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%%     journal="J. Differential Equations 128, 252-299 (1996)",
%%     doi="10.1006/jdeq.1996.0095"
%%     copyright="Academic Press". 
%%     }


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\begin{document}

\title{Commutation Methods for Jacobi Operators}

\author{F. Gesztesy}
\address{Department of Mathematics, University of Missouri,
Columbia, MO 65211, USA}
\email{gesztesyf@missouri.edu}

\author{G. Teschl}
\address{Department of Mathematics, University of Missouri,
Columbia, MO 65211, USA and Department of Theoretical Physics, 
Technical University of Graz, A-8010 Graz, Austria}
\curraddr{Institut f\"ur Mathematik\\
Strudlhofgasse 4\\ 1090 Wien\\ Austria}
\email{Gerald.Teschl@univie.ac.at}
\urladdr{http://www.mat.univie.ac.at/\string~gerald/}

\thanks{J. Differential Equations {\bf 128}, 252-299 (1996)}

\keywords{Commutation methods, Jacobi operators, eigenvalues, solitons}
\subjclass{Primary 47B39, 34B20; Secondary 35Q51, 35Q58}

\begin{abstract}
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. In addition we compute the (matrix-valued) Weyl 
$m$-functions of the
commuted operator in terms of the background Weyl $m$-functions.
 Finally we
show how to iterate the above methods and give explicit 
formulas for various
quantities (such as eigenfunctions and spectra) of the iterated 
operators in 
terms of the corresponding background quantities and scattering 
matrix. 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}

\maketitle




\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{gs}, \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{gs}, 
\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 
paper fills these gaps
and provides a complete spectral characterization of the 
single commutation
method (based on Deift's operator theoretic approach) in 
Sections 2 and 3 and
develops the corresponding results for the double commutation 
method in Sections
4-6. Section 7 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. Section 8 collects various 
appendices
on the Weyl-Titchmarsh theory for second-order difference 
operators.

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). Throughout this paper we denote by 
$\ell(I)=\ell((M,N))$, $I=\{ n \in \Z | M < n< N \}$,   
$M,N \in \Z \cup \{\pm\infty \}$ the set of complex-valued sequences
$\{ u(n) \}_{n\in I}$ and by $\ell^p(I)$, $1 \le p \le \infty$
the sequences $u \in \ell(I)$ such that $|u|^p$ is summable.
Furthermore, $\ell_0(I)$ denotes the set of 
sequences with only
finitely-many values being nonzero. The scalar product in 
the Hilbert space
$\ell^2(I)$ will be denoted by
\begin{equation}
\spr{u}{v} = \sum_{n \in I} \ol{u(n)} v(n), 
\quad u,v \in \ell^2(I).
\end{equation}
For brevity we focus in the following on the case $I=\Z$.

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{princips}), 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
\begin{equation}
H_{\sig_1} = A_{\sig_1} A_{\sig_1}^* + \lam_1.
\end{equation}
A straightforward calculation shows
\begin{equation}
(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),
\end{equation}
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}
\bay{rcl@{\quad}rcl}
\sigma(H_{\sig_1}) \!&=&\! \left\{ \bay{cl} 
\sigma(H) \cup \{ \lam_1\}, &
\sig_1 \in (-1,1)\\ \sigma(H), & 
\sig_1 \in \{-1,1\} \eay \right. ,&
\sig_{ac}(H_{\sig_1}) \!&=&\! \sig_{ac}(H), \\ 
\sigma_{p}(H_{\sig_1}) \!&=&\!
\left\{ \bay{cl} \sigma_{p}(H) \cup \{ \lam_1\}, 
& \sig_1 \in (-1,1)\\
\sigma_{p}(H), & \sig_1 \in \{-1,1\} \eay \right. , 
& \sig_{sc}(H_{\sig_1}) 
\!&=&\! \sig_{sc}(H).
\eay 
\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 2 and 3.

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
\begin{equation}
c_{\gam_\pm}(\lam_1,n) = 1 + 
\gam_\pm \sum_{j=\pm\infty}^{\genfrac{}{}{0pt}{}{n+1}{n}}
u_{\pm}(\lam_1,j)^2,
\end{equation}
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
\begin{equation}
H_{\pm1} = A_{\gam_\pm}^* A_{\gam_\pm} + \lam_1.
\end{equation}
Performing a second commutation yields the 
doubly commuted operator
\begin{equation}
H_{\gam_\pm} = 
A_{\gam_\pm} A_{\gam_\pm}^* + \lam_1.
\end{equation}
Explicitly, one verifies
\begin{equation}
\hspace*{5mm}(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),
\end{equation}
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 4. 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}


In this section we intend to give a detailed 
investigation of the single
commutation method. We will need the following 
condition on $a,b$ which will be
used throughout Sections 2 and 3.

\bigskip \noindent{\bf Hypothesis (H.2.1).} 
\addtocounter{thm}{1}
Suppose
\begin{equation}
a(n) < 0, \quad b(n) \in \R, \qquad b(n) \ge c, 
\quad c \in \R, \: n \in \Z.
\end{equation}

We shall consider (self-adjoint) Jacobi operators 
$H$ associated with the
difference expression
\begin{equation} \label{diffex}
(\tau f)(n) = a(n) f(n+1) + a(n-1) f(n-1) - b(n) f(n),
\end{equation}
in the Hilbert space $\lz$. 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}
Assume (H.2.1) and let $H$ be a Jacobi operator associated with the
difference expression (\ref{diffex}). Introduce $a_\eps$ by
\begin{equation}
a_\eps(n) = \eps(n)a(n), \; \eps(n) \in\{+1, -1\},
\quad n\in\Z
\end{equation}
and the unitary operator $U_\eps$ by
\begin{equation}
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).
\end{equation}
Then $H_\eps$ defined as
\begin{equation}
H_\eps = U_\eps^{-1} H U_\eps,
\end{equation}
is associated with the difference expression
\begin{equation}
(\tau_\eps f)(n) = a_\eps(n) f(n+1) + 
a_\eps(n-1) f(n-1) - b_\eps(n) f(n).	
\end{equation}
In particular, $H_\eps$ is unitarily equivalent 
to $H$. 
\end{lem}

As a preparation we prove

\begin{lem} \label{princips}
Assume (H.2.1). Let $H$ be a given Jacobi operator in 
$\lz$ and let
$\lam < \inf(\sig(H))$ ((H.2.1) implies that $H$ is 
semi-bounded, cf.\ 
\cite{crit}). Then there exist unique positive solutions 
$u_\pm(\lam,n)$ of
$\tau u = \lam u$ (up to constant multiples) which are 
square summable near
$\pm\infty$. (They are sometimes called principal 
solutions of $(H-z)u=0$ near
$\pm\infty$.)
\end{lem}

\begin{proof}
For the existence of square summable sequences 
$u_+(\lam)$ near $\infty$
consider the sequence $((H-\lam)^{-1} \delta_0)(n)$ for 
$n > 0$ and extend it
to a solution of $(\tau -\lam)u =0$ for all $n \in \Z$. 
Let $H_{+,n}$ be the
restriction of $H$ to $\ell^2((n,\infty))$ with a 
Dirichlet boundary condition at
$n$. 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}
which shows that $u_+(\lam)$ can be chosen to be positive. 
The existence of $u_-$
is proven similarly.
\end{proof}

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 $\ell_0(\N)$
\begin{equation}
(\dot{A}_+ f)(n) = \rho_{o,+}(n) f(n+1) + 
\rho_{e,+}(n) f(n), \quad f \in
\ell_0(\N)
\end{equation}
and denote its operator closure (in $\ell^2(\N)$) by 
$A_+$. One verifies,
\begin{equation}
\db(A_+) \subseteq \{ f \in \ell^2(\N) | \rho_{o,+}(n) 
f(n+1) + \rho_{e,+}(n)
f(n) \in \ell^2(\N) \}.
\end{equation}
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
\begin{equation}
\db(A_+ A_+^*) = 
\{ f \in \db(A_+^*) | A_+^* f \in \db(A_+) \}.
\end{equation}
A simple calculation shows 
$A_+ A_+^* f = (\tau -\lam_1)f$ and hence we may
define
\begin{equation}
H_+ = A_+ A_+^* + \lam_1, \quad \db(H_+) \subseteq \{ f \in
\ell^2(\N) | f(0)=0, \: \tau f \in \ell^2(\N) \},
\end{equation}
where equality in the last relation is equivalent 
to $\tau$ being limit point
($l.p.$) at $+\infty$. Similarly one defines 
for $n<0$
\begin{equation}
\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)}}
\end{equation}
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
\begin{equation}
H_{+,1} = A_+^* A_+ + \lam_1, \qquad H_{-,1} = 
A_- A_-^* + \lam_1.
\end{equation}
The next theorem characterizes $H_{\pm,1}$ in 
terms of $H_\pm$, but first we
need to introduce

\bigskip
\noindent{\bf 
Hypothesis (H.2.4).} \addtocounter{thm}{1}
Suppose $H_\pm$ satisfies one of the following spectral 
conditions.
\begin{list}{(\roman{me}).}{\usecounter{me}}
\item $\sig_{ess} (H_\pm) \ne \emptyset$.
\item $\sig(H_\pm)=\sig_{\operatorname{d}}(H_\pm)=
\{ \lam_{\pm,j} \}_{j \in J_\pm}$ with $\sum_{j\in J_\pm}
(1+\lam^2_{\pm,j})^{-1} =\infty$.
\end{list}

Hypothesis (H.2.4) 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
\begin{equation}
\db(H_\pm) = 
\{ f \in \ell^2(\pm\N) | f(0)=0, \: \tau f \in \ell^2(\pm\N)\}.
\end{equation}

\begin{thm} \label{thmpm}
Assume (H.2.1) and (H.2.4). Then the operators $H_{\pm,1}$ 
constructed above satisfy
(H.2.1) and (H.2.4) 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
\begin{equation}
\bay{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).
\eay 
\end{equation}
\end{thm}

\begin{proof}
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.2.4) and
hence $\tau_\pm$ and $\tau_{\pm,1}$ are $l.p.$ at 
$\pm\infty$. The rest are
straightforward calculations.
\end{proof}

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{princips} and set
\begin{equation}
u_{\sig_1}(\lam_1,n) = \frac{1+\sigma_1}{2} u_+(\lam_1,n)
 + \frac{1-\sigma_1}{2}
u_-(\lam_1,n).
\end{equation}
Now define sequences
\begin{equation} \hspace*{7mm}
\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}
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
\begin{equation}
\db(A_{\sig_1}^* A_{\sig_1}) = 
\{ f \in \db(A_{\sig_1}) | A_{\sig_1} f \in
\db(A_{\sig_1}^*) \}.
\end{equation}
A simple calculation shows $A_{\sig_1}^* A_{\sig_1} 
= \tau -\lam_1$ and we hence
may define
\begin{equation} \label{hpm}
H = A_{\sig_1}^* A_{\sig_1} + \lam_1, 
\qquad \db(H) \subseteq \{ f \in \lz |
\tau f \in \lz \}.
\end{equation}
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
\begin{equation}
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 \},
\end{equation}
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
\begin{equation}
\db(H) = \{ f \in \lz | \tau f \in \lz \}.
\end{equation}


\begin{thm} \label{thmsc}
Assume (H.2.1) and (H.2.4). 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.2.1). The 
equation $\tau_{\sig_1} v=
\lam_1 v$ has the positive solution
\begin{equation}
v_{\sig_1}(\lam_1,n) = 
\frac{1}{\sqrt{-a(n) u_{\sig_1}(\lam_1,n)
u_{\sig_1}(\lam_1,n+1)}}
\end{equation}
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}
\bay{rcl@{\quad}rcl}
\sigma(H_{\sig_1}) \!&=&\! \left\{ \bay{cl} 
\sigma(H) \cup \{ \lam_1\}, &
\sig_1 \in (-1,1)\\ \sigma(H), 
& \sig_1 \in \{-1,1\} \eay \right. ,&
\sig_{ac}(H_{\sig_1}) \!&=&\! \sig_{ac}(H), \\ 
\sigma_{p}(H_{\sig_1}) \!&=&\!
\left\{ \bay{cl} \sigma_{p}(H) \cup \{ \lam_1\}, 
& \sig_1 \in (-1,1)\\
\sigma_{p}(H), & \sig_1 \in \{-1,1\} \eay \right. , 
& \sig_{sc}(H_{\sig_1})
\!&=&\! \sig_{sc}(H).
\eay
\end{equation}
In addition, the sequence
\begin{equation}
(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)}}
\end{equation}
solves $\tau_{\sig_1} u = z u$ if $u(z)$ solves 
$\tau u = z u$ for some $z \in
\C$, where $W_n(u,v) = a(n)(u(n) v(n+1) - 
u(n+1) v(n))$ denotes the modified
Wronskian. Moreover, one obtains
\begin{equation} \label{wronsc}
W_{\sig_1,n}(A_{\sig_1} u(z), A_{\sig_1} v(z)) = 
(\lam_1-z) W_n(u(z),v(z))
\end{equation}
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
\begin{equation} \label{res}
(H_{\sig_1} - z)^{-1} = \frac{1}{z-\lam_1} 
\Big(A_{\sig_1} (H - z)^{-1}
A_{\sig_1}^* - 1\Big)
\end{equation}
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} &\text{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(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
\begin{equation}
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)}}.
\end{equation}
In addition, we have
\begin{equation} \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)
\end{equation}
and, if $\tau u(\lam) = \lam u(\lam)$, 
$u(\lam,.) \in \lz$,
\begin{equation} \label{normuz}
\sum_{n \in \Z}  (A_{\sig_1} u)(\lam,n)^2 = 
(\lam-\lam_1) \sum_{n \in \Z}
u(\lam,n)^2.
\end{equation}
\end{thm}

\begin{proof}
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.
\end{proof}

\begin{rem} \label{remagz}
(i). Hypothesis (H.2.4) 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.2.1) 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
\begin{equation} \hspace*{5mm}
\sum_{n=0}^{\pm\infty} 
\frac{1}{-a(n)u_{\sig_1}(\lam_1,n)u_{\sig_1}(\lam_1,n+1)} <
\infty \text{ if and only if }
 \sig_1 \in \bay{c}  [-1,1) \\ (-1,1]
\eay
\end{equation}
(cf.\ \cite{pat} 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.2.1). 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
\begin{equation}
\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).
\end{equation}
\end{lem}

\begin{proof}
The above formulas are straightforward calculations 
using (\ref{gfsc}) and
(\ref{mpl}), (\ref{mmi}).
\end{proof}

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{ta}) 
we now assume
\begin{equation} \label{decay}
a(n)>0,\: b(n)\in\R, \qquad n|1 - 2a(n)|, 
n b(n) \in \ell^1(\Z)
\end{equation}
(cf. Remark \ref{remagz}). This implies
\begin{equation} \hspace*{5mm}
\sig_{ac}(H) =[-1,1], \quad \sig_{sc}(H) = 
\emptyset, \quad \sig_{p}(H) = \{
\lam_j \}_{j \in J} \subseteq \R \backslash [-1,1],
\end{equation}
where $J \subseteq \N$ is a suitable (finite) 
index set,
and the existence of the so called Jost solutions 
$f_\pm(k,n)$,
\begin{equation}
\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.
\end{equation} 
Transmission $T(k)$ and reflection $R_\pm(k)$ 
coefficients are then defined
via
\begin{equation}
T(k) f_\mp(k,n) = f_\pm(k^{-1},n) + 
R_\pm(k) f_\pm(k,n), \quad |k|=1,
\end{equation}
and the norming constants $\gam_{\pm,j}$ 
corresponding to $\lam_j \in
\sig_{p}(H)$ are given by
\begin{equation} \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.
\end{equation}

\begin{lem} \label{scatsc}
Suppose $H$ satisfies {\em (\ref{decay})} and let 
$H_{\sig_1}$ be constructed as
in Theorem \ref{thmsc} with
\begin{equation}
u_{\sig_1}(\lam_1,n) = \frac{1+\sig_1}{2} f_+(k_1,n) + 
\frac{1-\sig_1}{2}
f_-(k_1,n).
\end{equation}
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
\begin{equation} \hspace*{10mm}
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),
\end{equation}
\begin{equation}
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\},
\end{equation}
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
\begin{equation}
\gam_{\sig_1,\pm,j} = |k_1^{\sig_1} - 
k_j^{\mp1}| \gam_{\pm,j}, \quad j \in J,
\: \sig_1 \in \{-1,1\}.
\end{equation}
\end{lem}

\begin{proof}
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})).
\end{proof}


\section{Iteration of the single commutation method}




By choosing $\lam_2<\lam_1$ and $\sigma_2 \in [-1,1]$ 
we can define
\begin{equation}
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)
\end{equation}
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
\begin{equation}
H_{\sig_1} = 
A_{\sig_1,\sig_2}^* A_{\sig_1,\sig_2} - \lam_2.
\end{equation}
A further commutation then yields the operator
\begin{equation}
H_{\sig_1,\sig_2} = 
A_{\sig_1,\sig_2}A_{\sig_1,\sig_2}^* - \lam_2
\end{equation}
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.2.1) and (H.2.4). Let $H$ be as in 
Section 2 and choose
\begin{equation}
\lam_N < \dots < \lam_2 < \lam_1 < \inf(\sigma(H)), 
\quad \sigma_\ell \in [-1,1],
\:\: 1\le \ell \le N, \: N \in \N.
\end{equation}
Then we have
\bea \hspace*{5mm}
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
\begin{equation}
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),
\end{equation}
and $C_n$ denotes the $n$-dimensional Casoratian
\begin{equation} \label{Nwon}
C_n(u_1,\dots,u_N) = 
\det\{ u_i(n+j-1)\}_{1\le i,j\le N}.
\end{equation}
Moreover, for $1 \le \ell \le N$, $\lam < \lam_\ell$
\begin{equation} \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) }},
\end{equation}
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
\begin{equation}
\sigma(H_{\sig_1,\dots,\sig_N}) = 
\sigma(H) \cup \{ \lam_\ell \; | \;
\sigma_\ell \in (-1,1), \; 1 \le \ell \le N \}.
\end{equation}
\end{thm}

\begin{proof}
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.
\end{proof}

\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
\begin{equation} \hspace*{5mm}
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).
\end{equation}
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_{\genfrac{}{}{0pt}{}{\ell=1}{\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}

\begin{proof}
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
\begin{equation}
\frac{1+\hat{\sig}_2}{1-\hat{\sig}_2} = 
\frac{1}{k_2} \frac{1+\sig_1}{1-\sig_1}.
\end{equation}
The claims now follow from Lemma \ref{scatsc} 
after extending this result by
induction.
\end{proof}


\section{The double commutation method}



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
\begin{equation}
c_\gam(n) = 1 + 
\gam \sum_{j=M_-}^n \psi(j)^2, \qquad n \ge M_-.
\end{equation}
(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
\begin{equation} \label{unitary1}
\bay{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). \eay
\end{equation}
By inspection, the sequence $f_\gam$ is also square 
summable near $M_-$ and the
inverse transformation is given by
\begin{equation} \label{transfUf}
\bay{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) \eay,
\end{equation}
where
\begin{equation} \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)}}.
\end{equation}

\begin{lem} \label{lemuni}
Define $\psi_\gam$ as in (\ref{psigam}). Then 
$\psi_\gam \in \hr$ and
\begin{equation}
\| \psi_\gam \|^2 = \frac{1}{\gam} 
\Big(1 - \lim_{n \to M_+} c_\gam(n)^{-1} \Big).
\end{equation}
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}

\begin{proof}
For the claims concerning $\psi$ we use
\begin{equation}
\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).
\end{equation}
Next we note that
\begin{equation}
c_\gam(n) \sum_{j=M_-}^n \psi_\gam(j) f_\gam(j) = 
\sum\limits_{j=M_-}^n \psi(j)
f(j)
\end{equation}
and a direct calculation shows
\begin{equation} \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.
\end{equation}
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).
\end{proof}

Using, e.g., the polarization identity, we 
further get
\begin{equation} \label{sprgam} \hspace*{5mm}
\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)} .
\end{equation}

Next we take two sequences $a,b$ satisfying

\bigskip
\noindent{\bf Hypothesis (H.4.2).}\addtocounter
{thm}{1}
Suppose
\begin{equation}
a,b \in \ell((M_--1,M_++1)), 
\qquad a(n) \in \R \bs \{ 0 \}, \: b(n) \in \R
\end{equation}

and introduce 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}
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.

\bigskip
\noindent{\bf Hypothesis (H.4.3).}\addtocounter{thm}{1}
Suppose $\psi(\lam)$ satisfies the following 
conditions.
\begin{list}{(\roman{me}).}{\usecounter{me}}
\item $\psi(\lam,n)$ is a real-valued solution of 
$\tau \psi(\lam) = \lam
\psi(\lam)$.
\item $\psi(\lam,n)$ is square summable near $M_-$ 
and fulfills the boundary
condition (of $H$) at $M_-$ (if any, i.e., if $\tau$ 
is $l.c.$ at $M_-$).
\item $\psi(\lam,n)$ also fulfills the boundary 
condition (of $H$) at $M_+$ if
$\tau$ is $l.c.$ at $M_+$ ($\psi(\lam,n)$ is then an 
eigenfunction of $H$).
\end{list}

Sufficient conditions for the above function to 
exist are
\begin{list}{(\alph{me}).}{\usecounter{me}}
\item $\lam \in \sigma_{p}(H)$, or
\item $\tau$ is $l.c.$ at $M_-$ but not at $M_+$, or
\item $\sigma(H) \ne \R$ (and $\lam \in \R \bs \sigma(H)$), or
\item $\sigma(H_-) \ne \R$ 
(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 $\hat{M}+1$.
\end{list}

It follows that $H$ is explicitly given by
\begin{equation} \label{defbh}
\hspace*{5mm} \db(H) = 
\{ f \in \hr | \bay[t]{l} \tau f \in \hr ; \:
W_{M_- -1}(\psi(\lam_1),f)=0 \text{ if $\tau$ 
is $l.c.$ at $M_-$} , \\
W_{M_+}(\psi(\lam_1),f)=0 \text{ if $\tau$ is 
$l.c.$ at $M_+$} \}. \eay
\end{equation}

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.4.2) and (H.4.3) and let $\tau_{\gam_1}$ be 
the difference expression
\begin{equation}
(\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),
\end{equation}
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
\begin{equation}
\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)}}
\end{equation}
associated with the eigenvalue $\lam_1$. If 
$\psi(\lam_1) \not\in \hr$ (and hence
$\tau$ is $l.p.$ at $M_+$) we have
\begin{equation}
(1-P_{\gam_1}(\lam_1)) H_{\gam_1} = 
U_{\gam_1} H U_{\gam_1}^{-1} (1-P_{\gam_1}(\lam_1)),
\end{equation}
where $U_{\gam_1}$ is the unitary transformation 
of Lemma \ref{lemuni} and thus
\begin{equation}
\bay{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).
\eay
\end{equation}
If $\psi(\lam_1) \in \hr$ there is a unitary operator 
$\ti{U}_{\gam_1} =
U_{\gam_1} \oplus \sqrt{1+\gam_1 \| \psi(\lam_1) \|^2}
 \text{\bf 1}$ on
$(1-P_{\gam_1}(\lam_1)) \hr \oplus P_{\gam_1}(\lam_1) \hr $ such that
\begin{equation}
H_{\gam_1} = \ti{U}_{\gam_1} H \ti{U}_{\gam_1}^{-1}
\end{equation}
and thus
\begin{equation}
\bay{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).
\eay
\end{equation}
\end{thm}

\begin{proof}
It suffices to prove
\begin{equation}
(1-P_{\gam_1}(\lam_1)) H_{\gam_1} = 
U_{\gam_1} H U_{\gam_1}^{-1} (1-P_{\gam_1}(\lam_1)).
\end{equation} 
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
\begin{equation}
\tau_{\gam_1} (U_{\gam_1} f) = U_{\gam_1} (\tau f)
\end{equation}
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
\begin{equation}
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)}
\end{equation}
shows that
\begin{equation} \label{WMm}
W_{\gam_1,M_- -1}(\psi_{\gam_1}(\lam_1),U_{\gam_1} f) =0, 
\qquad f \in \db(H).
\end{equation}
We further claim that
\begin{equation} \label{WMp}
W_{\gam_1,M_+}(\psi_{\gam_1}(\lam_1),U_{\gam_1} f) = 0, 
\qquad f \in \db(H).
\end{equation}
This is clear if $\psi(\lam_1) \in \hr$. Otherwise, 
i.e., if $\psi(\lam_1)
\not\in \hr$, we use
\begin{equation}
\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)}.
\end{equation}
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
\begin{equation}
(1-P_{\gam_1}(\lam_1)) U_{\gam_1} \db(H) 
\subseteq (1-P_{\gam_1}(\lam_1))
\db(H_{\gam_1}).
\end{equation}
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.
\end{proof}

\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.4.3)
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
\begin{equation} \hspace*{6mm}
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 \}
\end{equation}
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} \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
\begin{equation} \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)}.
\end{equation}
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}
\begin{proof}
All facts are tedious but straightforward calculations.
\end{proof}

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})).

\bigskip
\noindent{\bf Hypothesis 
(H.4.7).} \addtocounter{thm}{1}
Suppose $H_+$ satisfies one of the following 
spectral conditions:\\
(i). $\sig_{ess} (H_+) \ne \emptyset$.\\
(ii). $\sig(H_+)=\sig_{\operatorname{d}}(H_+)=
\{ \lam_{+,j} \}_{j\in J_+}$ with $\sum_{j\in J_+}
(1+\lam^2_{+,j})^{-1}
=\infty$.

Clearly Hypothesis (H.4.7) 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.4.2), (H.4.3), and (H.4.7). Then 
$\tau_{\gam_1}$ is $l.p.$ at
$M_+=\infty$.
\end{thm}

\begin{proof}
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.4.7). Hence $\tau_{\gam_1}$ is $l.p.$ 
at $\infty$ as claimed.
\end{proof}

\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
\begin{equation}
T_{\gam_1}(k) = 
\sgn(k_1) \frac{k \, k_1 -1}{k - k_1} T(k),
\end{equation}
\begin{equation}
R_{-,\gam_1}(k) = R_-(k), \qquad R_{+,\gam_1}(k) = 
\left( \frac{k - k_1}{k \, k_1
-1} \right)^2 R_+(k),
\end{equation}
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}

\begin{proof}
By Lemma \ref{addprophg} the Jost solutions 
$f_{\gam_1,\pm}(k,n)$ are up to a
constant given by 
\begin{equation}
\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)}}.
\end{equation}
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.
\end{proof}




\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{rem} \label{rembc}
We have restricted ourselves to a Dirichlet boundary 
condition since the
general boundary condition
\begin{equation}
\cos(\alpha) u(0) + \sin(\alpha) u(1) = 0
\end{equation}
can be reduced to the case $\alpha=0$ by the 
transformation $b(1) \to b(1) + a(0)
\tan(\alpha)$ for $\alpha \ne \pi/2$, whereas for 
$\alpha = \pi/2$ one can replace
$\ell^2(\N)$ by $\ell^2((1,\infty))$.
\end{rem}

\begin{thm}
Assume (H.4.2), $\psi(\lam_1,1)=1$ and let $m_+(z,0)$,
$m_{+,\gam_1}(z,0)$  denote the
Weyl $m$-functions of $H_+$, $H_{+,\gam_1}$. Then 
we have
\begin{equation}
m_{+,\gam_1}(z,0) = \frac{1}{1+\gam_1} \Big( m_+(z,0) - 
\frac{\gam_1}{z -
\lam_1} \Big).
\end{equation}
If $\mu_+$ and $\mu_{+,\gam_1}$ denote the 
corresponding spectral functions of
$H_+$ and $H_{+,\gam_1}$ it follows that
\begin{equation}
\mu_{+,\gam_1}(\lam) = \frac{1}{1+\gam_1} 
\Big( \mu_+(\lam) + \gam_1
\Theta(\lam-\lam_1)
\Big),
\end{equation}
where $\Theta(.)$ denotes the (right continuous) 
step function
\begin{equation}
\Theta(x) = \left\{ \bay{c@{\quad}l} 1, 
& x \ge 0 \\ 0, & x < 0 \eay
\right. .
\end{equation}
\end{thm}

\begin{proof}
As in Appendix B 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
\begin{equation}
\gam_{j,\gam_1}(N) = 
\frac{1}{1+\gam_1} \left\{ \bay{c@{\quad}l} \gam_j(N) + 
\gam_1, & \lam_j=\lam_1 \\ \gam_j(N), 
& \lam_j \ne \lam_1 \eay \right. .
\end{equation}
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
\begin{equation}
m_{N,\gam_1}(z,0) = 
\frac{1}{1+\gam_1} \Big( m_N(z,0) - 
\frac{\gam_1}{z - \lam_1}
\Big)
\end{equation}
and the theorem follows upon taking the limit 
$N \to \infty$.
\end{proof}

\begin{rem}
If we transform the operator $H_+$ into it's diagonal 
form as in Appendix C 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 Appendix C 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
\begin{equation} \label{initv}
\psi(\lam_1,0) = - \sin(\alpha), 
\quad \psi(\lam_1,0) = \cos(\alpha), \quad
\alpha \in [0,\pi).
\end{equation}

\begin{thm}
Assume (H.4.2) 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 Appendix B. Then we have
\begin{equation} \hspace*{5mm}
\ti{m}_{\pm,\gam_1}(z,\ti{\alpha}) = 
\frac{c_{\gam_1}(\lam_1,0)}{c_{\gam_1}(\lam_1,-1)}
\frac{1+\cot(\ti{\alpha})^2}{1+\cot(\alpha)^2}
\Big( \ti{m}_\pm(z,\alpha) - \frac{\ti{\gam_1}}{z - \lam_1} +
\frac{\cot(\alpha) c_{\gam_1}(\lam_1,1)^{-1}}{a(0)
(1+\cot(\ti{\alpha})^2)} \Big),
\end{equation}
where
\begin{equation}
\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).
\end{equation}
\end{thm}

\begin{proof}
Consider the sequences
\begin{equation}
\phi_{\alpha,\gam_1}(z,n), \quad \theta_{\alpha,\gam_1}(z,n) -
\Big( \frac{\ti{\gam}_1}{z-\lam_1} - \frac{\cot(\alpha)
c_{\gam_1}(\lam_1,1)^{-1}}{a(0)(1+\cot(\ti{\alpha})^2)}
\Big)\phi_{\alpha,\gam_1}(z,n)
\end{equation}
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}).
\end{proof}

The Weyl $M$-matrix and the corresponding spectral 
matrix can now be computed in a
straightforward manner (cf.\ Appendix D).



\section{Iteration of the double commutation method}


Finally we demonstrate how to iterate the double 
commutation method. We
choose a given background operator $H$ (with 
coefficients $a$, $b$ satisfying
(H.4.2)) and further $\gamma_1>0, \lam_1\in \R$. 
Next choose $\psi(\lam_1)$ as
in Hypothesis (H.4.3) 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.4.2) 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.4.3). 
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 $(1 \le \ell \le N)$
\begin{equation}
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},
\end{equation}
\begin{equation} \label{cij}
C^\ell_{i,j}(n) = \left\{ \bay{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 \eay 
\right\}_{1 \le r,s \le \ell},
\end{equation}
\begin{equation}
\Psi^\ell(\lam_j,n) = \left\{ \bay{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
\eay \right\}_{1 \le r,s \le \ell+1}.
\end{equation}
Then we have (set $C^0(n)=1$)
\begin{equation} \label{ckn}
c_{\gam_\ell}(\lam_\ell,n) = 1 + 
\gam_\ell \sum_{m=M_-}^n
\psi_{\gam_1,\dots,\gam_{\ell-1}} (\lam_\ell,m)^2 = 
\frac{\det C^\ell(n)}{\det
C^{\ell-1}(n)},
\end{equation}
and hence
\begin{equation} \label{prck}
\prod_{\ell=1}^N c_{\gam_\ell}(\lam_\ell,n) = 
\det C^N(n).
\end{equation}
Moreover,
\begin{equation} \label{sprk}
\sum_{m=M_-}^n \psi_{\gam_1,\dots,\gam_{\ell-1}}(\lam_i,m) 
\psi_{\gam_1,\dots,\gam_{\ell-1}}(\lam_j,m) = 
\frac{\det C^\ell_{i,j}(n)}{\det
C^{\ell-1}(n)}
\end{equation}
and
\begin{equation} \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)}}.
\end{equation}
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
\begin{equation}
\bay{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).
\eay 
\end{equation}
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}

\begin{proof}
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.
\end{proof}

\begin{rem}
(i). If $f$ 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.4.2) and (H.4.7). 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 
$u_{\sig_\ell}^\ell =
\frac{1+\sig_\ell}{2} k_\ell^n + 
(-1)^{\ell+1} \frac{1-\sig_\ell}{2}
k_\ell^{-n}$ as in Theorem \ref{thmscom} 
coincide if
\begin{equation}
\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_{
\genfrac{}{}{0pt}{}{\ell=1}{\ell \ne j}}^N |k_j-k_\ell|}, 
\quad 1 \le j \le N.
\end{equation}
For a direct proof compare \cite{TKvM}.
\end{rem}


\section{Applications}




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}))
\begin{equation}
\sig(H_\pm) = \sig(H) \cup \{ \mu_j | \sig_j = 
\pm, \: 1 \le j \le g \}.
\end{equation}
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
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))}},\\ \nn
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
\begin{equation}
R_{2g+2}(z)^{1/2} = \prod_{j=0}^{2g+1} (z-E_j)^{1/2}
\end{equation}
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
\begin{equation}
\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},
\end{equation}
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} \dfrac{\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
\begin{equation}
\underline{\ti{z}}(P,n) = \underline{z}(P,n) 
+ \underline{\hat{A}}_{P_0}(Q)
+ \underline{\hat{A}}_{P_0}(\infty_+).
\end{equation}
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
\begin{equation} \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_+)
\end{equation}
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 choosing
$Q=(\ti{\mu}_j,-\ti{\sig}_j R_{2g+2}(\ti{\mu}_j)^{1/2})$. The resulting
sequences
$a_{(\ti{\mu}_j,\ti{\sig}_j)}$, $b_{(\ti{\mu}_j,\ti{\sig}_j)}$ are
associated with
\begin{equation} \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}))
\end{equation}
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
\begin{equation}
a_{(\ti{\mu}_j,\ti{\sig}_j)}(n+1) = -
\sqrt{a(n) a(n+2)} \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}},
\end{equation}
\bea \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 factorizing $H$. 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
\begin{equation} \label{poft}
(P(t) f)(n) = a(n,t) f(n+1) - a(n-1,t) f(n-1).
\end{equation}
(\ref{systo}) and (\ref{systt}) then imply the Toda 
lattice equations,
\begin{equation} \label{tleq}
\bay{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,t-1)^2 - a(n,t)^2\Big)
\eay, \quad (n,t)
\in \Z \times \R
\end{equation}
which are well-known to be equivalent to the 
Lax equation
\begin{equation}
\frac{d}{dt} H(t) - [P(t),H(t)] =0, 
\quad t \in \R
\end{equation}
(where $[.,.]$ denotes the commutator).

Next, let $H(t)$ be as above and choose
\begin{equation}
\lam_N < \dots < \lam_1 < \inf(\sigma(H(0))), 
\quad \sigma_j \in
[-1,1], \quad 1\le j \le N \in \N.
\end{equation}
Then Theorem \ref{thmscom} implies
\bea \nn
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)},\\ \nn
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
\begin{equation}
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).
\end{equation}
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
\begin{equation}
\rho_{\sig_1,\dots,\sig_N}(n,t) = 
\left\{ \bay{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\eay \right. ,
\end{equation}
fulfills the Kac--van Moerbeke lattice equation
\begin{equation}
\frac{d}{dt} \rho(n,t) = \rho(n,t) \Big( \rho(n+1,t)^2 - 
\rho(n-1,t)^2
\Big).
\end{equation}

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
\begin{equation}
\frac{d}{dt} \psi(\lam,n,t) = P(t) \psi(\lam,n,t).
\end{equation}
As in Theorem \ref{dcommN} we define the following 
matrices
\begin{equation}
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},
\end{equation}
\begin{equation}
\Psi^N(\lam_j,n,t) = \left\{ \bay{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
\eay \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,
\begin{equation}
\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)}}
\end{equation}
satisfies
\begin{equation}
\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),
\end{equation}
where again $P_{\gam_1,\dots,\gam_N}(t)$ is 
defined as in (\ref{poft}) with $a$
replaced by $a_{\gam_1,\dots,\gam_N}$.


\section{Appendices}


Appendix A lists some formulas for Jacobi operators 
which are used in Sections 2
and 3. Appendices B--D contain some facts about 
Weyl--Titchmarsh theory for
Jacobi operators which are needed in Section 3. 
Finally Appendix D states a
$l.p.$ criterion which seems to be novel and of 
independent interest.

Appendices B--D 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 following material is 
essentially taken from
\cite{ak},\cite{at},\cite{be},\cite{cl}.


\appendix

\section{General Background}


Assume (H.4.2) and define the Jacobi difference 
expression
\begin{equation}
(\tau f)(n) = a(n) f(n+1) +a(n-1) f(n-1) -b(n) f(n).
\end{equation}
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}
where we have introduced the modified Wronskian
\begin{equation}
\label{wr} W_n(f,g) = a(n) \Big( f(n) g(n+1) - 
f(n+1) g(n) \Big), \:\: n \in \Z.
\end{equation}
The main object of our interest will be the equation
\begin{equation}
\label{gl}
\tau u = z \, u, \qquad u \in \lf, \: z \in \C.
\end{equation}
A glance at
(\ref{gf}) shows that the modified Wronskian of 
two solutions 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
\begin{equation} \label{weylkl}
[u(z)]_n = [u(z)]_{m-1} - \sum_{j=m}^n |u(z,j)|^2,
\end{equation}
where $[.]_n$ denotes the Weyl bracket
\begin{equation}
[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.
\end{equation}
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 $\pm\infty$.



\section{Weyl $m$-functions}


Let $\theta_\alpha(z,.),\phi_\alpha(z,.)$ be 
the fundamental system of (\ref{gl})
corresponding to the initial conditions
\begin{equation} \label{funsy}
\bay{l@{\qquad}l} \phi_\alpha(z,0) = -\sin(\alpha), 
& \phi_\alpha(z,1) =
\cos(\alpha), \\ \theta_\alpha(z,0) = 
\D \frac{\cos(\alpha)}{a(0)}, &
\theta_\alpha(z,1) = 
\D \frac{\sin(\alpha)}{a(0)} \eay
\end{equation}
such that
\begin{equation}
W(\theta_\alpha(z),\phi_\alpha(z)) =  1.
\end{equation}
Next pick $\lam_1 \in \R$ and define the following 
rational function with respect
to $z$,
\begin{equation} \label{mfun}
m_N(z,\alpha) = \frac{W_N(\phi_\alpha(\lam_1),
\theta_\alpha(z))}{W_N(\phi_\alpha(\lam_1), 
\phi_\alpha(z))}, \qquad N \in 
\Z \bs \{ 0 \},
\end{equation}
which has poles at the zeros $\lam_j(N) \in \R$, 
$\lam_1(N) \equiv \lam_1$ of $W_N(\phi_\alpha(\lam_1),
\phi_\alpha(.)) = 0$. 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(\phi_\alpha(\lam_j(N)), 
\theta_\alpha(z)) &=& -1, \\
\label{limw}
\lim_{z \to \lam_j(N)} 
\frac{W_N(\phi_\alpha(\lam_j(N)), \phi_\alpha(z))
}{z-\lam_j(N)} &=& W_N(\phi_\alpha(\lam_j(N)), 
\frac{d}{dz} 
\phi_\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
\begin{equation}
\gam_j(\alpha,N) = 
\Big( \sum_{n=\genfrac{}{}{0pt}{}{1}{N+1}}^{
\genfrac{}{}{0pt}{}{N}{0}}
\phi_\alpha(\lam_j(N),n)^2 \Big)^{-1}, 
\quad N \bay{c} \vspace*{-2.5mm} > \\
\vspace*{1.5mm} < \eay 0.
\end{equation}
The $\gam_j(\alpha,N)$ are called norming constants. 
Hence one gets
\begin{equation}
m_N(z,\alpha) = 
\sum_j \frac{\mp\gam_j(\alpha,N)}{z - \lam_j(N)} + \left\{
\bay{l} \frac{\pm\tan(\alpha)^{\pm1}}{a(0)}, \: \alpha \in
\genfrac{}{}{0pt}{}{[0,\pi)}{(0,\pi]}\\
\frac{\pm z - b(\genfrac{}{}{0pt}{}{1}{0})}{a(0)^2}, \:
\alpha = \genfrac{}{}{0pt}{}{\pi}{0}\eay \right. ,
\quad N \bay{c} \vspace*{-2.5mm} > \\
\vspace*{1.5mm} < \eay 0.
\end{equation}
(We note that $\lam_j(N)$ depend on 
$\alpha$ for $j>1$.) Furthermore, the
function
\begin{equation}
u_N(z,n) = \theta_\alpha(z,n) - 
m_N(z,\alpha) \phi_\alpha(z,n)
\end{equation}
satisfies
\begin{equation}
\sum_{n=\genfrac{}{}{0pt}{}{1}{N+1}}^{\genfrac{}{}{0pt}{}{N}{0}} 
|u_N(z,n)|^2 = \pm
\frac{\im(m_N(z,\alpha))}{\im(z)}, 
\quad N \bay{c} \vspace*{-2.5mm} > \\
\vspace*{1.5mm} < \eay 0,
\end{equation}
i.e., $\pm m_N(z,\alpha)$ are Herglotz functions 
for $N \bay{c} 
\vspace*{-2.5mm} > \\ \vspace*{1.5mm} < \eay 0$.

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
\begin{equation}
\{ m \in \C | [\theta_\alpha(z) - 
m \phi_\alpha(z)]_N =0 \}.
\end{equation}
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
\begin{equation}
\sum^{\pm\infty} |\phi_\alpha(z,n)|^2 = \infty, 
\quad\text{or}\quad
\sum^{\pm\infty} |\phi_\alpha(z,n)|^2 < \infty.
\end{equation}
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
\begin{equation}
\ti{m}_\pm(z,\alpha) = \lim_{N \to \pm\infty} m_N(z,\alpha).
\end{equation}

\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
\begin{equation}
u_\pm(z,n) = \theta_\alpha(z,n) - 
\ti{m}_\pm(z,\alpha) \phi_\alpha(z,n) 
\end{equation}
we get a function which is square summable near 
$\pm\infty$
\begin{equation}
\sum_{n= {\genfrac{}{}{0pt}{}{1}{-\infty}}^{
\genfrac{}{}{0pt}{}{\infty}{0}}} 
|u_\pm(z,n)|^2 = \pm
\frac{\im(\ti{m}_\pm(z,\alpha))}{\im(z)}.
\end{equation}
In addition,
\begin{equation}
W_{\pm\infty}(\phi_\alpha(\lam_1),u_\pm(z))=0,
\end{equation}
if $\tau$ is $l.c.$ at $\pm\infty$. We remark 
that (independently of the $l.c.$
and $l.p.$ case at $\pm\infty$)
\begin{equation}
\ti{m}_\pm(z) = \ti{m}_\pm(z,0) =
\frac{-u_\pm(z,1)}{a(0)u_\pm(z,0)}
\end{equation}
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
\begin{equation}
\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)}.
\end{equation}




\section{Weyl--Titchmarsh Theory on $\N$}


Let $H_+$ be a given self-adjoint operator 
associated with $\tau$ on $\N$
and a Dirichlet boundary condition at $n=0$. 
Abbreviate $\phi(z,n)=\phi_0(z,n)$
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
\begin{equation}
((H_+ - z)^{-1} f)(n) = \sum_{m \in \N}
G_+(z,m,n) f(m), \quad z \in \C \backslash \sig(H_+),
\end{equation}
where
\begin{equation}
G_+(z,m,n) = \frac{1}{W(\phi(z),
u_+(z))} \left\{ \bay{l@{,\quad}l} \phi(z,n)
u_+(z,m) & m \ge n \\ \phi(z,m) 
u_+(z,n) & m \le n \eay
\right. .
\end{equation}

Since $\phi(z,n)$ is a polynomial in $z$ we 
infer by induction 
\begin{equation} \label{relhl}
\phi(H_+,n) \delta_{1} = \delta_{n}, \qquad \delta_n(k) = \left\{
\bay{c@{\quad}l} 1, & k=n \\ 0, & k \ne n \eay \right. , 
\end{equation}
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
\begin{equation}
d\rho_+(.) = d\spr{\delta_1}{E_+(.) \delta_1}.
\end{equation}
Equation (\ref{relhl}) now shows that the 
polynomials $\phi(z,n), \: n \in \N$
are orthogonal with respect to this 
measure, i.e.,
\begin{equation}
\spr{\phi(j)}{\phi(k)} = 
\int\limits_{-\infty}^\infty \phi(\lam,j)
\phi(\lam,k) \, d\rho_+(\lam) = \delta_j(k) ,
\end{equation}
implying
\begin{equation} \label{polab}
a(n) = \spr{\phi(n+1)}{\lam \phi(n)}, 
\quad b(n)= -\spr{\phi(n)}{\lam \phi(n)},
\:\: n \in \N.
\end{equation}

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) \phi(\lam,n), 
\\ \label{unitary2} (U^{-1} F)(n)
 &=& \int_\R \phi(\lam,n) F(\lam) d\rho_+(\lam).
\eea
A simple calculation for $F(\lam) = (Uf)(\lam)$ 
shows that
\begin{equation}
\sum_{n=1}^\infty |f(n)|^2 = 
\int_\R |F(\lam)|^2 d\rho_+(\lam).
\end{equation}
Thus $U$ extends to a unitary transformation
\begin{equation}
\tilde{U}: \ell^2(\N) \to L^2(\R,d\rho_+)
\end{equation}
(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 $\lam$,
\begin{equation}
\tilde{U} H_+ \tilde{U}^{-1} = \tilde{H},
\end{equation}
where
\begin{equation} \hspace*{8mm}
\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_+)\}.
\end{equation}

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
\begin{equation}
d\rho_+ = d\rho_{+,p} + d\rho_{+,ac} + 
d\rho_{+,sc},
\end{equation}
then we have ($\rho_+(\lam) = 
\int_{(-\infty,\lam]} d\rho_+$)
\bea
\sig(H_+) &=& \{\lam \in \R |\text{$\lam$ is a 
growth point of $\rho_+$}\},\\
\sig_{p}(H_+) &=& \{\lam \in \R |\text{$\lam$ is 
a growth point of 
$\rho_{+,p}$}\},\\
\sig_{ac}(H_+) &=& \{\lam \in \R |\text{$\lam$ is 
a growth
point of $\rho_{+,ac}$}\},\\
\sig_{sc}(H_+) &=& \{\lam \in \R |\text{$\lam$ is 
a growth
point of $\rho_{+,sc}$}\}.
\eea

The Stieltjes transform of the spectral 
function $\rho_+$ is called the Weyl
$m$-function
\begin{equation}
m_+(z) = \int_\R  \frac{d\rho_+(\lam)}{\lam-z}, 
\qquad z
\in \C \bs \R.
\end{equation}
Conversely, the spectral function $\rho_+$ can be 
recovered from $m_+(z)$ by the Stieltjes inversion formula
\begin{equation}
\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.
\end{equation}
We have normalized $\rho_+$ such that it is right 
continuous and satisfies
$\lim\limits_{\lam \to -\infty} \rho_+(\lam) = 0$. 
One infers
\begin{equation} \label{mpl}
m_+(z) = G_+(z,1,1) = \frac{-u_+(1)}{a(0) u_+(0)} = \ti{m}_+(z), 
\end{equation}
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
\begin{equation}
\label{mmi} \hspace*{0.5cm}
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}.
\end{equation}
$m_\pm(z)$, like $\pm\ti{m}_\pm(z)$, are Herglotz functions.


\section{Weyl--Titchmarsh Theory on $\Z$}


In Appendix C we have dealt with the half-line $\N$. 
In this appendix 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
\begin{equation}
((H - z)^{-1} f)(n) = \sum_{m \in \Z} G(z,m,n) f(m),
\quad z \in \rho(H),
\end{equation}
where
\begin{equation}
G(z,m,n) = \frac{1}{W(u_-(z),u_+(z))} 
\left\{ \bay{l@{\quad}l} u_-(z,n) u_+(z,m),
& m \ge n \\ u_-(z,m) u_+(z,n), & m \le n \eay
\right. .
\end{equation}

Consider the vector-valued polynomials
\begin{equation}
\ul{\phi}(z,n) = 
\Big( \phi_1(z,n), \phi_2(z,n) \Big),
\end{equation}
where $\phi_{1,2}(z,n)$ are solutions of 
(\ref{gl}) satisfying the
initial conditions
\begin{equation}
\bay{c@{\qquad}c} \phi_1(z,0) = 0, & \phi_1(z,1) = 
1,\\ \phi_2(z,0)
= 1, & \phi_2(z,1) = 0. \eay
\end{equation}
The analog of (\ref{relhl}) reads
\begin{equation} \label{relwl}
\phi_1(H,n) \delta_1 + \phi_2(H,n) \delta_0 = 
\delta_n.
\end{equation}
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
\begin{equation}
d\rho_{j,k}(.) = d\spr{\delta_j}{E(.) \delta_k},
\end{equation}
and the (hermitian) matrix-valued measure
\begin{equation}
d\rho = \left( \bay{cc} d\rho_{1,1} 
& d\rho_{1,2} \\ d\rho_{2,1} & d\rho_{2,2}
\eay \right).
\end{equation}
By (\ref{relwl}) the vector-valued polynomials are 
orthogonal with respect to
$d\rho$
\bea \nn
\spr{\ul{\phi}(m)}{\ul{\phi}(n)} 
&=& \sum_{j,k=1}^2 \int_\R \phi_j(\lam,m)
\; \phi_k(\lam,n) d\rho_{j,k}(\lam)\\ 
&\equiv& \int_\R \ul{\phi}(\lam,m)
d\rho(\lam)
\; \ul{\phi}(\lam,n) = \delta_n(m).
\eea
The analogous formulas to (\ref{polab}) 
then read
\begin{equation}
a(n) = \spr{\ul{\phi}(n+1)}{\lam \ul{\phi}(n)}, 
\quad b(n) = \spr{\ul{\phi}(n)}
{\lam \ul{\phi}(n)}, \:\: n \in \Z.
\end{equation}

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{\phi}(\lam,n), \\
(U^{-1} \ul{F})(n) 
&=& \int_\R \ul{\phi}(\lam,n) d\rho(\lam) \ul{F}(\lam).
\eea
Again a simple calculation for 
$\ul{F}(\lam) = (Uf)(\lam)$ shows that
\begin{equation}
\sum_{n \in \Z} |f(n)|^2 = 
\int_\R \ol{\ul{F}(\lam)} d\rho(z) \ul{F}(\lam).
\end{equation}
Thus $U$ extends to a unitary transformation
\begin{equation}
\tilde{U}: \ell^2(\Z) \to L^2(\R,d\rho)
\end{equation}
which maps the operator $H$ to the 
multiplication operator by $\lam$,
\begin{equation}
\tilde{U} H \tilde{U}^{-1} = \tilde{H},
\end{equation}
where
\begin{equation} \hspace*{4mm}
\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)\},
\end{equation}
as in Appendix B.

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

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

The Weyl-matrix $M(z)$ is defined as
\begin{equation}
M(z) = \int\limits_{-\infty}^\infty  
\frac{d\rho(\lam)}{\lam-z}, \qquad z
\in \C \bs \R.
\end{equation}
Explicit evaluation yields
\bea \nn
&& M(z) = \left( \bay{cc} G(z,0,0) &
G(z,1,0) \\ 
G(z,0,1) & G(z,1,1) \eay \right)\\
&& = \frac{a(0)^{-2}}{\ti{m}_-(z) - \ti{m}_+(z)} \left( \bay{cc} 1 &
-a(0) \ti{m}_+(z) \\ -a(0) \ti{m}_+(z) & a(0)^2 \ti{m}_+(z) \ti{m}_-(z) \eay
\right).
\eea
Finally, assuming $\rho$ to be right 
continuous and normalizing
$\rho(-\infty)=0$ one obtains
\begin{equation}
\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.
\end{equation}


\section{A limit point criterion}

\begin{lem}
Let $w$, $a$, $b$ be real--valued sequences, 
$w>0$, $a<0$. Define
\begin{equation}
(\tau u)(n)=\frac{1}{w(n)} \Big(a(n)u(n+1) + 
a(n-1)u(n-1) - b(n) u(n) \Big)
\end{equation}
and suppose that $\tau$ is bounded from below. 
Then $\tau$ is in the $l.p.$
case at $\infty$ if 
$\sum^\infty |w(n) / a(n)|^{1/2} =\infty$.
\end{lem}

\begin{proof}
Since $\tau$ is bounded from below, there exists 
a principal
solution $u_+ >0$ of $\tau u=\lam u$ near 
$\infty$ for $\lam\in \R$
sufficiently small.  (See, e.g. \cite{crit}, 
\cite{pat} for the
definition and basic properties of (non)principal 
solutions associated
with $\tau$.) Then $\hat{u}_+$ defined by
\begin{equation}
\hat{u}_+(n) = 
u_+(n) \sum^n \frac{1}{a(m) u_+(m) u_+(m+1)}
\end{equation}
is nonprincipal near $\infty$, i.e.,
\begin{equation}
\sum^\infty \frac{1}{a(m) \hat{u}_+(m) 
\hat{u}_+(m+1)} <\infty.
\end{equation}
Now suppose that $\tau$ is l.c. at $\infty$ 
which implies
\begin{equation}
\sum^\infty w(m) 
\big|\hat{u}_+(m) \big|^2 <\infty.
\end{equation}
Then Cauchy's inequality yields the contradiction
\bea \nn
\infty = \sum^\infty |w(n) / a(n)|^{1/2} 
&=& \\ \nn
\sum^\infty |(w(m) \hat{u}_+(m) 
\hat{u}_+(m+1))/ (a(m) \hat{u}_+(m)
\hat{u}_+(m+1))|^{1/2} &\leq& \\ 
\left| \sum^\infty w(m)
|\hat{u}_+(m) |^2 \right|^{1/2} 
\left| \sum^\infty |a(m) \hat{u}_+(m)
\hat{u}_+(m+1) |^{-1}\right|^{1/2} 
&<& \infty.
\eea
\end{proof}

For further $l.p.$ criteria we refer the 
reader, e.g., to \cite{ak}, \cite{hl}.


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