam_1\}, &
\sig_1 \in (-1,1)\\ \sigma(H), & 
\sig_1 \in \{-1,1\} \ea \right. ,&
\sig_{ac}(H_{\sig_1}) \!&=&\! \sig_{ac}(H), \\ 
\sigma_{p}(H_{\sig_1}) \!&=&\!
\left\{ \ba{cl} \sigma_{p}(H) \cup \{ \lam_1\}, 
& \sig_1 \in (-1,1)\\
\sigma_{p}(H), & \sig_1 \in \{-1,1\} \ea \right. , 
& \sig_{sc}(H_{\sig_1}) 
\!&=&\! \sig_{sc}(H).
\ea 
\end{equation}
Here $\sigma_{p}(.),\sigma_{ac}(.)$, and 
$\sigma_{sc}(.)$ denote
the the point spectrum (i.e., the set of eigenvalues), 
absolutely, and singularly
continuous spectrum, respectively.

This method is known as the single commutation method 
\cite{TKvM} and we will
give a complete spectral characterization of it in 
Sections~\ref{secsc} and \ref{secscit}.

Our next aim is to remove the condition that $H$ is 
bounded from below and
thereby introduce the double commutation method. Fix
$\gam_\pm>0$ and define
\bea
\rho_{o,\gam_\pm}(n) &=& \rho_{e,\pm1}(n+1)
\sqrt{\frac{c_{\gam_\pm}(\lam_1,n)}
{c_{\gam_\pm}(\lam_1,n+1)}},\\
\rho_{e,\gam_\pm}(n) &=& \rho_{o,\pm1}(n)
\sqrt{\frac{c_{\gam_\pm}(\lam_1,n+1)}
{c_{\gam_\pm}(\lam_1,n)}},
\eea
where
\bea
c_{\gam_\pm}(\lam_1,n) = 1 + 
\gam_\pm \sum_{j=\pm\infty}^{n+1 \atop n}
u_{\pm}(\lam_1,j)^2,
\eea
and introduce corresponding operators 
$A_{\gam_\pm},A_{\gam_\pm}^*$ in $\lz$ by
\bea
(A_{\gam_\pm} f)(n) 
&=& \rho_{o,\gam_\pm}(n) f(n+1) + 
\rho_{e,\gam_\pm}(n)
f(n),\\ (A_{\gam_\pm}^* f)(n) 
&=& \rho_{o,\gam_\pm}(n-1) f(n-1) +
\rho_{e,\gam_\pm}(n) f(n).
\eea
A simple calculation shows that 
$A_{\gam_\pm}^* A_{\gam_\pm} = A_{\pm1}
A_{\pm1}^*$ and hence
\bea
H_{\pm1} = A_{\gam_\pm}^* A_{\gam_\pm} + \lam_1.
\eea
Performing a second commutation yields the 
doubly commuted operator
\bea
H_{\gam_\pm} = 
A_{\gam_\pm} A_{\gam_\pm}^* + \lam_1.
\eea
Explicitly, one verifies
\bea
(H_{\gam_\pm} f)(n) = 
a_{\gam_\pm}(n) f(n+1) + a_{\gam_\pm}(n-1)
f(n-1) - b_{\gam_\pm}(n) f(n),
\eea
with
\bea
a_{\gam_\pm}(n) &=& a(n+1)
\frac{\sqrt{c_{\gam_\pm}(\lam_1,n)
c_{\gam_\pm}(\lam_1,n+2)}}{c_{\gam_\pm}(\lam_1,n+1)},\\
\nn b_{\gam_\pm}(n) 
&=& b(n+1) \pm \gam_\pm \Big( \frac{a(n) u_{\pm}(\lam_1,n)
u_{\pm}(\lam_1,n+1)}{c_{\gam_\pm}(\lam_1,n)} 
\\&& {}- \frac{a(n+1)
u_{\pm}(\lam_1,n+1) u_{\pm}(\lam_1,n+2)}
{c_{\gam_\pm}(\lam_1,n+1)}\Big).
\eea

Now observe that $H_{\gam_\pm}$ remains well-defined 
even if $u_\pm$ is no
longer positive. This applies, in particular, in the 
case where $u_\pm(\lam_1)$
has zeros and hence all intermediate operators 
$A_{\pm1}, A_{\gam_\pm},
H_{\pm1}$, etc., become ill-defined. Thus to 
define $H_{\gam_\pm}$ it
suffices to assume the existence of a solution 
$u_\pm(\lam_1)$ which is square
summable near $\pm\infty$. This condition is much 
less restrictive than the
existence of a positive solution (e.g., 
$\sigma(H) \ne \R$, i.e., the existence
of a spectral gap for $H$ around $\lam_1$ is 
sufficient in this context).

One expects that formulas analogous to (\ref{sigsc}) 
will carry over to this
more general setup. That this is actually the case 
will be shown in our
principal Theorem \ref{thmdc} of Section~\ref{secdc}. 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 $H$.



\section{The Single Commutation Method}
\label{secsc}

In this section we intend to give a detailed 
investigation of the single
commutation method. We will assume $a,b$ to satisfy
(H.\ref{hypoaposb}) throughout Sections~\ref{secsc} and \ref{secscit}.

We shall consider (self-adjoint) Jacobi operators 
$H$ associated with the
difference expression
\bea \label{diffex}
(\tau f)(n) = a(n) f(n+1) + a(n-1) f(n-1) - b(n) f(n),
\eea
in the Hilbert space $\lz$. As a preparation we prove

We start with operators associated with the difference 
expression
(\ref{diffex}) on the half axis $\pm\N$. For simplicity we 
will do most
calculations only for $\ell^2(\N)$. Let $u(\lam_1)$ be a 
positive solution of
$\tau u = \lam_1 u$ and define
\bea
\rho_{o,+}(n) 
&=& -\sqrt{-\frac{a(n)u(\lam_1,n+1)}{u(\lam_1,n)}}, \\
\rho_{e,+}(n) 
&=& \sqrt{-\frac{a(n-1)u(\lam_1,n-1)}{u(\lam_1,n)}}, 
\quad n>0.
\eea
Define the operator $\dot{A}_+$ on $\ell_0(\N)$
\bea
(\dot{A}_+ f)(n) = \rho_{o,+}(n) f(n+1) + 
\rho_{e,+}(n) f(n), \quad f \in
\ell_0(\N)
\eea
and denote its operator closure (in $\ell^2(\N)$) by 
$A_+$. One verifies,
\bea
\db(A_+) \subseteq \{ f \in \ell^2(\N) | \rho_{o,+}(n) 
f(n+1) + \rho_{e,+}(n)
f(n) \in \ell^2(\N) \}.
\eea
The adjoint $A_+^*$ of $A_+$ is then given by
\bea
&(A_+^* f)(n) = \rho_{o,+}(n-1) f(n-1) + 
\rho_{e,+}(n) f(n), & \\ \nn
&\db(A_+^*) = \{ f \in \ell^2(\N) | f(0)=0; 
\rho_{o,+}(n-1) f(n-1) +
\rho_{e,+}(n) f(n) \in \ell^2(\N) \}. &
\eea
(The boundary condition $f(0)=0$ is only introduced 
so that we don't have to
specify $(A_+^* f)(1)$ separately.) Due to a well known 
result of von Neumann
(see, e.g., \cite{wd}, Theorem 5.39) the 
operator $A_+ A_+^*$ is a
positive self-adjoint operator when defined naturally
\bea
\db(A_+ A_+^*) = 
\{ f \in \db(A_+^*) | A_+^* f \in \db(A_+) \}.
\eea
A simple calculation shows 
$A_+ A_+^* f = (\tau -\lam_1)f$ and hence we may
define
\bea
H_+ = A_+ A_+^* + \lam_1, \quad \db(H_+) 
&\subseteq& \{ f \in
\ell^2(\N) | f(0)=0, \: \tau f \in \ell^2(\N) \},
\eea
where equality in the last relation is equivalent 
to $\tau$ being limit point
($l.p.$) at $+\infty$. Similarly one defines 
for $n<0$
\bea
\rho_{o,-}(n) = 
-\sqrt{-\frac{a(n)u(\lam_1,n)}{u(\lam_1,n+1)}}, \quad
\rho_{e,-}(n) = 
\sqrt{-\frac{a(n)u(\lam_1,n+1)}{u(\lam_1,n)}}
\eea
and operators $A_-$, and $A_-^*$ in $\ell^2(-\N)$ 
which satisfy $H_- = A_-^* A_-
+\lam_1$.

Commuting $A_\pm^*$ and $A_\pm$ yields a second 
positive
self-adjoint operator $A_- A_-^*$, respectively 
$A_+^* A_+$, and further the
commuted operators
\bea
H_{+,1} = A_+^* A_+ + \lam_1, \qquad H_{-,1} = 
A_- A_-^* + \lam_1.
\eea
The next theorem characterizes $H_{\pm,1}$ in 
terms of $H_\pm$, but first we
need to introduce

\bh \label{hyposc}
Suppose $H_\pm$ satisfies one of the following spectral 
conditions.\\
(i) $\sig_{ess} (H_\pm) \ne \emptyset$.\\
(ii) $\sig(H_\pm)=\sig_d(H_\pm)=
\{ \lam_{\pm,j} \}_{j \in J_\pm}$ with $\sum_{j\in J_\pm}
(1+\lam^2_{\pm,j})^{-1} =\infty$.
\eh

Hypothesis (H.\ref{hyposc}) is satisfied if $a,b$ are bounded 
near $\pm\infty$.

Either one of the conditions (i), (ii) implies that 
$\tau$ is $l.p.$ at
$\pm\infty$. This follows since otherwise the resolvent 
of $H_\pm$ would be a
Hilbert-Schmidt operator contradicting (i), (ii). This 
further implies that the
domain of $H_\pm$ is given by
\bea
\db(H_\pm) = 
\{ f \in \ell^2(\pm\N) | f(0)=0, \: \tau f \in \ell^2(\pm\N)\}.
\eea

\begin{thm} \label{thmpm}
Assume (H.\ref{hypoaposb}) and (H.\ref{hyposc}). Then the operators $H_{\pm,1}$ 
constructed above satisfy
(H.\ref{hypoaposb}) and (H.\ref{hyposc}) and are given by
\bea \nn
(H_{\pm,1} f)(n) &=& (\tau_{\pm,1} f)(n)\\ 
&=& a_{\pm,1}(n) f(n+1) + 
a_{\pm,1}(n-1) f(n-1) - b_{\pm,1}(n) f(n),\\
\nn \db(H_{\pm,1}) &=& \{ f \in \ell^2(\pm\N) | f(0)=0, \:
\tau_{\pm,1} f \in \ell^2(\pm\N) \},
\eea
with
\bea
a_{+,1}(n) &=& -\frac{\sqrt{a(n-1) a(n) u(\lam_1,n-1)
u(\lam_1,n+1)}}{u(\lam_1,n)},\quad n > 0, \\ \nn
b_{+,1}(n) 
&=& a(n-1) \Big(\frac{u(\lam_1,n)}{u(\lam_1,n-1)} +
\frac{u(\lam_1,n-1)}{u(\lam_1,n)}
\Big) - \lam_1,\quad n>1,\\
b_{+,1}(1) 
&=& a(0) \frac{u(\lam_1,0)}{u(\lam_1,1)} - \lam_1,
\eea
and
\bea
a_{-,1}(n) &=& -\frac{\sqrt{a(n) a(n+1) u(\lam_1,n)
u(\lam_1,n+2)}}{u(\lam_1,n+1)},\quad n <-1, \\ \nn
b_{-,1}(n) 
&=& a(n) \Big(\frac{u(\lam_1,n)}{u(\lam_1,n+1)} +
\frac{u(\lam_1,n+1)}{u(\lam_1,n)}
\Big) - \lam_1,\quad n<-1,\\
b_{-,1}(-1) &=& a(-1) \frac{u(\lam_1,0)}{u(\lam_1,-1)}- 
\lam_1.
\eea
Moreover, $H_\pm -\lam_1$ and $H_{\pm,1} -\lam_1$ 
restricted to the orthogonal
complements of their null-spaces are unitarily 
equivalent and hence
\bea
\ba{rcl@{\quad}rcl}
\sigma(H_{\pm,1}) \bs \{ \lam_1\}
 \!&=&\! \sigma(H_\pm) \bs \{ \lam_1\}, &
\sig_{ac}(H_{\pm,1}) \!&=&\! \sig_{ac}(H_\pm), 
\\  \sigma_{p}(H_{\pm,1}) \bs \{
\lam_1\} \!&=&\! \sig_{p}(H_\pm) \bs \{ \lam_1\}, 
& \sig_{sc}(H_{\pm,1}) \!&=&\!
\sig_{sc}(H_\pm).
\ea 
\eea
\end{thm}

\bpf
The unitary equivalence follows from \cite{deift}, 
Theorem 1 and clearly settles
the spectral claims. Thus both $H_\pm$ and $H_{\pm,1}$ 
satisfy (H.\ref{hyposc}) and
hence $\tau_\pm$ and $\tau_{\pm,1}$ are $l.p.$ at 
$\pm\infty$. The rest are
straightforward calculations.
\epf

Next we turn to the case of the whole lattice $\lz$. We 
pick $\sigma_1 \in [-1,1]$
and $\lam_1 < \inf(\sigma(H))$. Further denote by 
$u_\pm(\lam,n)$ (for $\lam <
\inf(\sigma(H))$) the solutions constructed in Lemma~\ref{lemaeps}
and set
\bea
u_{\sig_1}(\lam_1,n) = \frac{1+\sigma_1}{2} u_+(\lam_1,n)
 + \frac{1-\sigma_1}{2}
u_-(\lam_1,n).
\eea
Now define sequences
\bea
\rho_{o,\sig_1}(n) = 
-\sqrt{-\frac{a(n)u_{\sig_1}(\lam_1,n)
}{u_{\sig_1}(\lam_1,n+1)}}, \quad \rho_{e,\sig_1}(n) =
\sqrt{-\frac{a(n)u_{\sig_1}(\lam_1,n+1)}
{u_{\sig_1}(\lam_1,n)}},
\eea
and the corresponding operator $A_{\sig_1}$ 
(first on $\ell_0(\Z)$ and then take
the closure in $\lz$ as before) together with its adjoint
$A_{\sig_1}^*$,
\bea
&(A_{\sig_1} f)(n) = \rho_{o,\sig_1}(n) f(n+1) + 
\rho_{e,\sig_1}(n) f(n),&\\ \nn
&\db(A_{\sig_1}) 
\subseteq \{ f \in \lz | \rho_{o,\sig_1}(n) f(n+1) +
\rho_{e,\sig_1}(n) f(n) \in \lz \},
&\\ &(A_{\sig_1}^* f)(n) = \rho_{o,\sig_1}(n-1)
f(n-1) + \rho_{e,\sig_1}(n) f(n),&\\ \nn
&\db(A_{\sig_1}^*) = 
\{ f \in \lz | \rho_{o,\sig_1}(n-1) f(n-1) +
\rho_{e,\sig_1}(n) f(n) \in \lz \}.&
\eea
Again by von Neumann's result $A_{\sig_1}^* A_{\sig_1}$ 
is a positive self-adjoint
operator when defined naturally by
\bea
\db(A_{\sig_1}^* A_{\sig_1}) = 
\{ f \in \db(A_{\sig_1}) | A_{\sig_1} f \in
\db(A_{\sig_1}^*) \}.
\eea
A simple calculation shows $A_{\sig_1}^* A_{\sig_1} 
= \tau -\lam_1$ and we hence
may define
\bea \label{hpm}
H = A_{\sig_1}^* A_{\sig_1} + \lam_1, 
\qquad \db(H) \subseteq \{ f \in \lz |
\tau f \in \lz \}.
\eea
Commuting $A_{\sig_1}^*$ and $A_{\sig_1}$ yields
 a second positive
self-adjoint operator $A_{\sig_1} A_{\sig_1}^*$ and 
further the commuted
operator
\bea
H_{\sig_1} = A_{\sig_1} A_{\sig_1}^* + \lam_1, 
\qquad \db(H_{\sig_1}) \subseteq
\{ f \in \lz | \tau_{\sig_1} f \in \lz \},
\eea
where $\tau_{\sig_1}$ is the difference expression 
corresponding to $H_{\sig_1}$.
The next theorem characterizes $H_{\sig_1}$ under 
Assumption (H.2.2) for
$H_+$ and $H_-$ implying that $\tau$ is $l.p.$ at 
$\pm\infty$ and hence that
\bea
\db(H) = \{ f \in \lz | \tau f \in \lz \}.
\eea


\begin{thm} \label{thmsc}
Assume (H.\ref{hypoaposb}) and (H.\ref{hyposc}). Then the operator 
$H_{\sig_1}$,
\bea\nn
(H_{\sig_1} f)(n) &=& (\tau_{\sig_1}f)(n)\\ 
&=& a_{\sig_1}(n) f(n+1) + 
a_{\sig_1}(n-1) f(n-1) - b_{\sig_1}(n) f(n),\\ \nn
\db(H_{\sig_1}) 
&=& \{ f \in \lz | \tau_{\sig_1} f \in \lz \},
\eea
is self-adjoint. Moreover,
\bea
a_{\sig_1}(n) 
&=& -\frac{\sqrt{a(n) a(n+1) u_{\sig_1}(\lam_1,n)
u_{\sig_1}(\lam_1,n+2)}}{u_{\sig_1}(\lam_1,n+1)}, \\
b_{\sig_1}(n) &=& a(n) \Big( \frac{u_{\sig_1}(\lam_1,n)}
{u_{\sig_1}(\lam_1,n+1)} +
\frac{u_{\sig_1}(\lam_1,n+1)}
{u_{\sig_1}(\lam_1,n)} \Big) - \lam_1
\eea
and $a_{\sig_1}$, $b_{\sig_1}$ satisfy (H.\ref{hypoaposb}). The 
equation $\tau_{\sig_1} v=
\lam_1 v$ has the positive solution
\bea
v_{\sig_1}(\lam_1,n) = 
\frac{1}{\sqrt{-a(n) u_{\sig_1}(\lam_1,n)
u_{\sig_1}(\lam_1,n+1)}}
\eea
which is an eigenfunction of $H_{\sig_1}$ if and only 
if $\sigma_1 \in (-1,1)$.
$H-\lam_1$ and $H_{\sig_1}-\lam_1$ restricted to the 
orthogonal complements of
their corresponding one-dimensional null-spaces are 
unitarily equivalent and hence
\begin{equation}
\ba{rcl@{\quad}rcl}
\sigma(H_{\sig_1}) \!&=&\! \left\{ \ba{cl} 
\sigma(H) \cup \{ \lam_1\}, &
\sig_1 \in (-1,1)\\ \sigma(H), 
& \sig_1 \in \{-1,1\} \ea \right. ,&
\sig_{ac}(H_{\sig_1}) \!&=&\! \sig_{ac}(H), \\ 
\sigma_{p}(H_{\sig_1}) \!&=&\!
\left\{ \ba{cl} \sigma_{p}(H) \cup \{ \lam_1\}, 
& \sig_1 \in (-1,1)\\
\sigma_{p}(H), & \sig_1 \in \{-1,1\} \ea \right. , 
& \sig_{sc}(H_{\sig_1})
\!&=&\! \sig_{sc}(H).
\ea 
\end{equation}
In addition, the sequence
\bea
(A_{\sig_1} u)(z,n) = 
\frac{W_n(u_{\sig_1}(\lam_1),u(z))}{\sqrt{-a(n)
u_{\sig_1}(\lam_1,n) u_{\sig_1}(\lam_1,n+1)}}
\eea
solves $\tau_{\sig_1} u = z u$ if $u(z)$ solves 
$\tau u = z u$ for some $z \in
\C$, where
\begin{equation}
W_n(u,v) = a(n)(u(n) v(n+1) - 
u(n+1) v(n))
\end{equation}
denotes the modified Wronskian. Moreover, one obtains
\bea \label{wronsc}
W_{\sig_1,n}(A u(z), A v(z)) = 
(\lam_1-z) W_n(u(z),v(z))
\eea
for solutions $u,v$ of $\tau u = z u$, where 
$W_{\sig_1,n}(u,v) =
a_{\sig_1}(n)(u(n) v(n+1) - u(n+1) v(n))$.
The resolvents of $H,H_{\sig_1}$ for 
$z \in \C \bs (\sig(H) \cup \{\lam_1\})$
are related via
\bea \label{res}
(H_{\sig_1} - z)^{-1} = \frac{1}{z-\lam_1} 
\Big(1 - A_{\sig_1} (H - z)^{-1}
A_{\sig_1}^* \Big)
\eea
or, in terms of Green's functions for $n \ge m$, 
$z \in \C \bs (\sig(H) \cup
\{\lam_1\})$, 
\bea \nn 
&G(z,n,m) = \frac{\D u_+(z,n)u_-(z,m)}
{\D W_n(u_+(z),u_-(z))}&\\
\label{gfsc} &\mb{implies}\quad G_{\sig_1}(z,n,m) = 
\frac{\D (A_{\sig_1}
u_+)(z,n) (-A_{\sig_1} u_-)(z,m)}
{\D (z-\lam_1) W_n(u_+(z),u_-(z))}. &
\eea
Furthermore, $u_{\sig_1,\pm}(z,n)$, the principal 
solutions of $(H_{\sig_1} - z)
u=0$ for $z<\lam_1$, are given by
\bea
u_{\sig_1,\pm}(z,n) = \pm A_{\sig_1} u_\pm(z,n) = 
\frac{ \mp
W_n(u_{\sig_1}(\lam_1),u_\pm(z))}
{\sqrt{-a(n) u_{\sig_1}(\lam_1,n)
u_{\sig_1}(\lam_1,n+1)}}.
\eea
In addition, we have
\bea \label{normv}
\sum_{n \in \Z} v_{\sig_1}(\lam_1,n)^2 = \frac{4}{1-\sig_1^2}
W(u_-(\lam_1),u_+(\lam_1))^{-1}, \quad \sig_1 \in (-1,1)
\eea
and, if $\tau u(\lam) = \lam u(\lam)$, 
$u(\lam,.) \in \lz$,
\bea \label{normuz}
\sum_{n \in \Z}  (A_{\sig_1} u)(\lam,n)^2 = 
(\lam-\lam_1) \sum_{n \in \Z}
u(\lam,n)^2.
\eea
\end{thm}

\bpf
The unitary equivalence together with equation 
(\ref{res}) follow from \cite{deift}, Theorem 1. That 
$H_{\sig_1}$ is $l.p.$
at $\pm\infty$ follows upon looking at the restrictions 
$H_\pm$, $H_{\pm,1}$ and
using Theorem \ref{thmpm}. Equation (\ref{res}) 
together with (\ref{wronsc})
imply (\ref{gfsc}). The facts concerning the point 
spectrum follow since
$G_{\sig_1}(z,n,n)$ has a pole at $z=\lam_1$ if and 
only if $\sig_1 \in (-1,1)$.
(\ref{normv}) can be obtained by investigating the 
residue of
$G_{\sig_1}(z,n,n)$ at $z=\lam_1$. The rest are 
straightforward calculations.
\epf

\begin{rem} \label{remagz}
(i). Hypothesis (H.\ref{hyposc}) is only needed in 
Theorem \ref{thmsc} to characterize the
domains of $H$ and $H_{\sig_1}$ explicitly.\\
(ii). Multiplying $u_{\sig_1}$ with a positive 
constant leaves all
formulas and, in particular, $H_{\sig_1}$ 
invariant.\\
(iii). If $H$ is bounded from above we can insert 
eigenvalues into the
highest spectral gap, i.e., above the spectrum of 
$H$, upon considering $-H$.
Then $\lam>\sup(\sigma(H))$ implies that we don't 
have positive but rather
alternating solutions and all our previous calculations 
carry over with minor
changes.\\ 
(iv). We can weaken (H.\ref{hypoaposb}) by requiring $a(n) \ne 0$ 
instead of $a(n)<0$.
Everything stays the same with the only difference that 
$u_\pm$ are not positive
but change sign in such a way that (\ref{posauou}) 
stays positive. Moreover, the
signs of $a_{\sig_1}(n)$ can also be prescribed 
arbitrarily by altering the
signs of $\rho_{o,\sig_1}$ and $\rho_{e,\sig_1}$.\\
(v). The fact that $v_{\sig_1} \in \ell^2(\Z)$ if 
and only if $\sig_1 \in
(-1,1)$ gives an alternate proof of
\bea
\sum_{n=0}^{\pm\infty} 
\frac{1}{-a(n)u_{\sig_1}(\lam_1,n)u_{\sig_1}(\lam_1,n+1)} <
\infty \mb{ if and only if }
\sig_1 \in \ba{c}  [-1,1) \\ (-1,1] \ea
\eea
(cf.\ \cite{pat2} and \cite{crit}, Lemma 2.10, 
Remark 2.11).
\end{rem}

At the end of this section we will show some connections 
between the single
commutation method and some other theories. We start
with the Weyl-Titchmarsh
theory and freely use the definitions of Appendices 
B and C. 

\begin{lem}
Assume (H.\ref{hypoaposb}). The Weyl $\ti{m}$-functions $\ti{m}_{\pm,\sig_1}(z)$ 
of $H_{\sig_1}$, $\sig_1 \in [-1,1]$ in terms of $\ti{m}_{\pm}(z)$,  
the ones of $H$, read
\bea
\ti{m}_{\pm,\sig_1}(z) = \frac{-u_{\sig_1}(\lam_1,1)}
{a(1) u_{\sig_1}(\lam_1,2)} 
\Big( 1 + \frac{(z-\lam_1) \ti{m}_\pm(z)}{1 + 
\frac{a(0) u_{\sig_1}(\lam_1,0)
}{u_{\sig_1}(\lam_1,1)} \ti{m}_\pm(z)} \Big).
\eea
\end{lem}

\bpf
The above formulas are straightforward calculations 
using (\ref{gfsc}) and
(\ref{mpl}), (\ref{mmi}).
\epf

Finally we turn to scattering theory. In order to 
facilitate comparison with the
standard literature on (inverse) scattering theory 
for second-order 
difference operators (cf.\ \cite{dinv1},
\cite{dinv2}, \cite{fad}, \cite{conl}, \cite{gu}, \cite{ta}) 
we now assume
\bea
a(n)>0,\: b(n)\in\R, \qquad n|1 - 2a(n)|, 
n b(n) \in \ell^1(\Z)
\eea
(cf. Remark \ref{remagz}). This implies
\bea
\sig_{ac}(H) =[-1,1], \quad \sig_{sc}(H) = 
\emptyset, \quad \sig_{p}(H) = \{
\lam_j \}_{j \in J} \subseteq \R \backslash [-1,1],
\eea
where $J \subseteq \N$ is a suitable (finite) 
index set,
and the existence of the so called Jost solutions 
$f_\pm(k,n)$,
\bea
\Big( \tau - 
\frac{k+k^{-1}}{2} \Big) f_\pm(k,n) = 0, \quad
\lim_{n\to\pm\infty} k^{\mp n} f_\pm(k,n) = 1, 
\quad |k| \le 1.
\eea 
Transmission $T(k)$ and reflection $R_\pm(k)$ 
coefficients are then defined
via
\bea
T(k) f_\mp(k,n) = f_\pm(k^{-1},n) + 
R_\pm(k) f_\pm(k,n), \quad |k|=1,
\eea
and the norming constants $\gam_{\pm,j}$ 
corresponding to $\lam_j \in
\sig_{p}(H)$ are given by
\bea \label{norming}
\gam_{\pm,j}^{-1} = 
\sum_{n \in \Z} |f_\pm(k_j,n)|^2, \quad k_j =
\lam_j + \sqrt{\lam_j^2 -1} \in (-1,0),\: j \in J.
\eea

\begin{lem} \label{scatsc}
Suppose $H$ satisfies {\em (\ref{decay})} and let 
$H_{\sig_1}$ be constructed as
in Theorem \ref{thmsc} with
\bea
u_{\sig_1}(\lam_1,n) = \frac{1+\sig_1}{2} f_+(k_1,n) + 
\frac{1-\sig_1}{2}
f_-(k_1,n).
\eea
Then the transmission $T_{\sig_1}(k)$ and reflection 
coefficients
$R_{\pm,\sig_1}(k)$ of $H_{\sig_1}$ in terms of the 
corresponding scattering 
data $T(k),R_\pm(k)$ of $H$ are given by
\bea
T_{\sig_1}(k) = \frac{1 - k \, k_1}{k - k_1} T(k), 
\quad  R_{\pm,\sig_1}(k) =
k^{\pm 1} \frac{k - k_1}{1 - 
k \, k_1} R_\pm(k), \:\: \sig_1 \in
(-1,1),
\eea
\bea
T_{\sig_1}(k) = T(k), \quad  R_{\pm,\sig_1}(k) =
\frac{k_1^{\sig_1} - k^{\mp 1}}{k_1^{\sig_1} - 
k^{\pm 1}} R_\pm(k), \quad 
\sig_1 \in \{-1,1\},
\eea
where $k_1 = \lam_1 + \sqrt{\lam_1^2 -1} \in 
(-1,0)$. Moreover, the norming
constants $\gam_{\sig_1,\pm,j}$ associated with 
$\lam_j \in \sig_p(H_{\sig_1})$
in terms of $\gam_{\pm,j}$ corresponding to 
$H$ read
\bea \nn
\gam_{\sig_1,\pm,j} &=& |k_j|^{\pm1} \frac{1 - k_j k_1}
{(k_j - k_1)}
\gam_{\pm,j}, \quad j \in J, \: \sig_1 \in (-1,1),\\
\gam_{\sig_1,\pm,1} &=& \left( \frac{1-\sig_1}
{1+\sig_1} \right)^{\pm1} |1 -
k_1^{\mp2}| \, T(k_1), \quad \sig_1 \in (-1,1),
\eea
\bea
\gam_{\sig_1,\pm,j} = |k_1^{\sig_1} - 
k_j^{\mp1}| \gam_{\pm,j}, \quad j \in J,
\: \sig_1 \in \{-1,1\}.
\eea
\end{lem}

\bpf
The claims follow easily after observing that up 
to normalization the Jost
solutions of $H_{\sig_1}$ are given by 
$A_{\sig_1} f_\pm(k,n)$ (compare
(\ref{gfsc})).
\epf


\section{Iteration of the Single Commutation Method}
\label{secscit}



By choosing $\lam_2<\lam_1$ and $\sigma_2 \in [-1,1]$ 
we can define
\bea
u_{\sig_1,\sig_2}(\lam_2,n) = 
\frac{1+\sig_2}{2} u_{\sig_1,+}(\lam_2,n) +
\frac{1-\sig_2}{2} u_{\sig_1,-}(\lam_2,n)
\eea
and repeat the process of the previous section by 
defining
$\rho_{o,\sig_1,\sig_2}$, $\rho_{e,\sig_1,\sig_2}$ and 
corresponding operators
$A_{\sig_1,\sig_2}$, $A_{\sig_1,\sig_2}^*$ which 
satisfy
\bea
H_{\sig_1} = 
A_{\sig_1,\sig_2}^* A_{\sig_1,\sig_2} - \lam_2.
\eea
A further commutation then yields the operator
\bea
H_{\sig_1,\sig_2} = 
A_{\sig_1,\sig_2}A_{\sig_1,\sig_2}^* - \lam_2
\eea
associated with sequences $a_{\sig_1,\sig_2}$, 
$b_{\sig_1,\sig_2}$. The result
after $N$ steps is summarized in

\begin{thm} \label{thmscom}
Assume (H.\ref{hypoaposb}) and (H.\ref{hyposc}). Let $H$ be as in 
Section~\ref{secsc} and choose
\bea
\lam_N < \dots < \lam_2 < \lam_1 < \inf(\sigma(H)), 
\quad \sigma_\ell \in [-1,1],
\:\: 1\le \ell \le N, \: N \in \N.
\eea
Then we have
\bea
a_{\sig_1,\dots,\sig_N}(n) \!\!&=&\!\! - 
\sqrt{a(n) a(n+N)} \frac{ \sqrt{
C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)
 C_{n+2}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)
}}{C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)},\\ \nn
b_{\sig_1,\dots,\sig_N}(n) \!\!&=&\!\! - \lam_N 
+ a(n)
\frac{C_{n+2}(u_{\sig_1}^1,\dots,u_{\sig_{N-1}}^{N-1})
C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}
{C_{n+1}(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1}) 
C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)} \\ && {} +
a(n+N-1) \frac{C_n(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1})
C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}
{C_{n+1}(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1}) 
C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)},
\eea
where
\bea
u_{\sig_\ell}^\ell(n) = 
\frac{1+\sigma_\ell}{2} u_+(\lam_\ell,n) +
(-1)^{\ell+1} 
\frac{1-\sigma_\ell}{2} u_-(\lam_\ell,n),
\eea
and $C_n$ denotes the $n$-dimensional Casoratian
\bea \label{Nwon}
C_n(u_1,\dots,u_N) = 
\det\{ u_i(n+j-1)\}_{1\le i,j\le N}.
\eea
Moreover, for $1 \le \ell \le N$, $\lam < \lam_\ell$
\bea \label{defuj}
u_{\sig_1,\dots,\sig_\ell,\pm}(\lam,n) =  
\frac{ \pm \sqrt{
\prod\limits_{j=0}^{\ell-1} (-a(n+j))}
C_n(u_{\sig_1}^1,\dots,u_{\sig_\ell}^\ell,u_\pm(\lam))}
{\sqrt{
C_n(u_{\sig_1}^1,\dots, u_{\sig_\ell}^\ell)
C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_\ell}^\ell) }},
\eea
are the principal solutions of 
$\tau_{\sig_1,\dots,\sig_\ell} u = \lam u$
and
\begin{equation}
u_{\sig_1,\dots,\sig_\ell}(\lam_\ell,n) = 
\frac{1+\sig_\ell}{2}
u_{\sig_1,\dots,\sig_{\ell-1},+}(\lam_\ell,n) +
\frac{1-\sig_\ell}{2} 
u_{\sig_1,\dots,\sig_{\ell-1},-}(\lam_\ell,n)
\end{equation}
is used to define $H_{\sig_1,\dots,\sig_\ell}$. 
We also have
\bea \label{rhoo}
&\rho_{o,\sig_1,\dots,\sig_N}(n) =
-\sqrt{-a(n) \frac{C_{n+2}(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1})
C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}
{C_{n+1}(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1}) 
C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}},&\\
\label{rhoe} &\rho_{e,\sig_1,\dots,\sig_N}(n) =
\sqrt{-a(n+N-1) \frac{C_n(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1})
C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}
{C_{n+1}(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1}) 
C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}}.&
\eea
The spectrum of $H_{\sig_1,\dots,\sig_N}$ is 
given by
\bea
\sigma(H_{\sig_1,\dots,\sig_N}) = 
\sigma(H) \cup \{ \lam_\ell \; | \;
\sigma_\ell \in (-1,1), \; 1 \le \ell \le N \}.
\eea
\end{thm}

\bpf
It is enough to prove the formulas for 
$a_{\sig_1,\dots,\sig_N}(n)$ and
$u_{\sig_1,\dots,\sig_N}(n)$, the remaining 
assertions then follow easily. We
will use a proof by induction on $N$. They are 
valid for $N=1$ and we need to
show
\begin{equation}
u_{\sig_1,\dots,\sig_{N+1},\pm}(\lam,n) =
\frac{\sqrt{-a_{\sig_1,\dots,\sig_N}(n)} 
C_n(u_{\sig_1,\dots,\sig_N}(\lam_N),
u_{\sig_1,\dots,\sig_N,\pm1}(\lam))}{ \pm
\sqrt{u_{\sig_1,\dots,\sig_N}(\lam_N,n) 
u_{\sig_1,\dots,\sig_N}(\lam_N,n+1)}},
\end{equation}
\bea \nn
a_{\sig_1,\dots,\sig_{N+1}}(n) 
&=& \sqrt{a_{\sig_1,\dots,\sig_N}(n)
a_{\sig_1,\dots,\sig_N}(n+1)} \times \\ 
&& \frac{\sqrt{
u_{\sig_1,\dots,\sig_N}(\lam_N,n) 
u_{\sig_1,\dots,\sig_N}(\lam_N,n+1)}
}{u_{\sig_1,\dots,\sig_N}(\lam_N,n+1)}.
\eea
The first relation follows after a straightforward 
calculation using
Sylvester's determinant identity 
(cf.\ \cite{gant}, Sect.\ II.3)
\bea \nn
&C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N,u_\pm(\lam))
C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_{N+1}}^{N+1})& 
\\ \nn &
{}- C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N,u_\pm(\lam))
C_n(u_{\sig_1}^1,\dots,u_{\sig_{N+1}}^{N+1})&\\ & =
C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)
C_n(u_{\sig_1}^1,\dots,
u_{\sig_{N+1}}^{N+1},u_\pm(\lam)),&
\eea
and the second is a simple calculation.
\epf

\begin{rem}
If $u(z,n)$ is any solution of  $\tau u = z u$, $z \in \C$ define
$u_{\sig_1,\dots,\sig_N}(z,n)$ as in (\ref{defuj}) but with 
$\ell=N$ and $u_\pm(\lam,n)$ replaced by $u(z,n)$. Then
$u_{\sig_1,\dots,\sig_N}(z,n)$ solves $\tau_{\sig_1,\dots,\sig_N} u = z u$.
\end{rem}

Finally we extend Lemma \ref{scatsc} and assume 
for brevity $\sig_\ell \in (-1,1)$.

\begin{lem} \label{scatscN}
Suppose $H$ satisfies {\em (\ref{decay})} and 
let $H_{\sig_1,\dots,\sig_N}$,
$\sig_\ell \in (-1,1,)$, $1 \le \ell \le N$ be 
constructed as in Theorem
\ref{thmscom} with
\bea
u_{\sig_\ell}^\ell(n) = 
\frac{1+\sig_\ell}{2} f_+(k_\ell,n) + (-1)^{\ell+1}
\frac{1-\sig_\ell}{2} f_-(k_\ell,n).
\eea
Then the transmission $T_{\sig_1,\dots,\sig_N}(k)$ 
and reflection coefficients
$R_{\pm,\sig_1,\dots,\sig_N}(k)$ of the operator 
$H_{\sig_1,\dots,\sig_N}$ in
terms of the corresponding scattering data 
$T(k),R_\pm(k)$ of $H$ are given by
\bea
&T_{\sig_1,\dots,\sig_N}(k) = 
\left( \prod\limits_{\ell=1}^N \D \frac{1 - k \,
k_\ell }{k - k_\ell} \right) T(k),
&\\ &R_{\pm,\sig_1,\dots,\sig_N}(k) =
k^{\pm N} \left( \prod\limits_{\ell=1}^N 
\D \frac{k - k_\ell}{1 - k \, k_\ell}
\right) R_\pm(k),&
\eea
where $k_\ell = \lam_\ell + 
\sqrt{\lam_\ell^2 -1} \in (-1,0)$, $1 \le \ell \le N$.
Moreover, the norming  constants 
$\gam_{\sig_1,\dots,\sig_N,\pm,j}$ associated with $\lam_j 
\in \sig_p(H_{\sig_1,\dots,\sig_N})$ in terms of 
$\gam_{\pm,j}$ corresponding to $H$ read
\bea \nn
&\gam_{\sig_1,\dots,\sig_N,\pm,j} = 
\left( \frac{1-\sig_j}{1+\sig_j}
\right)^{\pm1} |k_j|^{-2 \mp(N-1)} 
\D \frac{\prod_{\ell=1}^N |1 - k_j
k_\ell|}{\prod_{\ell=
1 \atop \ell \ne j}^N |k_j-k_l|} T(k_j), \quad 1 \le j \le
N,&\\ &\gam_{\sig_1,\dots,\sig_N,\pm,j} 
= |k_j|^{\pm N} \prod\limits_{\ell=1}^N \D
\frac{1 - k_j k_\ell}{|k_j - k_\ell|} 
\gam_{\pm,j}, \quad j \in J.&
\eea
\end{lem}

\bpf
Observe that
\bea \nn
u_{\sig_1,\sig_2}(\lam_2,n) 
&=& \frac{1+\sig_2}{2} A_{\sig_1} f_+(k_2,n) + 
\frac{1-\sig_2}{2} A_{\sig_1} f_-(k_2,n)\\ 
&=& c \Big( \frac{1+\hat{\sig}_2}{2}
f_{\sig_1,+}(k_2,n) + 
\frac{1-\hat{\sig}_2}{2} f_{\sig_1,-}(k_2,n) \Big),
\eea
where $c>0$ and $\sig_2,\hat{\sig}_2$ are 
related via
\bea
\frac{1+\hat{\sig}_2}{1-\hat{\sig}_2} = 
\frac{1}{k_2} \frac{1+\sig_1}{1-\sig_1}.
\eea
The claims now follow from Lemma \ref{scatsc} 
after extending this result by
induction.
\epf


\section{The Double Commutation Method}
\label{secdc}


In this section we provide a complete characterization 
of the double commutation
method for Jacobi operators. We start with a linear 
transformation which turns
out to be unitary when restricted to proper 
subspaces of our Hilbert space. We 
use this transformation to construct an operator 
$H_{\gam_1}$ from a given
background operator $H$. This operator 
$H_{\gam_1}$ will be the doubly
commuted operator of $H$ as discussed in the 
Introduction. The results of
Sections 4-6 appear to be without precedent.

Let $\hr=\ell^2(M_--1,M_++1)$ be the underlying 
Hilbert space ($-\infty \le M_-
< M_+ \le \infty$) and let $\psi(n)$ be a given 
real-valued sequence which is
square summable near
$M_-$. Choose a positive constant $\gam>0$ and define
\bea
c_\gam(n) = 1 + 
\gam \sum_{j=M_-}^n \psi(j)^2, \qquad n \ge M_-.
\eea
(We set in addition $c_\gam(M_- -1) =1$ if 
$M_-$ is finite.)
Denote the set of sequences in $\ell(M_--1,M_++1)$ 
which are square summable
near $M_-$ by $\hr_-$ and consider the 
following (linear)
transformation
\bea \label{unitary1}
\ba{llcl} U_\gam: &\hr_- & \to & \hr_- \\
& f(n) &\mapsto& f_\gam(n) = \sqrt{\frac{c_\gam(n)}
{c_\gam(n-1)}} f(n) - \gam \psi_\gam(n)
\sum\limits_{j=M_-}^n \psi(j) f(j). \ea
\eea
By inspection, the sequence $f_\gam$ is also square 
summable near $M_-$ and the
inverse transformation is given by
\bea \label{transfUf}
\ba{llcl} U_\gam^{-1}: & \hr_- &\to& \hr_- \\
& g(n) &\mapsto& \sqrt{\frac{d_\gam(n)}{d_\gam(n-1)}} g(n) 
+ \gam \psi(n) \sum\limits_{j=M_-}^n \psi_\gam(j)g(j) \ea,
\eea
where
\bea \label{psigam}
d_\gam(n) = c_\gam(n)^{-1} = 1 -  \gam \sum_{j=M_-}^n
\psi_\gam(j)^2, \qquad \psi_\gam(n) = 
\frac{\psi(n)}{\sqrt{c_\gam(n-1)c_\gam(n)}}.
\eea

\begin{lem} \label{lemuni}
Define $\psi_\gam$ as in (\ref{psigam}). Then 
$\psi_\gam \in \hr$ and
\bea
\| \psi_\gam \|^2 = \frac{1}{\gam} 
\Big(1 - \lim_{n \to M_+} c_\gam(n)^{-1} \Big).
\eea
If $P,P_\gam$ denote the orthogonal projections onto 
the one-dimensional
subspaces of $\hr$ spanned by $\psi, 
\psi_\gam$ (set $P = 0$ if $\psi \not\in
\hr$) the operator $U_\gam$ is unitary 
from $(1-P) \hr$ onto
$(1-P_\gam) \hr$.
\end{lem}

\bpf
For the claims concerning $\psi$ we use
\bea
\sum_{j=M_-}^n |\psi_\gam(j)|^2 = \frac{1}{\gam}
\sum_{j=M_-}^n \Big( \frac{1}{c_\gam(j-1)} -
\frac{1}{c_\gam(j)} \Big) =  \frac{1}{\gam} 
\Big( 1 - \frac{1}{c_\gam(n)} \Big).
\eea
Next we note that
\bea
c_\gam(n) \sum_{j=M_-}^n \psi_\gam(j) f_\gam(j) = 
\sum\limits_{j=M_-}^n \psi(j)
f(j)
\eea
and a direct calculation shows
\bea \label{un}
\sum_{j=M_-}^n |f_\gam(j)|^2 = 
\sum_{j=M_-}^n |f(j)|^2 -
\frac{\gam}{c_\gam(n)} |\sum_{j=M_-}^n 
f(j) \psi(j)|^2.
\eea
This clearly proves the lemma if $\psi \in \hr$. 
Otherwise, i.e., if $\psi
\not\in \hr$, consider $U_\gam$, $U_\gam^{-1}$ on the 
dense subspace
$\ell_0((M_-,M_+))$ and take closures 
(cf$.$, e.g$.$, \cite{wd}, Theorem 6.13).
\epf

Using, e.g., the polarization identity, we 
further get
\bea \label{sprgam}
\sum_{j=M_-}^n \ol{g_\gam(j)} f_\gam(j) = 
\sum_{j=M_-}^n
\ol{g(j)} f(j) - \frac{\gam}
{c_\gam(n)} \sum_{j=M_-}^n \psi(j) f(j)
\sum_{j=M_-}^n \psi(j) \ol{g(j)} .
\eea

Next we take two sequences $a,b$ satisfying

\bh \label{hypoabf}
Suppose
\bea
a,b \in \ell(M_- -1,M_+ +1), 
\qquad a(n) \in \R \bs \{ 0 \}, \: b(n) \in \R
\eea
\eh

and introduce the difference expression
\bea
(\tau f) (n) = a(n) f(n+1) +a(n-1) f(n-1) -b(n) f(n).
\eea
We want to consider a self-adjoint operator $H$ 
associated with $\tau$ and
separated boundary conditions at $M_\pm$ and assume 
the existence of a sequence
$\psi(\lam_1,n)$ of the following kind.

\bh \label{hypodc}
Suppose $\psi(\lam)$ satisfies the following 
conditions.\\
(i) $\psi(\lam,n)$ is a real-valued solution of 
$\tau \psi(\lam) = \lam
\psi(\lam)$.\\
(ii) $\psi(\lam,n)$ is square summable near $M_-$ 
and fulfills the boundary
condition (of $H$) at $M_-$ (if any, i.e., if $\tau$ 
is $l.c.$ at $M_-$).\\
(iii) $\psi(\lam,n)$ also fulfills the boundary 
condition (of $H$) at $M_+$ if
$\tau$ is $l.c.$ at $M_+$ ($\psi(\lam,n)$ is then an 
eigenfunction of $H$).
\eh

Sufficient conditions for the above function to 
exist are\\
(i) $\lam \in \sigma_{p}(H)$, or\\
(ii) $\tau$ is $l.c.$ at $M_-$ but not at $M_+$, or\\
(iii) $\sigma(H) \ne \R$ (and $\lam \in \R \bs \sigma(H)$), or\\
(iv) $\sigma(H_-) \ne \R$ 
(and $\lam \in \R \bs \sigma(H_-)$), where
$H_-$ is a restriction of $H$ to $\ell^2(M_--1,\hat{M}+1)$ 
with $\hat{M} \in \Z$
and (for instance) a Dirichlet boundary condition 
at $\hat{M}+1$.

It follows that $H$ is explicitly given by
\bea \label{defbh}
\db(H) = 
\{ f \in \hr | \ba[t]{l} \tau f \in \hr ; \:
W_{M_- -1}(\psi(\lam_1),f)=0 \mb{ if $\tau$ 
is $l.c.$ at $M_-$} , \\
W_{M_+}(\psi(\lam_1),f)=0 \mb{ if $\tau$ is 
$l.c.$ at $M_+$} \}. \ea
\eea

We now use Lemma \ref{lemuni} with $\psi(n) = 
\psi(\lam_1,n)$, $\gam=\gam_1$,
$U_\gam=U_{\gam_1}$ to prove

\begin{thm} \label{thmdc}
Suppose (H.\ref{hypoabf}) and (H.\ref{hypodc}) and let $\tau_{\gam_1}$ be 
the difference expression
\bea
(\tau_{\gam_1} f)(n) = a_{\gam_1}(n) f(n+1) + 
a_{\gam_1}(n-1) f(n-1) -
b_{\gam_1}(n) f(n),
\eea
where
\bea
a_{\gam_1}(n) &=& a(n) \frac{\sqrt{c_{\gam_1}
(\lam_1,n-1)c_{\gam_1}(\lam_1,n+1)}}
{c_{\gam_1}(\lam_1,n)},\\
\nn b_{\gam_1}(n) &=& b(n) + 
\gam_1 \Big( \frac{a(n-1) \psi(\lam_1,n-1)
\psi(\lam_1,n)}{c_{\gam_1}(\lam_1,n-1)} \\ &&{}- 
\frac{a(n) \psi(\lam_1,n)
\psi(\lam_1,n+1)}{c_{\gam_1}(\lam_1,n)}\Big).
\eea
Then the operator $H_{\gam_1}$ defined by
\bea
& H_{\gam_1} f = \tau_{\gam_1} f,& \\ 
\nn &\db(H_{\gam_1}) = \{ f \in \hr | 
\tau_{\gam_1} f \in \hr ; W_{\gam_1,M_- -1}
(\psi_{\gam_1}(\lam_1),f) =
W_{\gam_1,M_+}(\psi_{\gam_1}(\lam_1),f)=0  \},&
\eea
where $W_{\gam_1,n}(u,v) = 
a_{\gam_1}(n)(u(n) v(n+1) - u(n+1) v(n))$, is
self-adjoint and has the eigenfunction
\bea
\psi_{\gam_1}(\lam_1,n) =
\frac{\psi(\lam_1,n)}{\sqrt{c_{\gam_1}(\lam_1,n-1)
c_{\gam_1}(\lam_1,n)}}
\eea
associated with the eigenvalue $\lam_1$. If 
$\psi(\lam_1) \not\in \hr$ (and hence
$\tau$ is $l.p.$ at $M_+$) we have
\bea
(1-P_{\gam_1}(\lam_1)) H_{\gam_1} = 
U_{\gam_1} H U_{\gam_1}^{-1} (1-P_{\gam_1}(\lam_1)),
\eea
where $U_{\gam_1}$ is the unitary transformation 
of Lemma \ref{lemuni} and thus
\bea
\ba{rcl@{\qquad}rcl}
\sigma(H_{\gam_1}) &=& \sigma(H) \cup \{ \lam_1\}, 
& \sigma_{ac}(H_{\gam_1})
&=& \sigma_{ac}(H), \\  \sigma_{p}(H_{\gam_1})
 &=& \sigma_{p}(H) \cup \{
\lam_1\}, & \sigma_{sc}(H_{\gam_1}) &=& \sigma_{sc}(H).
\ea 
\eea
If $\psi(\lam_1) \in \hr$ there is a unitary operator 
$\ti{U}_{\gam_1} =
U_{\gam_1} \oplus \sqrt{1+\gam_1 \| \psi(\lam_1) \|^2}
 \mb{\bf 1}$ on
$(1-P_{\gam_1}(\lam_1)) \hr \oplus P_{\gam_1}(\lam_1) \hr $ such that
\bea
H_{\gam_1} = \ti{U}_{\gam_1} H \ti{U}_{\gam_1}^{-1}
\eea
and thus
\bea
\ba{rcl@{\qquad}rcl}
\sigma(H_{\gam_1}) &=& \sigma(H), 
& \sigma_{ac}(H_{\gam_1})
&=& \sigma_{ac}(H), \\  \sigma_{p}(H_{\gam_1}) 
&=& \sigma_{p}(H), &
\sigma_{sc}(H_{\gam_1}) &=& \sigma_{sc}(H).
\ea
\eea
\end{thm}

\bpf
It suffices to prove
\bea
(1-P_{\gam_1}(\lam_1)) H_{\gam_1} = 
U_{\gam_1} H U_{\gam_1}^{-1} (1-P_{\gam_1}(\lam_1)).
\eea 
Let $f$ be a sequence which is square summable 
near $M_-$ such that $\tau f$ is
also square summable near $M_-$ and assume that $f$ 
fulfills the boundary
condition at $M_-$, if any. Then a straightforward 
calculation shows
\bea
\tau_{\gam_1} (U_{\gam_1} f) = U_{\gam_1} (\tau f)
\eea
and we only have to check the boundary conditions 
at $M_\pm$. Equation (\ref{un})
shows that $\tau_{\gam_1}$ is $l.c.$ at $M_-$ if and 
only if $\tau$ is and
that $\tau_{\gam_1}$ is $l.c.$ at $M_+$ if $\tau$ is. 
The formula
\bea
W_{\gam_1,n}(\psi_{\gam_1}(\lam_1),U_{\gam_1} f) =
\frac{W_n(\psi(\lam_1),f)}{c_{\gam_1}(\lam_1,n)}
\eea
shows that
\bea \label{WMm}
W_{\gam_1,M_- -1}(\psi_{\gam_1}(\lam_1),U_{\gam_1} f) =0, 
\qquad f \in \db(H).
\eea
We further claim that
\bea \label{WMp}
W_{\gam_1,M_+}(\psi_{\gam_1}(\lam_1),U_{\gam_1} f) = 0, 
\qquad f \in \db(H).
\eea
This is clear if $\psi(\lam_1) \in \hr$. Otherwise, 
i.e., if $\psi(\lam_1)
\not\in \hr$, we use
\bea
\frac{W_n(\psi(\lam_1),f)}{c_{\gam_1}(\lam_1,n)} = 
\frac{\sum_{j=M_-}^n
\psi(\lam_1,j) (\lam_1 - 
\tau) f(j)}{c_{\gam_1}(\lam_1,n)}.
\eea
The right hand side tends to zero for $ f \in \db(H)$ 
as can be seen from
(\ref{un}) and the fact that $U_{\gam_1}$ is unitary. 
Combining (\ref{WMm}) and
(\ref{WMp}) yields
\bea
(1-P_{\gam_1}(\lam_1)) U_{\gam_1} \db(H) 
\subseteq (1-P_{\gam_1}(\lam_1))
\db(H_{\gam_1}).
\eea
But $(1-P_{\gam_1}(\lam_1)) U_{\gam_1} \db(H)$  cannot 
be properly contained in
$(1-P_{\gam_1}(\lam_1)) \db(H_{\gam_1})$ by the 
property of self-adjoint
operators being maximally defined.
\epf

\begin{rem}
(i). By choosing $\lam_1 \in \sigma_{ac}(H) 
\cup \sigma_{sc}(H)$ (provided
the continuous spectrum is not empty and a 
solution satisfying (H.\ref{hypodc})
exists) we can use the double commutation method 
to construct operators with
eigenvalues embedded in the continuous spectrum.\\
(ii). If $M_+=\infty$ and $H$ has an eigenfunction
$\psi(\lam_1)$ one can remove this eigenfunction 
from the spectrum upon choosing
$\gam_1 = - \| \psi(\lam_1) \|^{-2}$. The 
corresponding function
$\psi_{\gam_1}(\lam_1)$ is then no longer in $\hr$, 
implying that
$\tau_{\gam_1}$ is $l.p.$  at $M_+$.\\
(iii). Especially, removing an eigenvalue from an 
operator which
is $l.c.$ at $\infty$ yields an operator which is 
$l.p.$. Thus $\tau_{\gam_1}$ is
not necessary $l.p.$ if $\tau$ is. Moreover, this 
shows that one cannot insert
additional eigenvalues into an operator which is 
$l.c.$ at $M_+$ (remove this
eigenvalue again to obtain a contradiction).\\
(iv). The limiting case $\gam_1 =\infty$ can be handled 
analogously producing a unitarily equivalent operator if
$\psi(\lam_1) \not\in \hr$ and removes the eigenvalue
$\lam_1$ otherwise.
\end{rem}

The previous theorem tells us only how to transfer 
solutions of $\tau u = z u$
into solutions of $\tau_{\gam_1} v = z v$ if $u$ is 
square summable near $M_-$.
The following lemma treats the general case.

\begin{lem} \label{addprophg}
The sequence
\bea
u_{\gam_1}(z,n) = \frac{c_{\gam_1}(\lam_1,n) u(z,n) - 
\frac{\gam_1}{z-\lam_1} \psi(\lam_1,n)
W_n(\psi(\lam_1),u(z))}{\sqrt{c_{\gam_1}(\lam_1,n-1)
c_{\gam_1}(\lam_1,n)}},
\:\: z \in \C \bs \{ \lam_1 \}
\eea
solves $\tau_{\gam_1} u = z u$ if $u(z)$ solves 
$\tau u = z u$. If
$u(z)$ is square summable near $M_-$ and fulfills the boundary 
condition at $M_-$ (if any)
we have $u_{\gam_1}(z,n)=(U_{\gam_1} u)(z,n)$ justifying our notation.
Furthermore, we note
\bea \nn
|u_{\gam_1}(z,n)|^2 &=& |u(z,n)|^2 -
\frac{\gam_1}{|z-\lam_1|^2} \times\\ &&{} 
\Big( \frac{|W_n(\psi(\lam_1),u(z))|^2}
{c_{\gam_1}(\lam_1,n)} -
\frac{|W_{n-1}(\psi(\lam_1),u(z))|^2}
{c_{\gam_1}(\lam_1,n-1)}\Big),
\eea
and
\bea \label{wrongamp}
W_{\gam_1,n}(\psi_{\gam_1}(\lam_1),u_{\gam_1}(z)) =
\frac{W_n(\psi(\lam_1),u(z))}{c_{\gam_1}(\lam_1,n)}.
\eea
Hence $u_{\gam_1}$ is square summable near $M_+$ if $u$ is.
If $\hat{u}_\gam(\hat{z})$ is constructed analogously then
\bea \nn
W_{\gam_1,n}(u_{\gam_1}(z),\hat{u}_{\gam_1}(\hat{z})) &=&
W_n(u(z),\hat{u}(\hat{z})) + \frac{\gam_1}{c_{\gam_1}(\lam_1,n)}
\frac{z-\hat{z}}{(z-\lam_1)(\hat{z}-\lam_1)}
\times \\ \label{wrongam} &&
W_n(\psi(\lam_1),u(z))W_n(\psi(\lam_1),\hat{u}(\hat{z})).
\eea
\end{lem}
\bpf
All facts are tedious but straightforward calculations.
\epf

Next we want to give some conditions implying the 
$l.p.$ case of $\tau_{\gam_1}$
at $M_+$, assuming $M_+=\infty$. Let 
$M_-<\hat{M}<\infty$ and let $H_+$ denote a
self-adjoint operator associated with $\tau$ on 
$(\hat{M}-1,\infty)$ and the boundary
condition induced by $\psi(\lam_1)$ at $\hat{M}$ 
(cf. equation (\ref{defbh})).

\bh \label{hypodcsp}
Suppose $H_+$ satisfies one of the following 
spectral conditions:\\
(i). $\sig_{ess} (H_+) \ne \emptyset$.\\
(ii). $\sig(H_+)=\sig_d(H_+)=
\{ \lam_{+,j} \}_{j\in J_+}$ with $\sum_{j\in J_+}
(1+\lam^2_{+,j})^{-1}
=\infty$.
\eh

Clearly Hypothesis (H.\ref{hypodcsp}) is satisfied if $a,b$ are 
bounded near $\infty$ (which
is equivalent to $H_+$ being bounded) since then 
$\tau$ is $l.p.$ at $\infty$.

\begin{thm} \label{thmlp}
Assume (H.\ref{hypoabf}), (H.\ref{hypodc}), and (H.\ref{hypodcsp}). Then 
$\tau_{\gam_1}$ is $l.p.$ at
$M_+=\infty$.
\end{thm}

\bpf
Let $\gam_{1,+} = c_{\gam_1}(\lam_1,\hat{M})^{-1}\gam_1$ 
and consider the doubly
commuted operator $H_{+,\gam_{1,+}}$ of $H_+$. Then
$\tau_{\gam_1}|_{(\hat{M},\infty)} = \tau_{\gam_{1,+}}$ 
and $H_{+,\gam_{1,+}}$
also satisfies (H.\ref{hypodcsp}). Hence $\tau_{\gam_1}$ is $l.p.$ 
at $\infty$ as claimed.
\epf

\begin{rem}
We can interchange the role of $M_-$ and $M_+$ 
in this section by
substituting $M_- \leftrightarrow M_+$, 
$\sum_{j=M_-}^n \to \sum_{j=n+1}^{M_+}$
and $\gam_1 \to -\gam_1$.
\end{rem}

Let $M_\pm=\pm\infty$ and $H$ be a given Jacobi 
operator satisfying
(\ref{decay}). Our next aim is to show how the scattering 
data of the operators
$H, H_{\gam_1}$ are related, where $H_{\gam_1}$ is defined 
as in Theorem
\ref{thmdc}.

\begin{lem} \label{sctdc}
Let $H$ be a given Jacobi operator satisfying 
(\ref{decay}). Then the doubly 
commuted operator $H_{\gam_1}$, defined via $\psi(\lam_1,n) 
= f_-(k_1,n), \quad
\lam_1=(k_1+k_1^{-1})/2$ as in Theorem \ref{thmdc}, 
has the transmission and
reflection coefficients
\bea
T_{\gam_1}(k) = 
\sgn(k_1) \frac{k \, k_1 -1}{k - k_1} T(k),
\eea
\bea
R_{-,\gam_1}(k) = R_-(k), \qquad R_{+,\gam_1}(k) = 
\left( \frac{k - k_1}{k \, k_1
-1} \right)^2 R_+(k),
\eea
where $k$ and $z$ are related via $z = (k + k^{-1})/2$.
Furthermore, the norming  constants $\gam_{-,j}$ 
corresponding to $\lam_j \in \sig_{p}(H)$, $j \in J$ 
(cf.~$(\ref{norming})$) remain
unchanged except for an additional eigenvalue $\lam_1$ 
with norming constant
$\gam_{-,1}=\gam_1$ if $\psi(\lam_1) \not\in \hr$  
respectively with norming
constant $\ti{\gam}_{-,1}= \gam_{-,1} + \gam_1$ if 
$\psi(\lam_1) \in \hr$ and
$\gam_{-,1}$ denotes the original norming constant 
of $\lam_1 \in \sig_{p}(H)$.
\end{lem}

\bpf
By Lemma \ref{addprophg} the Jost solutions 
$f_{\gam_1,\pm}(k,n)$ are up to a
constant given by 
\bea
\frac{c_{\gam_1}(\lam_1,n-1) f_\pm(k,n) - 
\frac{\gam_1}{z-\lam_1} \psi(\lam_1,n)
W_{n-1}(\psi(\lam_1),f_\pm(k))}
{\sqrt{c_{\gam_1}(\lam_1,n-1)c_{\gam_1}(\lam_1,n)}}.
\eea
This constant is easily seen to be 1 for 
$f_{\gam_1,-}(k,n)$. Thus we can compute
$R_-(\lam)$ using (\ref{wrongam}) (the second 
unknown constant cancels). The rest
follows by a straightforward calculation.
\epf




\section{Double Commutation and Weyl--Titchmarsh 
Theory}


In this section we want to reveal the connections 
between Weyl--Titchmarsh
theory and the double commutation method. Without 
loss of generality we consider
only the cases $\ell^2(\N)$ and $\ell^2(\Z)$. We 
start with the half-line
$\N$ and freely use the notation employed in 
Appendices A--D.

Let $H_+$ be a self-adjoint operator associated 
with $\tau$ on
$\N$ and a Dirichlet boundary condition at 0. 
Without loss of generality
we assume $\psi(\lam_1,1)=1$.

\begin{thm}
Assume (H.\ref{hypoabf}), $\psi(\lam_1,1)=1$ and let $m_+(z,0)$,
$m_{+,\gam_1}(z,0)$  denote the
Weyl $m$-functions of $H_+$, $H_{+,\gam_1}$. Then 
we have
\bea
m_{+,\gam_1}(z,0) = \frac{1}{1+\gam_1} \Big( m_+(z,0) - 
\frac{\gam_1}{z -
\lam_1} \Big).
\eea
If $\mu_+$ and $\mu_{+,\gam_1}$ denote the 
corresponding spectral functions of
$H_+$ and $H_{+,\gam_1}$ it follows that
\bea
\mu_{+,\gam_1}(\lam) = \frac{1}{1+\gam_1} 
\Big( \mu_+(\lam) + \gam_1
\Theta(\lam-\lam_1)
\Big),
\eea
where $\Theta(.)$ denotes the (right continuous) 
step function
\bea
\Theta(x) = \left\{ \ba{c@{\quad}l} 1, 
& x \ge 0 \\ 0, & x < 0 \ea
\right. .
\eea
\end{thm}

\bpf
As in Section~\ref{secwm} we use the finite approximations 
$m_N(z,0)$ and
$m_{N,\gam_1}(z,0)$. If $\gam_j(N)$, 
$\gam_{j,\gam_1}(N)$ are the
corresponding norming constants we have
\bea
\gam_{j,\gam_1}(N) = 
\frac{1}{1+\gam_1} \left\{ \ba{c@{\quad}l} \gam_j(N) + 
\gam_1, & \lam_j=\lam_1 \\ \gam_j(N), 
& \lam_j \ne \lam_1 \ea \right. .
\eea
This follows since $\psi(z,0)=0$, $\psi(z,1)=1$ 
implies $\psi_{\gam_1}(z,0)=0$,
$\psi_{\gam_1}(z,1)= (1+\gam_1)^{-1/2}$. Hence we 
infer
\bea
m_{N,\gam_1}(z,0) = 
\frac{1}{1+\gam_1} \Big( m_N(z,0) - 
\frac{\gam_1}{z - \lam_1}
\Big)
\eea
and the theorem follows upon taking the limit 
$N \to \infty$.
\epf

\begin{rem}
If we transform the operator $H_+$ into it's diagonal 
form as in Section~\ref{secwtn} the
double commutation method gets particularly  
transparent: it corresponds to
adding a step function to the spectral function. 
This approach can also be used
to derive the unitary transformation stated in 
Section 2 in the following way.
Take the spectral function $\mu_+$ of a given Jacobi 
operator, switch to
$\mu_{+,\gam_1}$, and compute the orthogonal polynomials 
with respect to this new
measure (compare Section~\ref{secwtn} and \cite{ak}, Ch. 1). Now 
take a sequence $f(n)$ and its
transform $F(z)$ and use (\ref{unitary2}) to obtain 
(\ref{unitary1}).
\end{rem}


Next we turn to operators in $\ell^2(\Z)$. Without 
loss of generality we assume
\bea \label{initv}
\psi(\lam_1,0) = - \sin(\alpha), 
\quad \psi(\lam_1,0) = \cos(\alpha), \quad
\alpha \in [0,\pi).
\eea

\begin{thm}
Assume (H.\ref{hypoabf}) and let $\ti{m}_\pm(z,\alpha)$, 
$\ti{m}_{\pm,\gam_1}(z,\alpha)$ denote the
Weyl $\ti{m}$-functions of $H$, $H_{\gam_1}$ as introduced 
in Section~\ref{secwm}. Then we have
\bea
\ti{m}_{\pm,\gam_1}(z,\ti{\alpha}) = 
\frac{(1+\ti{\gam}_1 (\cos(\alpha)^4 -
\sin(\alpha)^4))^{-1/2}}{((1+\ti{\gam}_1\cos(\alpha)^2)
(1-\ti{\gam}_1\sin(\alpha)^2))^{1/2}}
\Big( \ti{m}_\pm(z,\alpha) - \frac{\ti{\gam}_1}{z - \lam_1} \Big),
\eea
where
\bea
\ti{\gam}_1 = \frac{\gam_1}{c_{\gam_1}(\lam_1,0)}, 
\qquad \tan(\ti{\alpha}) =
\sqrt{\frac{c_{\gam_1}(\lam_1,1)}
{c_{\gam_1}(\lam_1,-1)}} \tan(\alpha).
\eea
\end{thm}

\bpf
Consider the sequences
\bea
\phi_{\alpha,\gam_1}(z,n), \quad \theta_{\alpha,\gam_1}(z,n) -
\frac{\ti{\gam}_1}{z-\lam_1} \phi_{\alpha,\gam_1}(z,n)
\eea
constructed from the fundamental system $\theta_\alpha(z,n)$,
$\phi_\alpha(z,n)$ for $\tau$ (cf. (\ref{funsy})) as in Lemma
\ref{addprophg}. They form a fundamental system for $\tau_{\gam_1}$
corresponding to the initial conditions associated with
$\ti{\alpha}$ up to constant multiples. Now use (\ref{wrongamp}) to
evaluate (\ref{mfun}).
\epf

The Weyl $M$-matrix and the corresponding spectral 
matrix can now be computed in a
straightforward manner (cf.\ Section~\ref{secwtz}).



\section{Iteration of the double commutation method}
\label{secdcit}

Finally we demonstrate how to iterate the double 
commutation method. We
choose a given background operator $H$ (with 
coefficients $a$, $b$ satisfying
(H.\ref{hypoabf})) and further $\gamma_1>0, \lam_1\in \R$. 
Next choose $\psi(\lam_1)$ as
in Hypothesis (H.\ref{hypodc}) to define the transformation 
$U_{\gam_1}$ and the operator
$H_{\gam_1}$. In the second step, we choose 
$\gamma_2>0, \lam_2 \in \R$ and 
another function $\psi(\lam_2)$ to define 
$\psi_{\gam_1}(\lam_2) = U_{\gam_1}
\psi(\lam_2)$, a corresponding transformation 
$U_{\gam_1,\gam_2}$, and an 
operator $H_{\gam_1,\gam_2}$. Applying this 
procedure $N$-times results in

\begin{thm} \label{dcommN}
Assuming (H.\ref{hypoabf}) let $H$ be a given background 
Jacobi operator in $\hr =
\ell^2(M_--1,M_++1)$ and let $\gam_j>0$, 
$\lam_j$, $1 \le j \le N$
be such that there exist corresponding solutions 
$\psi(\lam_j,n)$ of $\tau
\psi = \lam_j \psi$ satisfying Hypothesis (H.\ref{hypodc}). 
We set
$\psi_{\gam_1,\dots,\gam_k}(\lam_j) = 
U_{\gam_1,\dots,\gam_k} \cdots U_{\gam_1}
\psi(\lam_j)$ and define the following 
matrices $(1 \le \ell \le N)$
\bea
C^\ell(n) =  \left\{ \delta_r(s) + 
\sqrt{\gam_r \gam_s} \sum_{m=M_-}^n
\psi(\lam_r,m) \psi(\lam_s,m) \right\}_{1 \le r,s \le \ell},
\eea
\bea \label{cij}
C^\ell_{i,j}(n) = \left\{ \ba{c@{\quad}l} 
C^{\ell-1}(n)_{r,s} & \scriptstyle r,s
\le \ell-1\\
\sqrt{\gam_s} \sum\limits_{m=M_-}^n \psi(\lam_i,m) 
\psi(\lam_s,m) & \scriptstyle s
\le \ell-1, r=\ell\\ \sqrt{\gam_r} \sum\limits_{m=M_-}^n 
\psi(\lam_r,m)
\psi(\lam_j,m) & \scriptstyle r \le \ell-1, s=\ell\\
\sum\limits_{m=M_-}^n \psi(\lam_i,m) \psi(\lam_j,m) 
& \scriptstyle r=s=\ell \ea 
\right\}_{1 \le r,s \le \ell},
\eea
\bea
\Psi^\ell(\lam_j,n) = \left\{ \ba{c@{\quad}l} 
C^\ell(n)_{r,s} & \scriptstyle
r,s \le \ell\\ \sqrt{\gam_s} \sum\limits_{m=M_-}^n 
\psi(\lam_j,m)
\psi(\lam_s,m) 
& \scriptstyle s \le \ell, r=\ell+1\\
\sqrt{\gam_r}\psi(\lam_r,n) &
\scriptstyle r \le \ell, s=\ell+1\\ 
\psi(\lam_j,n) & \scriptstyle r=s=\ell+1
\ea \right\}_{1 \le r,s \le \ell+1}.
\eea
Then we have (set $C^0(n)=1$)
\bea \label{ckn}
c_{\gam_\ell}(\lam_\ell,n) = 1 + 
\gam_\ell \sum_{m=M_-}^n
\psi_{\gam_1,\dots,\gam_\ell} (\lam_\ell,m)^2 = 
\frac{\det C^\ell(n)}{\det
C^{\ell-1}(n)},
\eea
and hence
\bea \label{prck}
\prod_{\ell=1}^N c_{\gam_\ell}(\lam_\ell,n) = 
\det C^N(n).
\eea
Moreover,
\bea \label{sprk}
\sum_{m=M_-}^n \psi_{\gam_1,\dots,\gam_\ell}(\lam_i,m) 
\psi_{\gam_1,\dots,\gam_\ell}(\lam_j,m) = 
\frac{\det C^\ell_{i,j}(n)}{\det
C^{\ell-1}(n)}
\eea
and
\bea \label{phik}
\psi_{\gam_1,\dots,\gam_\ell}(\lam_j,n) = 
\frac{\det
\Psi^\ell(\lam_j,n)}
{\sqrt{\det C^\ell(n-1)\det C^\ell(n)}}.
\eea
In addition, we get
\bea
a_{\gam_1,\dots,\gam_N}(n) 
&=& a(n) \frac{\sqrt{\det C^N(n-1) \det
C^N(n+1)}}{\det C^N(n)}, \\ \nn
b_{\gam_1,\dots,\gam_N}(n) &=& b(n) - 
\sum_{\ell=1}^N \gam_\ell \left( a(n)
\frac{ \det \Psi^\ell(\lam_\ell,n) 
\det \Psi^\ell(\lam_\ell,n+1)}{\det
C^{\ell-1}(n)\det C^\ell(n)} \right. \\ \nn &&
\left. {} -  a(n-1) \frac{ \det
\Psi^\ell(\lam_\ell,n-1) \det
\Psi^\ell(\lam_\ell,n)}
{\det C^{\ell-1}(n-1) \det C^\ell(n-1)} \right)\\ \nn
&=& - \lam_N + a(n) \frac{\det C^N(n-1)}
{\det C^N(n)}\frac{\det
\Psi^N(\lam_N,n+1)}
{\det \Psi^N(\lam_N,n)}\\ &&{}+ a(n-1) \frac{\det
C^N(n)}{\det C^N(n-1)}\frac{\det \Psi^N(\lam_N,n-1)}
{\det \Psi^N(\lam_N,n)},
\eea
the last equation only being valid if $\det \Psi^N(\lam_N,n) \ne 0$ (e.g.,
if $\lam_N\le\inf \sig(H)$). The spectrum of $H_{\gam_1,\dots,\gam_N}$ is
given by
\bea
\ba{rcl@{\quad}rcl}
\sig(H_{\gam_1,\dots,\gam_N}) \!\!&=&\!\! \sig(H) 
\cup \{ \lam_j\}_{j=1}^N, &
\sig_{ac}(H_{\gam_1,\dots,\gam_N}) 
\!\!&=&\!\! \sig_{ ac}(H),\\ \sig_{p}(
H_{\gam_1,\dots,\gam_N}) \!\!&=&\!\! \sig_{p}(H) 
\cup \{ \lam_j \}_{j=1}^N , & 
\sig_{sc}(H_{\gam_1,\dots,\gam_N}) 
\!\!&=&\!\! \sig_{sc}(H).
\ea 
\eea
Moreover,
\bea \nn
&& H_{\gam_1,\dots,\gam_N} 
(1-\sum_{j=1}^N P_{\gam_1,\dots,\gam_N}(\lam_j)) 
\\ &&
\quad = (U_{\gam_1,\dots,\gam_N} \cdots U_{\gam_1})
 H (U_{\gam_1}^{-1} \cdots
U_{\gam_1,\dots,\gam_N}^{-1}) (1-\sum_{j=1}^N 
P_{\gam_1,\dots,\gam_N}(\lam_j)),
\eea
where $P_{\gam_1,\dots,\gam_N}(\lam_j)$ 
denotes the projection onto the 
one-dimensional subspace of $\hr$ spanned by 
$\psi_{\gam_1,\dots,\gam_N}(\lam_j)$.
\end{thm}

\bpf
We start with (\ref{sprk}). Using Sylvester's 
determinant identity
(cf.\ \cite{gant}, Sect.\ II.3) we obtain
\bea \nn
&& \det C^{\ell-1}(n) \det C^{\ell+
1}_{i,j}(n)\\ &&=
\det C^\ell(n) \det C^\ell_{i,j}(n) - 
\gam_\ell \det C^\ell_{\ell,j}(n) \det
C^\ell_{i,\ell}(n),
\eea
which proves (\ref{sprk}) together with a 
look at (\ref{sprgam}) by induction
on $N$. Next, (\ref{ckn}) easily follows from 
(\ref{sprk}). Similarly,
\bea \nn
&& \det C^\ell(n) \det \Psi^{\ell+
1}(\lam_j,n) \\ && = \det C^{\ell+1}(n) \det
\Psi^\ell(\lam_j,n) - 
\gam_{\ell} \det \Psi^\ell(\lam_\ell,n) \det
C^\ell_{j,\ell}(n),
\eea
and (\ref{transfUf}) prove (\ref{phik}). The rest 
follows in a straightforward
manner.
\epf

\begin{rem}
(i). For a sequence $f$, which is square summable 
near $M_-$,
$f_{\gam_1,\dots,\gam_j} = 
U_{\gam_1,\dots,\gam_j} \cdots U_{\gam_1} f$ is
given by substituting $\psi(\lam_j) \to f$ in 
(\ref{phik}). Similarly we get the
scalar product of $f_{\gam_1,\dots,\gam_i}$ and 
$g_{\gam_1,\dots,\gam_j}$ from
(\ref{sprk}) by substituting $f \to \psi(\lam_i)$ 
and $g \to \psi(\lam_j)$ in
(\ref{cij}).\\ (ii). Equation (\ref{phik}) can be 
rephrased as
\bea \nn
&& (\gam_1 \psi_{\gam_1,\dots,\gam_\ell}(\lam_1,n), 
\dots, \gam_\ell
\psi_{\gam_1,\dots,\gam_\ell}(\lam_\ell,n)) =\\
&& \sqrt{\frac{\det C^\ell(n)}
{\det C^\ell(n-1)}}(C^\ell(n))^{-1} (\gam_1
\psi(\lam_1,n), \dots, \gam_\ell \psi(\lam_\ell,n)),
\eea
where $(C^\ell(n))^{-1}$ is the inverse matrix 
of $C^\ell(n)$.

\end{rem}

Clearly Theorem \ref{thmlp} extends (by induction) 
to this more general
situation.

\begin{thm}
Assume (H.\ref{hypoabf}) and (H.\ref{hypodcsp}). Then 
$\tau_{\gam_1,\dots,\gam_N}$ is $l.p.$ at $M_+$.
\end{thm}

Finally we also extend Lemma \ref{sctdc}. For 
simplicity we assume
$\psi(\lam_j,n) \not\in \hr$, $1 \le j \le N$.

\begin{lem} 
Let $H$ be a given Jacobi operator satisfying 
(\ref{decay}). Then
$H_{\gam_1,\dots,\gam_N}$, defined via 
$\psi(\lam_\ell,n) = f_-(k_\ell,n),
\quad \lam_\ell=(k_\ell+k_\ell^{-1})/2 
\in \R \bs \sig(H_{\gam_1,\dots,
\gam_{\ell-1}})$, $1 \le \ell \le N$ has the 
transmission and reflection
coefficients
\bea
&T_{\gam_1,\dots,\gam_N}(k) = 
\prod_{\ell=1}^N \sgn(k_\ell) \frac{k \, k_\ell
-1}{k - k_\ell} T(k),&\\
&R_{-,\gam_1,\dots,\gam_N}(k) = R_-(k), \quad
R_{+,\gam_1,\dots,\gam_N}(k) = 
\left( \prod\limits_{\ell=1}^N \Big( \D \frac{k -
k_\ell}{k \, k_\ell -1} \Big)^2 \right) R_+(k),&
\eea
where $z = (k + k^{-1})/2$. Furthermore, the 
norming constants $\gam_{-,j}$ 
corresponding to 
$\lam_j \in \sig_{p}(H)$, $j \in J$
(cf.\ $(\ref{norming})$) remain unchanged and 
the additional eigenvalues $\lam_\ell$ have norming
constants $\gam_{-,\ell}=\gam_\ell$.
\end{lem}

\begin{rem}
Of special importance is the case $a(n) =1/2$, 
$b(n)=0$. Here we have $f_\pm(k,n)
= k^{\pm n}$, $T(k)=1$, and $R_\pm(k)=0$. It is 
well known from inverse
scattering theory that $R_\pm(k)$, $|k|=1$ together 
with the point spectrum and
corresponding norming constants uniquely determine 
$a(n),b(n)$. Hence we infer
from Lemma \ref{scatscN} that 
$H_{\gam_1,\dots,\gam_N}$ constructed from
$\psi(\lam_\ell,n) = f_-(k_\ell,n)$ as in 
Theorem \ref{dcommN} and
$H_{\sig_1,\dots,\sig_N}$ constructed  from 
$u_{\sig_\ell}^\ell =
\frac{1+\sig_\ell}{2} f_+(k_\ell,n) + 
(-1)^{\ell+1} \frac{1-\sig_\ell}{2}
f_-(k_\ell,n)$ as in Theorem \ref{thmscom} 
coincide if
\bea
\gam_j = \left( \frac{1-\sig_j}{1+\sig_j}
\right)^{-1} |k_j|^{-1-N} \frac{\prod_{\ell=1}^N
 |1 - k_j k_\ell|}{\prod_{
{\ell=1 \atop \ell \ne j}}^N |k_j-k_\ell|} T(k_j), 
\quad 1 \le j \le N.
\eea
For a direct proof compare \cite{TKvM}.
\end{rem}


\section{Applications}
\label{secappc}


First we state the discrete analogue of the 
FIT-formula derived in \cite{fit}
for the isospectral torus of periodic Schr\"odinger 
operators. This yields an
explicit realization of the isospectral torus of all 
algebro-geometric
quasi-periodic finite-gap Jacobi operators.

Let $a(n),b(n)$ be given algebro-geometric 
quasi-periodic $g$-gap sequences
characterized by the band-edges 
$E_0 < E_1 < \dots < E_{2g+1}$ and Dirichlet
data $\{ (\mu_j,\sig_j) \}_{j=1}^g$ at the 
reference point $n=0$ (cf.\
\cite{bght}), where $\mu_j \in [E_{2j-1},E_{2j}]$ 
and $\sig_j \in \{\pm\}$, $1
\le j \le g$. Then the spectrum of the associate 
Jacobi operator $H$ is of the
type
\bea \nn
&\sig(H) = \sig_{ac}(H) = 
\bigcup_{n=1}^{g+1} [E_{2n-2},E_{2n-1}],&\\
\label{ggapspec} &\sig_{sc}(H) = 
\sig_p(H) = \emptyset.&
\eea
and (cf.\ (\ref{hpm}))
\bea
\sig(H_\pm) = \sig(H) \cup \{ \mu_j | \sig_j = 
\pm, \: 1 \le j \le g \}.
\eea
Then considerations as in Theorem \ref{thmscom} 
readily yield that all
other isospectral algebro-geometric $g$-gap 
sequences can be realized in the
following way
\bea \nn
\lefteqn{a_{(\ti{\mu}_1,\ti{\sig}_1),
\dots,(\ti{\mu}_g,\ti{\sig}_g)}(n) = -
\sqrt{a(n-g) a(n-g+2)} \times} \\ 
\nn &&  \sqrt{\frac{
C_{n-g}(\psi_{\sig_1}(\mu_1),
\psi_{-\ti{\sig}_1}(\ti{\mu}_1),\dots,
\psi_{\sig_g}(\mu_g), 
\psi_{-\ti{\sig}_g}(\ti{\mu}_g))}{
C_{n-g+1}(\psi_{\sig_1}(\mu_1),
\psi_{-\ti{\sig}_1}(\ti{\mu}_1),\dots,
\psi_{\sig_g}(\mu_g), 
\psi_{-\ti{\sig}_g}(\ti{\mu}_g))}} \times \\ 
\label{fita}
&& \sqrt{\frac{ C_{n-g+2}
(\psi_{\sig_1}(\mu_1),\psi_{-\ti{\sig}_1}
(\ti{\mu}_1), \dots, \psi_{\sig_g}(\mu_g), 
\psi_{-\ti{\sig}_g}(\ti{\mu}_g))
}{C_{n-g+1}(\psi_{\sig_1}(\mu_1),
\psi_{-\ti{\sig}_1}(\ti{\mu}_1),\dots,
\psi_{\sig_g}(\mu_g), 
\psi_{-\ti{\sig}_g}(\ti{\mu}_g))}},
\eea
\bea \nn
\lefteqn{b_{(\ti{\mu}_1,\ti{\sig}_1),
\dots,(\ti{\mu}_g,\ti{\sig}_g)}(n) = a(n-g)
\frac{ C_{n-g+2}(\psi_{\sig_1}(\mu_1), 
\psi_{-\ti{\sig}_1}(\ti{\mu}_1),
\dots, \psi_{\sig_g}(\mu_g)) }{ C_{n-g+1}
(\psi_{\sig_1}(\mu_1),
\psi_{-\ti{\sig}_1}(\ti{\mu}_1),
\dots,\psi_{\sig_g}(\mu_g))} \times} \\ 
&& \nn
\frac{C_{n-g}(\psi_{\sig_1}
(\mu_1),\psi_{-\ti{\sig}_1}(\ti{\mu}_1), \dots,
\psi_{\sig_g}(\mu_g), 
\psi_{-\ti{\sig}_g}(\ti{\mu}_g))}{C_{n-g+1}(
\psi_{\sig_1}(\mu_1), 
\psi_{-\ti{\sig}_1}(\ti{\mu}_1), \dots,
\psi_{\sig_g}(\mu_g), 
\psi_{-\ti{\sig}_g}(\ti{\mu}_g))} \\ \nn && {} +
a(n+1) \frac{C_{n-g}
(\psi_{\sig_1}(\mu_1),
\psi_{-\ti{\sig}_1}(\ti{\mu}_1),\dots ,
\psi_{\sig_g}(\mu_g))}{
C_{n-g+1}(\psi_{\sig_1}(\mu_1),
\psi_{-\ti{\sig}_1}(\ti{\mu}_1),
\dots, \psi_{\sig_g}(\mu_g)) } \times \\
&& \label{fitb} \frac{C_{n-g+1}
(\psi_{\sig_1}(\mu_1), 
\psi_{-\ti{\sig}_1}(\ti{\mu}_1),
\dots,\psi_{\sig_g}(\mu_g), 
\psi_{-\ti{\sig}_g}(\ti{\mu}_g))}{
C_{n-g}(\psi_{\sig_1}(\mu_1),
\psi_{-\ti{\sig}_1}(\ti{\mu}_1), \dots,
\psi_{\sig_g}(\mu_g), 
\psi_{-\ti{\sig}_g}(\ti{\mu}_g))} - \ti{\mu}_g,
\eea
where $\psi_\pm(z,n)$ are the branches of 
the Baker-Akhiezer function associated
with $a,b$ (i.e., the solutions of $\tau \psi = z \psi$ 
which are square summable
near $\pm\infty$) and the new sequences are 
associated with the new Dirichlet
data $\{ (\ti{\mu}_j,\ti{\sig}_j) \}_{j=1}^g$ at 
the same reference point $n=0$.
Even though $\psi_\pm(z,n)$ is not necessarily
positive as required in our
Theorem \ref{thmscom}, the above sequences can be 
shown to be well-defined by
using the explicit theta-function representations for
$\psi_\pm(z,n)$ (cf., e.g., \cite{bght}) as long as 
$\ti{\mu}_j \in [E_{2j-1},E_{2j}]$ and $\ti{\sig}_j
\in \{\pm\}$, $1 \le j \le g$. In fact, consider the hyperelliptic
Riemann surface $K_g$ associated with the function
\bea
R_{2g+2}(z)^{1/2} = \prod_{j=0}^{2g+1} (z-E_j)^{1/2}
\eea
and branch points $E_0 < E_1 < \dots < E_{2g+1}$. A point $P\in
K_g$ will be denoted by $P = (z,\pm R_{2g+2}(z)^{1/2})$ and
we add two points $\infty_\pm \in K_g$ such that $K_g$ is
compact. Introduce
\bea
\underline{z}(P,n)  =\underline{\hat{A}}_{P_0}(P) 
- \sum_{j=1}^g \underline{\hat{A}}_{P_0}(\hat{\mu}_j) 
+ 2 n \underline{\hat{A}}_{P_0}(\infty_+)
- \hat{\underline{\Xi}}_{P_0},
\eea
where $\underline{\hat{A}}_{P_0}$ is Abel's map with base point
$P_0=(E_0,0)$ and $\hat{\underline{\Xi}}_{P_0}$ is the vector of
Riemann constants (cf.\ \cite{bght} for more details). Then
\bea \label{thetaa}
a(n) &=& \ti{a} [\theta (\underline{z}(\infty_+,n-1)) 
\theta (\underline{z}(\infty_+,n+1))/ \theta (\underline{z}
(\infty_+,n))^2 ] ^{1/2},\\ \nn
b(n) &=& -E_0 + 
\ti{a} \frac{\theta (\underline{z} (\infty_+,n-1))\theta
(\underline{z} (P_0, n+1))}{\theta (\underline{z} (\infty_+,n))
\theta (\underline{z}(P_0, n))}\\ \label{thetab} && {} + \ti{a}
\frac{\theta (\underline{z} (\infty_+,n)) \theta
(\underline{z}(P_0,n-1))}{\theta (\underline{z}(\infty_+,n-1))\theta
(\underline{z}(P_0,n))},
\eea
where $\theta$ is Riemann's theta function associated with
$K_g$ and $\ti{a}$ is a constant depending only on $K_g$ (i.e.,on $ \{
E_j\}_{j=0}^{2g+1}$). Performing one single commutation at a point
$Q=(z,\sig R_{2g+2}(z)^{1/2})\in K_g$ (i.e., choosing $\psi_\sig(z,n)$
to perform the commutation) it is shown in \cite{bght}, Chapter 9
that the new sequences are again given by (\ref{thetaa}),
(\ref{thetab}) if $\underline{z}(P,n)$ is replaced by
\bea
\underline{\ti{z}}(P,n) = \underline{z}(P,n) 
+ \underline{\hat{A}}_{P_0}(Q)
+ \underline{\hat{A}}_{P_0}(\infty_+).
\eea
As a consequence we note that for the standard procedure as in
Theorem \ref{thmsc} (i.e., with
$Q=(\lam_1,\sig_1 R_{2g+2}(\lam_1)^{1/2})$, $\sig_1 \in \{ \pm1\}$)
the corresponding commuted operator $H_{\sig_1}$ is again
quasi-periodic and isospectral to $H$.

Hence, choosing $Q=\hat{\mu}_j$ we obtain
\bea \label{utizbla}
\underline{\ti{z}}(P,n) = \underline{z}(P,n) 
+ \underline{\hat{A}}_{P_0}(\hat{\mu}_j)
+ \underline{\hat{A}}_{P_0}(\infty_+)
\eea
and the Dirichlet eigenvalue at $\hat{\mu}_j$ is formally replaced by
one at $\infty_-$ (since $\underline{\hat{A}}_{P_0}(\infty_-) = -
\underline{\hat{A}}_{P_0}(\infty_+)$). The corresponding sequences are
neither real-valued nor well-defined. To repair this we perform a
second single commutation making the following choice
$Q=(\ti{\mu}_j,-\ti{\sig}_j R_{2g+2}(\ti{\mu}_j)^{1/2})$. The resulting
sequences
$a_{(\ti{\mu}_j,\ti{\sig}_j)}$, $b_{(\ti{\mu}_j,\ti{\sig}_j)}$ are
associated with
\bea \label{divisors}
\underline{z}_{(\ti{\mu}_j,\ti{\sig}_j)}(P,n) =
\underline{z}(P,n+1) + \underline{\hat{A}}_{P_0}(\hat{\mu}_j)
-\underline{\hat{A}}_{P_0}((\ti{\mu}_j,\ti{\sig}_j
R_{2g+2}(\ti{\mu}_j)^{1/2}))
\eea
and are again real-valued. Moreover, we have replaced the Dirichlet
eigenvalue $(\mu_j,\sig_j)$ by $(\ti{\mu}_j,\ti{\sig}_j)$ and we have
shifted the reference point for the Dirichlet boundary
condition by one (since $\underline{z}(P,n+1)$ and not
$\underline{z}(P,n)$ occurs in (\ref{divisors})) whereas everything
else remains unchanged. From Section 3 we know that
$a_{(\ti{\mu}_j,\ti{\sig}_j)}$, $b_{(\ti{\mu}_j,\ti{\sig}_j)}$
are equivalently given by
\bea\nn
a_{(\ti{\mu}_j,\ti{\sig}_j)}(n+1) \!\!&=&\!\! -
\sqrt{a(n) a(n+2)} \times\\ &&\sqrt{\frac{
C_{n}(\psi_{\sig_j}(\mu_j), \psi_{-\ti{\sig}_j}(\ti{\mu}_j))
C_{n+2} (\psi_{\sig_j}(\mu_j),\psi_{-\ti{\sig}_j}
(\ti{\mu}_j))}{ C_{n+1}(\psi_{\sig_j}(\mu_j),
\psi_{-\ti{\sig}_j}(\ti{\mu}_j))^2}},\\ \nn
b_{(\ti{\mu}_j,\ti{\sig}_j)}(n+1) \!\!&=&\!\! a(n)
\frac{\psi_{\sig_j}(\mu_j,n+2) C_{n}(\psi_{\sig_j}
(\mu_j),\psi_{-\ti{\sig}_j}(\ti{\mu}_j))}{
\psi_{\sig_j}(\mu_j,n+1) C_{n+1}( \psi_{\sig_j}(\mu_j), 
\psi_{-\ti{\sig}_j}(\ti{\mu}_j))} + \\ &&
a(n+1) \frac{\psi_{\sig_j}(\mu_j,n) C_{n+1}(\psi_{\sig_j}
(\mu_j),\psi_{-\ti{\sig}_j}(\ti{\mu}_j))}{
\psi_{\sig_j}(\mu_j,n+1) C_{n}( \psi_{\sig_j}(\mu_j), 
\psi_{-\ti{\sig}_j}(\ti{\mu}_j))} -\ti{\mu}_j,
\eea
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 proving (\ref{fita}),
(\ref{fitb}).

The reader might be puzzled by the fact that the Dirichlet
eigenvalue $\hat{\mu}_j$ is shifted to $\infty_-$ (as opposed to
$\infty_+$) which seemingly distinguishes $\infty_-$ from 
$\infty_+$. However, this apparent asymmetry between $\infty_+$ and
$\infty_-$ is related to our way of factoring $H$. If we would 
instead split up $H$ as
\begin{equation}
H = \ti{A}_{\sig_j}^* \ti{A}_{\sig_j} + \mu_j,
\end{equation}
where
\begin{equation}
(\ti{A}_{\sig_j}) f(n) = - \sqrt{-\frac{a(n-1)
\psi_{\sig_j}(\mu_j,n)}{\psi_{\sig_j}(\mu_j,n-1)}} f(n-1) +
\sqrt{-\frac{a(n-1)
\psi_{\sig_j}(\mu_j,n-1)}{\psi_{\sig_j}(\mu_j,n)}} f(n),
\end{equation}
with $\ti{A}_{\sig_j}^*$ being the adjoint of $\ti{A}_{\sig_j}$, the 
role of $\infty_+$ and $\infty_-$ would be interchanged.


We stress again that (\ref{fita}), (\ref{fitb}) represent an 
explicit realization of the
isospectral torus of all algebro-geometric quasi-periodic 
$g$-gap Jacobi
operators with spectrum (\ref{ggapspec}).

Next we turn to bounded solutions $(a(n,t),b(n,t))$ of 
the Toda equations
and construct $N$-soliton solutions on these (arbitrary) 
background solutions
using the single commutation method.

The corresponding Jacobi operators $H(t)$ satisfy
$\inf(\sig(H(t))) = \inf(\sig(H(0))) > -\infty$ for all
$t\in \R$. Furthermore, this implies the existence of
principal solutions $u_\pm(\lam,n,t)$ which satisfy
\bea \label{systo}
H(t) u_\pm(\lam,n,t) &=& \lam u_\pm(\lam,n,t),\\ 
\label{systt}
\frac{d}{dt} u_\pm(\lam,n,t) &=& P(t) u_\pm(\lam,n,t), 
\quad (n,t) \in \Z 
\times \R,
\eea
where the difference expression $P(t)$ associated with 
$(a(t),b(t))$ is defined
by
\bea \label{poft}
(P(t) f)(n) = a(n,t) f(n+1) - a(n-1,t) f(n-1).
\eea
(\ref{systo}) and (\ref{systt}) then imply the Toda 
lattice equations,
\bea \label{tleq}
\ba{rcl} \D \frac{d}{dt} a(n,t) &=& a(n,t)
\Big( b(n,t) - b(n+1,t)\Big)\\
\D \rule{0pt}{4ex} \frac{d}{dt} b(n,t) 
&=& 2 \Big( a(n-1,t)^2 - a(n,t)^2\Big)
\ea, \quad (n,t)
\in \Z \times \R
\eea
which are well-known to be equivalent to the 
Lax equation
\bea
\frac{d}{dt} H(t) - [P(t),H(t)] =0, 
\quad t \in \R
\eea
(where $[.,.]$ denotes the commutator).

Next, let $H(t)$ be as above and choose
\bea
\lam_N < \dots < \lam_1 < \inf(\sigma(H(0))), 
\quad \sigma_j \in
[-1,1], \quad 1\le j \le N \in \N.
\eea
Then Theorem \ref{thmscom} implies
\bea \nn
\lefteqn{a_{\sig_1,\dots,\sig_N}(n,t) = - 
\sqrt{a(n,t) a(n+N,t)} \times}\\
&& \frac{ \sqrt{ C_n(u_{\sig_1}^1,\dots,
u_{\sig_N}^N)
C_{n+2}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)
}}{C_{n+1}(u_{\sig_1}^1,\dots,
u_{\sig_N}^N)},
\eea
\bea \nn
\lefteqn{b_{\sig_1,\dots,\sig_N}(n,t) = - \lam_N} \\ 
&& \nn
+ a(n,t) \frac{C_{n+2}(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1})
C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}
{C_{n+1}(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1}) C_{n+1}(u_{\sig_1}^1,\dots,
u_{\sig_N}^N)} \\ && {} +
a(n+N-1,t) \frac{C_n(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1})
C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}
{C_{n+1}(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1}) C_n(u_{\sig_1}^1,\dots,
u_{\sig_N}^N)},
\eea
where
\bea
u_{\sig_\ell}^\ell(n,t) = 
\frac{1+\sigma_\ell}{2} u_+(\lam_\ell,n,t) +
(-1)^{\ell+1} \frac{1-\sigma_\ell}{2} 
u_-(\lam_\ell,n,t).
\eea
Moreover, for $\lam < \lam_N$,
\begin{equation}
u_{\sig_1,\dots,\sig_N,\pm}(\lam,n,t) =  
\frac{ \pm \sqrt{
\prod\limits_{j=0}^{N-1} (-a(n+j,t))}
C_n(u_{\sig_1}^1,\dots,
u_{\sig_N}^\ell,u_\pm(\lam))}{\sqrt{
C_n(u_{\sig_1}^1,\dots, u_{\sig_N}^N)
C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N) }}
\end{equation}
are the principal solutions of 
$\tau_{\sig_1,\dots,\sig_N}(t) u = \lam u$
satisfying
\begin{equation}
\frac{d}{dt} u_{\sig_1,\dots,\sig_N,\pm}(\lam,n,t) =
P_{\sig_1,\dots,\sig_N}(t) 
u_{\sig_1,\dots,\sig_N,\pm}(\lam,n,t),
\end{equation}
where $P_{\sig_1,\dots,\sig_N}(t)$ is defined 
as in (\ref{poft}) with $a$
replaced by $a_{\sig_1,\dots,\sig_N}$. We also 
have (cf.\ (\ref{rhoo}),
(\ref{rhoe}))
\bea
&\rho_{o,\sig_1,\dots,\sig_N}(n,t) =
-\sqrt{-a(n,t) \frac{C_{n+2}
(u_{\sig_1}^1,\dots,u_{\sig_{N-1}}^{N-1})
C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}
{C_{n+1}(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1}) C_{n+1}
(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}},&\\
&\rho_{e,\sig_1,\dots,\sig_N}(n,t) =
\sqrt{-a(n+N-1,t) \frac{C_n(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1})
C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}
{C_{n+1}(u_{\sig_1}^1,\dots,
u_{\sig_{N-1}}^{N-1}) C_n(u_{\sig_1}^1,\dots,
u_{\sig_N}^N)}}.&
\eea

Finally, the sequences 
$a_{\sig_1,\dots,\sig_N}(n,t)$,
$b_{\sig_1,\dots,\sig_N}(n,t)$ fulfill the 
Toda lattice equations (\ref{tleq})
and the sequence
\bea
\rho_{\sig_1,\dots,\sig_N}(n,t) = 
\left\{ \ba{c@{\quad}l}
\rho_{e,\sig_1,\dots,\sig_N}(m,t), & n=2m \\
\rho_{o,\sig_1,\dots,\sig_N}(m,t), 
& n=2m+1\ea \right. ,
\eea
fulfills the Kac--van Moerbeke lattice equation
\bea
\frac{d}{dt} \rho(n,t) = \rho(n,t) \Big( \rho(n+1,t)^2 - 
\rho(n-1,t)^2
\Big).
\eea

At the end we derive the $N$-soliton solutions relative 
to  an arbitrary Toda 
background solution $(a(t),b(t))$ 
using the double commutation method.

Denote by $\psi(\lam,n,t)$ the solutions of 
$\tau(t) \psi = \lam \psi$ which
are square summable near $-\infty$ and satisfy
\bea
\frac{d}{dt} \psi(\lam,n,t) = P(t) \psi(\lam,n,t).
\eea
As in Theorem \ref{dcommN} we define the following 
matrices
\bea
C^N(n,t) =  \left\{ \delta_r(s) + 
\sqrt{\gam_r \gam_s} \sum_{m=M_-}^n
\psi(\lam_r,m,t) \psi(\lam_s,m,t) 
\right\}_{1 \le r,s \le N},
\eea
\begin{equation}
\Psi^N(\lam_j,n,t) = \left\{ \ba{c@{\quad}l}
 C^N(n,t)_{r,s} & \scriptstyle
r,s \le N\\ \sqrt{\gam_s} 
\sum\limits_{m=M_-}^n \psi(\lam_j,m,t)
\psi(\lam_s,m,t) 
& \scriptstyle s \le \ell, r=N+1\\
\sqrt{\gam_r}\psi(\lam_r,n,t) &
\scriptstyle r \le \ell, s=N+1\\ 
\psi(\lam_j,n,t) & \scriptstyle r=s=N+1
\ea \right\}_{1 \le r,s \le N+1}.
\end{equation}
Then the sequences
\bea
a_{\gam_1,\dots,\gam_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)}, \\
b_{\gam_1,\dots,\gam_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)}.
\eea
satisfy the Toda lattice equations (\ref{tleq}). 
Moreover,
\bea
\psi_{\gam_1,\dots,\gam_N}(\lam_j,n,t) = 
\frac{\det
\Psi^N(\lam_j,n,t)}
{\sqrt{\det C^N(n-1,t)\det C^N(n,t)}}
\eea
satisfies
\bea
\frac{d}{dt} \psi_{\gam_1,\dots,\gam_N}(\lam_j,n,t) 
= P_{\gam_1,\dots,\gam_N}(t)
\psi_{\gam_1,\dots,\gam_N}(\lam_j,n,t),
\eea
where again $P_{\gam_1,\dots,\gam_N}(t)$ is 
defined as in (\ref{poft}) with $a$
replaced by $a_{\gam_1,\dots,\gam_N}$.






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\newpage
\addcontentsline{toc}{chapter}{Vita}
\vspace*{1cm}
\begin{center}
\large VITA
\end{center}
\vspace*{0.5cm}

\begin{tabular}{ll}
{\bf Name:} & Gerald Eric Teschl \\  \vspace*{1mm}
{\bf Date of Birth:} &  May 12, 1970 in Graz, Austria\\  \vspace*{1mm}
{\bf Nationality:} & Austria \\  \vspace*{1mm}
{\bf Education:} &  \\  \vspace*{1mm}
1995 December & \begin{minipage}[t]{9cm} Ph.D., 
University of Missouri, Columbia
\end{minipage}\\ \vspace*{1mm}
1994-1995& \begin{minipage}[t]{9cm} Ph.D student, Department of
Mathematics, University of Missouri, Columbia. Doctoral dissertation:
``Spectral Theory for Jacobi Operators''. Thesis advisor: Prof.~F.~Gesztesy
\end{minipage}\\ 1993-1994 & \begin{minipage}[t]{9cm} Visiting
the Department of Mathematics, University of Missouri, Columbia
(December 93 - March 94) \end{minipage}\\
\vspace*{1mm}
1993 July & \begin{minipage}[t]{9cm} Dipl.~Ing. (MS), with
distinction, Technical University of Graz, Austria
\end{minipage}\\ \vspace*{1mm}
1988-1993 & \begin{minipage}[t]{9cm}
Undergraduate studies at the Technical University of Graz.
Diploma Thesis: ``Schr\"odinger Operators with Interaction Concentrated on a
Spheroid'', May 93.
Advisor: Prof. W. Bulla
\end{minipage}
\end{tabular}

\end{document}
