%% @texfile{
%%     filename="TraceJac.tex",
%%     version="1.2",
%%     date="Jan-1998",
%%     cdate="19960712",
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%%     journal="Comm. Math. Phys. 196-1, 175-202 (1998)",
%%     doi="10.1007/s002200050419",
%%     copyright="Springer".   
%%     }


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\begin{document}

\title[Trace Formulas for Jacobi Operators]{Trace Formulas and Inverse Spectral
Theory for Jacobi Operators}


\author{Gerald Teschl}
\address{Institut f\"ur Reine und Angewandte Mathematik\\
RWTH Aachen\\ 52056 Aachen\\ Germany}
\curraddr{Institut f\"ur Mathematik\\
Strudlhofgasse 4\\ 1090 Wien\\ Austria}
\email{\href{mailto:Gerald.Teschl@univie.ac.at}{Gerald.Teschl@univie.ac.at}}
\urladdr{\href{http://www.mat.univie.ac.at/~gerald/}{http://www.mat.univie.ac.at/\string~gerald/}}

\thanks{{\it Comm. Math. Phys. {\bf 196-1}, 175--202 (1998)}}

\keywords{Trace formulas, Jacobi operators, inverse spectral theory, scattering theory}
\subjclass{Primary 39A10, 39A70; Secondary 34B20, 35Q58}

\begin{abstract}
Based on high energy expansions and Herglotz properties of Green and Weyl
$m$-functions we develop a self-contained theory of trace formulas for Jacobi
operators. In addition, we consider connections with inverse spectral theory, in
particular uniqueness results. As an application we work out a new
approach to the inverse spectral problem of  a class of reflectionless operators
producing explicit formulas for the coefficients in terms of minimal spectral data.
Finally, trace formulas are applied to scattering theory with periodic backgrounds.
\end{abstract}

\maketitle


\section{Introduction}


Trace formulas have a long history in the theory of one-dimensional
second order equations. One case of particular importance are
periodic potentials. Let
\begin{equation}
(H f)(n) = a(n) f(n+1) + a(n-1) f(n-1) +b(n) f(n), \quad n\in\Z
\end{equation}
be our Jacobi operator with $a(n+N)=a(n)$, $b(n+N)=b(n)$ for some $N\in\N$.
Then, using Floquet theory (cf., e.g., \cite{bght}, Appendix B, \cite{kr1}, \cite{vm})
one can show that the spectrum $\sig(H)$ of $H$ consists of $N$ bands
(some of which might collide)
\begin{equation}
\sig(H) = \bigcup_{j=1}^N [E_{2j-2},E_{2j-1}], \quad E_0 < E_1 \le E_2 \cdots<
E_{2N-1}.
\end{equation}
Next, we consider finite matrices associated with $H$ obtained by
restricting $H$ to finite intervals from $n_0$ to $n_0+N$ and imposing
boundary conditions at the endpoints. Denote the matrix obtained
with Dirichlet boundary conditions (i.e., $f(n_0)=0$, $f(n_0+N)=0$) by
$\ti{H}_{n_0}^\infty$ and the one obtained with periodic/anti periodic
boundary conditions (i.e., $f(n_0)=\pm f(n_0+N)$, $f(n_0+1)=\pm f(n_0+N+1)$)
by $\ti{H}_{n_0}^\pm$. The eigenvalues of $\ti{H}_{n_0}^+$, $\ti{H}_{n_0}^-$
are precisely the even, odd band edges $E_{2j-2}$, $E_{2j-1}$,
$1\le j \le N$, respectively. The eigenvalues of $\ti{H}_{n}^\infty$
are denoted by $\mu_j(n)$, $1\le j \le N-1$. Since
$\tr(\ti{H}_n^\pm)= \sum_{j=0}^{N-1} b(n+j)$ and 
$\tr(\ti{H}_n^\infty)= \sum_{j=1}^{N-1} b(n+j)$ we infer from
$b(n)= \tr(\ti{H}_n^\pm - \ti{H}_n^\infty) = \tr(\ti{H}_n^+ +\ti{H}_n^-) /2 -
\tr\ti{H}_n^\infty$ by elementary linear algebra 
\begin{equation} \label{tfper}
b(n) = \frac{1}{2} \sum_{j=0}^{2N-1} E_j - \sum_{j=1}^{N-1} \mu_j(n).
\end{equation}
Similarly, considering $\tr((\ti{H}_n^+)^\ell +(\ti{H}_n^-)^\ell) /2 -
\tr(\ti{H}_n^\infty)^\ell$, $\ell \in \N$ one can obtain higher order trace relations.

The corresponding formulas for $\ell=1$ (i.e., (\ref{tfper})) and $\ell=2$ were first
given in \cite{vm}. Formula (\ref{tfper}) plays a key role in the inverse spectral
theory of periodic operators and the reconstruction of $a,b$ from suitable
spectral data. Those ingredients form the basis for the
solution of the periodic initial value problem of the Toda equations
(cf., e.g., \cite{bght}, \cite{dt}, \cite{ta}). Moreover, relation (\ref{tfper}) was
extended to certain reflectionless operators in \cite{akr} and successfully used
in \cite{akr}, \cite{gkt} to solve inverse spectral problems for these operators.

To generalize trace formulas to arbitrary operators one invokes the measure
$d\rho_\delta$  of $H$ associated with the vector $\delta\in\lz$ 
(cf.\ Lemma~\ref{lemexhg}) by the spectral theorem. Choosing, e.g.,
$\delta=\delta_n$ (the standard basis of $\lz$) we immediately obtain
\begin{equation} \label{tfnonex}
\spr{\delta_n}{H^\ell \delta_n} = \int_\R \lam^\ell d\rho_{\delta_n}(\lam),
\end{equation}
connecting the matrix elements $\spr{\delta_n}{H^\ell \delta_n}$ with the
moments of the measure $d\rho_{\delta_n}$. In the special case where $H$ has
purely discrete spectrum, the integral can be evaluated,
\begin{equation}
\spr{\delta_n}{H^\ell \delta_n} = \sum_{\lam\in\sig(H)} \gam(\lam,n,n) \lam^\ell,
\end{equation}
where $-\gam(\lam,n,m)$ is the residue of $G(z,n,m)$ at
$z=\lam\in\sig(H)$, that is,
\begin{equation}
\gam(\lam,n,n) = \frac{u(\lam,n)u(\lam,m)}{\|u(\lam)\|^2},
\quad \lam\in\sig(H),
\end{equation}
where $u(\lam)$ is the eigenvector corresponding to $\lam\in\sig(H)$.
In particular, for $\ell=1$ this gives the interesting result that (for
$H$ with purely discrete spectrum) $b(n)$ is equal to the sum over all
eigenvalues of $H$ weighted by $\gam(\lam,n,n)$.

However, generalizations of (\ref{tfper}) cannot be obtained in this way.
This can be done by using the {\em exponential measure} $\xi d\lam$ 
(cf.\ Appendix~A) associated with $d\rho(\lam)$ as was discovered 
by F.\ Gesztesy and B.\ Simon in \cite{gsxi}. There they extended the analog of
(\ref{tfper}) for Schr\"odinger operators to a much larger class of potentials (in
essence, only semiboundedness of the potential is needed)  based on the theory of
the Krein spectral shift \cite{kr}. In a subsequent series of papers \cite{gsun},
\cite{gsro}, \cite{ghsas}, \cite{ghsztr}, and \cite{ghszmt} they, together with H.\
Holden and Z.\ Zhao, exploit the ideas of \cite{gsxi} and extend them in various
directions. In \cite{gsxi} they also give a generalization of (\ref{tfper}) to 
arbitrary bounded Jacobi operators. However, a comprehensive treatment  of
trace formulas for Jacobi operators is still missing. Since it is desirable, for
further work in inverse spectral theory, to have these powerful tools at
one's disposal, one goal of the present paper is to fill this gap.

Furthermore, we want to  point out an annoying mismatch in formula
(\ref{tfper}). In order to express $b(n)$ for all $n\in\Z$ one needs
$\{ E_j \}_{0 \le j \le 2N-1}$, $\{ \mu_j(n) \}_{1 \le j \le N}$ for {\em all}
$n\in\Z$. On the other hand, it is well-known that the spectral data
$\{ E_j \}_{0 \le j \le 2N-1}$, $\{ \mu_j(n_0) \}_{1 \le j \le N}$ plus
some additional signs $\{ \sig_j(n_0) \}_{1 \le j \le N}$ for {\em one}
fixed $n_0\in\Z$  already determine $a(n)^2,b(n)$ for all $n\in\Z$.
Hence it must be, in principle, possible to express $a(n)^2,b(n)$ in terms of
these spectral data for one $n_0\in\Z$. This naturally raises the question whether
one might be able to find explicit expressions of $a(n)^2,b(n)$ in terms of suitable
minimal spectral data for certain classes of operators. To the best of our
knowledge a problem of this kind has not been solved yet. Combining the approach
of (\ref{tfnonex}), the theory of \cite{gsxi}, Weyl--Titchmarsh
theory, and the moment problem we will add a new wrinkle to the theory of
trace formulas and give a solution to this problem for a certain class of
bounded reflectionless Jacobi operators in Section~6.

To give the reader an overview of the results
established, we briefly summarize the content of the remaining sections.

Section 2 introduces all the necessary notation and is mainly added to make
the paper self-contained and easier to read.

Section 3 contains a comprehensive treatment of asymptotic expansions for
Weyl~$m$ and Green functions. We establish that expansions for these
objects always exist up to arbitrary order. In addition, recursion relations
for the expansion coefficients are derived.

Section 4 contains an alternate (recursive) approach to inverse spectral theory
which gives simple proofs for standard uniqueness theorems. Moreover,  new
uniqueness results are established as well.

In Section 5 we derive an infinite series of trace formulas for Jacobi
operators in the spirit of \cite{gsxi}, \cite{ghsztr}. The basic
ingredients are the asymptotic expansions of Section~3 and Herglotz
properties of these objects. In particular, we extend (\ref{tfper}) to
\begin{enumerate} \renewcommand{\theenumi}{\roman{enumi}}
\item arbitrary order $\ell \in\N$,
\item arbitrary Jacobi operators, and
\item general boundary conditions.
\end{enumerate}

Section 6 applies the results of Section 5 to the theory of reflectionless
Jacobi operators, producing formulas of type (\ref{tfper}) plus an
explicit expression of the coefficients $a^2,b$ in terms of minimal spectral
data.

Section 7 considers scattering theory with periodic backgrounds. Basic objects
like transmission and reflection coefficients are introduced. In addition,
the analog of a trace formula for Schr\"odinger operators involving the
reflection coefficient is obtained.

Finally, an appendix collects some properties of Herglotz functions needed in
the main body of the paper.



\section{Jacobi operators, resolvents, Green's functions and all that}
\setcounter{equation}{0}
\setcounter{thm}{0}



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 over $I$.
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}

We will be concerned with operators on $\lz$ associated with the
difference expression
\begin{equation}
(\tau f)(n) = a(n) f(n+1) + a(n-1) f(n-1) +b(n) f(n),
\end{equation}
where $a,b \in \ell(\Z)$ satisfy
\bh \label{hab}
Suppose
\begin{equation}
a(n) \in \R \bs \{ 0\}, \quad b(n) \in \R, \quad n \in \Z.
\end{equation}
\eh
If $\tau$ is limit point ($l.p.$) at both $\pm\infty$ (cf., e.g., \cite{at},
\cite{be}), then $\tau$ gives rise to a unique self-adjoint operator $H$ when
defined maximally. Otherwise, we need to fix a boundary condition at each
endpoint where $\tau$ is limit circle ($l.c.$) (cf., e.g., \cite{at}, \cite{be}).
Throughout this paper we denote by $u_\pm(z,.)$, $z \in \C$, nontrivial solutions
of $\tau u = z u$ which satisfy the boundary condition at $\pm\infty$ (if any) with
$u_\pm(z,.) \in \ell^2_\pm(\Z)$, respectively. Here $\ell^2_\pm(\Z)$ denotes the
sequences in $\ell(\Z)$ being $\ell^2$ near $\pm\infty$. The solution $u_\pm(z,.)$
might not exist for $z \in \R$ (cf.\ \cite{gtosc}, Lemma~A.1), but if it exists it is
unique up to a constant multiple.

Picking a fixed $z_0 \in \C \bs \R$ we can characterize $H$ by
\begin{equation}
\ba{lccl} H :& \db(H) & \to & \lz \\ & f &\mapsto& \tau f \ea ,
\end{equation}
where the domain of $H$ is explicitly given by
\begin{equation}
\db(H) = \{ f \in \lz | \ba[t]{l} \tau f \in \lz, \: \lim_{n \to +\infty}
W_n(u_+(z_0),f) = 0, \\ \lim_{n \to -\infty}
W_n(u_-(z_0),f) = 0 \} \ea
\end{equation}
and
\begin{equation}
W_n(f,g) = a(n) \Big( f(n)g(n+1) - f(n+1)g(n) \Big)
\end{equation}
denotes the (modified) Wronskian. The boundary condition at $\pm\infty$ imposes
no additional restriction on $f$ if $\tau$ is $l.p.$ at $\pm\infty$ and can hence be
omitted in this case.

Next, consider the sequence
\begin{equation}
\delta^\beta_{n_0} = \cos(\alpha) \delta_{n_0} + \sin(\alpha)
\delta_{n_0+1}, \quad \beta = \cot(\alpha),\alpha\in[0,\pi),
\end{equation}
where $\delta_{n_0}(n)$ is $1$ for $n=n_0$ and $0$ otherwise. Restrict $H$ to
the orthogonal complement of $\delta^\beta_{n_0}$ in $\lz$ and denote
this restriction by $H^\beta_{n_0}$, that is,
\begin{equation}
H^\beta_{n_0} f = \tau f, \quad f\in \db(H^\beta_{n_0}) =
\{ f\in \db(H) | \spr{\delta^\beta_{n_0}}{f}=0\}.
\end{equation}
Clearly $H^\beta_{n_0}$ is self-adjoint on the subspace $\{ f\in \lz |
\spr{\delta^\beta_{n_0}}{f}=0\}$ but not on $\lz$ since $\db(H^\beta_{n_0})$
is not dense.

Now we turn to resolvents and introduce the Green's function
\bea\nn
G(z,m,n) &=& \spr{\delta_m}{(H-z)^{-1}\delta_n}\\
&=& \frac{1}{W(u_-(z),u_+(z))} \left\{ \ba{l@{\quad \text{ for }}l} u_+(z,n)
u_-(z,m) & m \le n \\ u_+(z,m) u_-(z,n) & n \le m \ea \right. ,
\eea
where $z\in\C\bs\sig(H)$ and $\sig(H)$ denotes the spectrum of $H$.
For later use we also introduce the convenient abbreviations
\bea \label{gzn}
g(z,n) &=& G(z,n,n)= \frac{u_+(z,n) u_-(z,n)}{W(u_-(z),u_+(z))},\\ \nn
h(z,n) &=& 2a(n)G(z,n,n+1) -1 \\ \label{hzn}
&=& \frac{a(n)(u_+(z,n) u_-(z,n+1) + u_+(z,n) u_-(z,n+1))}{W(u_-(z),u_+(z))}.
\eea

Similarly, the corresponding object for $H^\beta_{n_0}$ 
(viewed as a self-adjoint operator on $\{ f\in \lz |
\spr{\delta^\beta_{n_0}}{f}=0\}$) reads
\bea\nn
&& G^\beta_{n_0}(z,m,n) = \spr{\delta_m}{(H^\beta_{n_0}-z)^{-1}\delta_n}
= G(z,m,n) + \gam^\beta(z,n_0)^{-1}\times\\
&& \qquad \Big( G(z,m,n_0+1) + \beta G(z,m,n_0)\Big) \Big( G(z,n_0+1,n) +
\beta G(z,n,n_0)\Big),
\eea
where
\bea \nn
\gam^\beta(z,n) &=& \frac{\Big( u_+(z,n+1) + \beta u_+(z,n)\Big)
\Big( u_-(z,n+1) + \beta u_-(z,n)\Big)}{W(u_-(z),u_+(z))}\\ \label{defgambet}
&=& g(z,n+1) + \frac{\beta}{a(n)} h(z,n) + \beta^2 g(z,n).
\eea
The quantities $g(z,n)$ and $\gam^\beta(z,n)$ are most important for
our purpose and satisfy the following recurrence equations which can be
verified by tedious but straightforward calculations. We use the
shortcuts $(f^-)(n)=f(n-1)$, $(f^+)(n)=f(n+1)$,  $(f^{++})(n)=f(n+2)$,
etc..

\bl
Let $u,v$ be two solutions of $\tau u = z u$. Then $g(n) = u(n)v(n)$ satisfies 
\begin{equation} \label{ldedgf}
\frac{(a^+)^2 g^{++} - a^2 g}{z-b^+} + \frac{a^2 g^+ - (a^-)^2 g^-}{z-b} =
(z-b^+) g^+ - (z-b) g,
\end{equation}
and
\begin{equation} \label{nldedgf}
\Big( a^2 g^+ - (a^-)^2  g^- + (z-b)^2 g \Big)^2
= (z-b)^2 \Big( W(u,v)^2 + 4 a^2 g g^+ \Big).
\end{equation}
Moreover, set $\gam^\beta(n) = (u(n+1) + \beta u(n))(v(n+1) + \beta v(n))$,
then we have
\bea \nn
\lefteqn{\Big( (a^+A^-)^2 (\gam^\beta)^+ - (a^-A)^2  (\gam^\beta)^- +
B^2 \gam^\beta \Big)^2}\\ \label{nldedgam}
&=& (A^-B)^2 \Big( (\frac{A}{a} W(u,v))^2 + 4 (a^+)^2
\gam^\beta (\gam^\beta)^+ \Big),
\eea
with
\bea
A &=& a +\beta (z-b^+) + \beta^2 a^+,\\ \nn
B &=& a^- (z-b^+) + \beta( (z-b^+)(z-b) + a^+a^- - a^2)\\
&& {} + \beta^2 a^+(z-b).
\eea
\el

\br
Equations (\ref{ldedgf}) and (\ref{nldedgf}) are the analogs of
well-known differential equations for the diagonal Green function
of one-dimensional Schr\"odinger operators (cf., e.g., \cite{gd}, \cite{grt},
equations (5.19) and (5.20)). Equation (\ref{nldedgam}) is the analog
of equation (5.18) in \cite{grt}.
\er

Finally, we turn to half line restrictions $H_{\pm,n_0}$ of $H$ defined by
\begin{equation}
\ba{lccl} H_{\pm,n_0} :& \db(H_{\pm,n_0}) & \to & \ell^2(n_0,\pm\infty) \\
& f &\mapsto& \tau f \ea ,
\end{equation}
and
\begin{equation}
\db(H_{\pm,n_0}) = \{ f \in \ell^2(n_0,\pm\infty) | \tau f
\in \ell^2(n_0,\pm\infty), \: \lim_{n \to \pm\infty}
W_n(u_\pm(z_0),f) = 0 \},
\end{equation}
where we set $f(n_0)=0$ in the definition of $(\tau f)(n_0\pm1)$. The
corresponding Green functions read
\begin{equation}
G_{\pm,n_0}(z,m,n) = \frac{\pm 1}{W(s(z),u_\pm(z))} \left\{
\ba{l@{\quad \text{ for }}l} s(z,n,n_0) u_\pm(z,m) & m \geqleq n \\
s(z,m,n_0) u_\pm(z,n) & n \geqleq m \ea \right. ,
\end{equation}
where $s(z,.,n_0)$ is the solution of $\tau u = z u$ satisfying the
Dirichlet boundary condition $s(z,n_0,n_0)=0$. The analogous quantities of
$g(z,n)$ are the Weyl $m$-functions
\bea \nn
m_\pm(z,n) &=& \spr{\delta_{n\pm 1}}{(H_{\pm,n} - z)^{-1}
\delta_{n\pm 1}} = G_{\pm,n}(z,n\pm 1,n\pm1)\\
&=& -\frac{u_\pm(z,n\pm 1)}{a(n - \genfrac{}{}{0pt}{}{0}{1})u_\pm(z,n)},
\eea
which satisfy
\begin{equation} \label{riccwm}
a(n - {\textstyle \genfrac{}{}{0pt}{}{0}{1}})^2 m_\pm(z,n) +
\frac{1}{m_\pm(z,n\mp1)} = b(n) -z.
\end{equation}

\br \label{rembc}
We can also consider half line operators $H_{\pm,n_0}^\beta$ on
$\ell^2(n_0,\pm\infty)$ associated with the general boundary condition
\begin{equation} \label{boundcon}
f(n_0+1) + \beta f(n_0)=0, \qquad \beta \in \R \cup \{ \infty \}
\end{equation}
 at $n_0$ rather than only the Dirichlet boundary condition $f(n_0)=0$. 
We set
\begin{equation} \label{hbetann}
H^0_{+,n_0} = H_{+,n_0+1}, \quad
H^\beta_{+,n_0} = H_{+,n_0} - a(n_0) \beta^{-1}
\spr{\delta_{n_0+1}}{.}\delta_{n_0+1}, \quad \beta \ne 0,
\end{equation}
and
\begin{equation}
H^\infty_{-,n_0} = H_{-,n_0}, \quad
H^\beta_{-,n_0} = H_{-,n_0+1} - a(n_0) \beta
\spr{\delta_{n_0}}{.}\delta_{n_0}, \quad \beta \ne \infty,
\end{equation}
implying $H^\beta_{n_0} \cong H^\beta_{-,n_0} \oplus H^\beta_{+,n_0}$.
\er


\section{Asymptotic expansions}
\setcounter{equation}{0}
\setcounter{thm}{0}



In the sequel, asymptotic expansions for $g(z,n)=G(z,n,n)$ and $\gam^\beta(z,n)$ will
turn out to be very useful. Both quantities are Herglotz functions as can be seen
from
\bea
g(z,n) &=& \spr{\delta_n}{(H-z)^{-1} \delta_n},\\
\gam^\beta(z,n) &=& (1+\beta^2)\spr{\delta_n^\beta}{(H-z)^{-1} \delta_n^\beta}
-\frac{\beta}{a(n)}
\eea
(we note that, by (\ref{defgambet}), $h(z,n)$ is the difference of two Herglotz
functions) and the following lemma which is immediate from the spectral theorem.

\bl \label{lemexhg}
Suppose $\delta \in \lz$ with $\|\delta\| =1$. Then
\begin{equation}
g(z) = \spr{\delta}{(H-z)^{-1} \delta}
\end{equation}
is Herglotz, that is,
\begin{equation}
g(z) = \int_\R \frac{1}{\lam-z} d\rho_\delta(\lam),
\end{equation}
where $d\rho_\delta(\lam)=d\spr{\delta}{P_{(-\infty,\lam]}(H) \delta}$
is the spectral measure of $H$ associated to the sequence $\delta$.
Moreover,
\begin{equation}
\im(g(z)) = \im(z)\| (H - z)^{-1} \delta\|^2
\end{equation}
and
\begin{equation}
g(\ol{z}) = \ol{g(z)}, \qquad |g(z)| 
\le \| (H - z)^{-1} \| \le \frac{1}{|\im(z)|}.
\end{equation}
\el

Next, we turn to asymptotic expansions for $g(z,n)$, $h(z,n)$, and
$\gam^\beta(z,n)$.

\bth \label{lemaghg}
The quantities $g(z,n)$, $h(z,n)$, and $\gam^\beta(z,n)$ have the
following asymptotic expansions for arbitrary $\eps>0$
\bea \label{asyptgf}
g(z,n) &\asympt& - \sum_{j=0}^\infty \frac{g_j(n)}{z^{j+1}}, \quad g_0=1,\\
h(z,n) &\asympt& -1 - \sum_{j=0}^\infty \frac{h_j(n)}{z^{j+1}}, \quad h_0=0,\\
\gam^\beta(z,n) &\asympt& -\frac{\beta}{a(n)} - \sum_{j=0}^\infty
\frac{\gam^\beta_j(n)}{z^{j+1}}, \quad \gam^\beta_0=1+\beta^2.
\eea
Moreover, the coefficients are given by
\bea \label{fjspr}
g_j(n) &=& \spr{\delta_n}{H^j \delta_n}, \quad j \in \N_0,\\ \label{gjspr}
h_j(n) &=& 2a(n) \spr{\delta_{n+1}}{H^j \delta_n}, \quad j \in \N_0,\\ \nn
\gam^\beta_j(n) &=& \spr{(\delta_{n+1} +\beta
\delta_n)}{H^j(\delta_{n+1} +\beta \delta_n)}\\ \label{gamjspr}
&=& g_j(n+1) + \frac{\beta}{a(n)} h_j(n)+ \beta^2 g_j(n), \quad j \in \N_0.
\eea
\eth

\bpf
We only carry out the proof for $g(z,n)$ since the remaining expansions are
similar. Rewriting $g(z,n)$ as
\bea \nn
g(z,n) &=& \spr{\delta_n}{(H - z)^{-1} \delta_n}\\
&=& -\sum_{j=0}^{N-1} \frac{\spr{\delta_n}{H^j \delta_n}}{z^{j+1}} +
\frac{1}{z^N}\spr{\delta_n}{H^N(H - z)^{-1} \delta_n}, \:\: N\in\N
\eea
shows that it suffices to vindicate that the last term is $O(z^{-N})$. This
follows from
\begin{equation}
|\spr{\delta_n}{H^N(H - z)^{-1} \delta_n}| \le \frac{\|H^N \delta_n \|}{|\im(z)|}
\le \frac{\|H^N \delta_n \|}{\eps}.
\end{equation}
\epf

\br \label{remsigna}
(i). If $H$ is bounded, the above expansions are in fact Laurent series
converging for $|z| >\| H \|$.\\
(ii). Pick $\eps(n) \in\{-1,+1\}$ and introduce $a_\eps(n) = \eps(n)a(n)$
and $b_\eps(n) = b(n)$. Then the operator $H_\eps$ associated with
$a_\eps,b_\eps$ is unitarily equivalent to $H$. Indeed, take
the unitary operator $U_\eps = \{ \ti{\eps}(n) \,
\delta_{m,n} \}_{m,n\in\Z}$, where $\eps(n+1)\ti{\eps}(n)=\eps(n)$, then
$H_\eps = U_\eps H U_\eps^{-1}$. In particular, this shows that
$g(n),h(n)$ do not depend on the sign of $a$, that is, they only depend
on $a^2$.
\er

The following lemma (\cite{bght}, Lemma~2.1) shows how to compute
$g_j,h_j$ recursively.

\bl \label{recexpgf}
The coefficients $g_j(n)$ and $h_j(n)$ for $j \in \N_0$
satisfy the following recursion relation
\bea \label{recfj}
g_{j+1} &=& \frac{h_j + h_j^-}{2} + b g_j,\\ \label{recgj}
h_{j+1} - h_{j+1}^- &=& 2 \Big( a^2 g_j^+ - (a^-)^2 g_j^- \Big)
+ b \Big(h_j - h_j^- \Big).
\eea
\el

\bpf
The first equation follows from
\begin{equation}
g_{j+1}(n) = \spr{H \delta_n}{ H^j \delta_n} =
\frac{h_j(n)+h_j(n-1)}{2} + b(n) g_j(n).
\end{equation}
Similarly,
\bea \nn
h_{j+1}(n) &=& b(n) h_j(n) + 2a(n)^2 g_j(n+1) + 2 a(n-1) a(n)\spr{\delta_{n+1}}{H^j
\delta_{n-1}}\\
&=& b(n+1) h_j(n) +2a^2 g_j(n) +2 a(n) a(n+1)\spr{\delta_{n+2}}{H^j \delta_{n}}.
\eea
Eliminating $\spr{\delta_{n+1}}{H^j \delta_{n-1}}$ completes the proof.
\epf

This system does not determine $g_j(n)$, $h_j(n)$ uniquely since it requires
solving a first-order recurrence relation at each step, producing an unknown
summation constant each time. To determine these constants we
assign the weight one to $a(n)$ and $b(n)$, $n\in\Z$. Then $g_{j+1}(n)$ and
$h_j(n)$ have weight $j+1$, fixing the summation constants. 

To avoid this drawback we advocate a different approach using (\ref{nldedgf}).
First observe that $h_j(n)$ can be determined if $g_j(n)$ is known using
\begin{equation} \label{altrecgj}
h_{j+1} = b h_j + g_{j+2} - 2b g_{j+1} + a^2 g_j^+ - (a^-)^2 g_j^- + b^2 g_j,
\quad j\in\N_0,
\end{equation}
which follows after inserting (\ref{recfj}) into (\ref{recgj}).  In addition,
inserting the expansion (\ref{asyptgf}) for $g(z,n)$ into (\ref{nldedgf}) and
comparing coefficients of $z^j$ one infers
\bea \nn
&& g_0 = 1,\quad g_1 = b,\quad g_2 = a^2 + (a^-)^2 + b^2,\\
&& g_3 = a^2(b^+ + 2b) + (a^-)^2 (2b + b^-) + b^3,
\eea
and
\bea \nn
g_{j+1} &=& 2b g_j - a^2 g_{j-1}^+ + (a^-)^2 g_{j-1}^- - b^2
g_{j-1} - \frac{1}{2} \sum_{\ell=0}^{j-1} k_{j-\ell-1} k_\ell \\
\label{recnlf} &&{} +2a^2 \Big( \sum_{\ell=0}^{j-1} g_{j-\ell-1} g_\ell^+ 
- 2b \sum_{\ell=0}^{j-2} g_{j-\ell-2} g_\ell^+ + b^2
\sum_{\ell=0}^{j-3} g_{j-\ell-3} g_\ell^+ \Big),\quad
\eea
for $j \ge 3$, where $k_0(n) = -b(n)$ and
\begin{equation}
k_j = a^2 g_{j-1}^+ - (a^-)^2 g_{j-1}^- + b^2 g_{j-1} - 2b g_j +g_{j+1}, \quad j
\in\N.
\end{equation}

Analogously, one can get a recurrence relation for $\gam^\beta_j$
using (\ref{nldedgam}). Since this approach gets too cumbersome we
omit further details at this point but note that $\gam^\beta_j$
can be computed from (\ref{gamjspr}). Invoking (\ref{altrecgj})
one explicitly obtains
\begin{equation}
h_0 = 0,\quad h_1 = 2a^2,\quad h_2 = 2a^2(b^+ + b)
\end{equation}
and hence
\bea \nn
\gam^\beta_0 &=& 1 + \beta^2,\quad \gam^\beta_1 = b^+ +2 a
\beta + b \beta^2,\\
\gam^\beta_2 &=& (a^+)^2 + a^2 + (b^+)^2 + 2a (b^+ + b)\beta + (a^2 + (a^-)^2 + b^2)
\beta^2.
\eea

\br
Instead of (\ref{altrecgj}) and (\ref{recnlf}) one can also use (\ref{recfj})
and
\begin{equation}
h_{j+1} = 2a^2 \sum_{\ell=0}^j g_{j-\ell} g_{\ell}^+ - \frac{1}{2} 
\sum_{\ell=0}^j h_{j-\ell} h_{\ell}, \quad j\in\N,
\end{equation}
together with (\ref{recfj}) to determine $g_j,h_j$. The
above equation follows as before using (\ref{detgf}) below.
\er

Next we turn to Weyl $m$-functions.  As before we obtain

\bl
The quantities $m_\pm(z,n)$ have the asymptotic expansions
\begin{equation} \label{asephi}
m_\pm(z,n) \asympt - \sum_{j=0}^\infty
\frac{m_{\pm,j}(n)}{z^{j+1}}, \quad m_{\pm,0}(n)=1.
\end{equation}
The coefficients $m_{\pm,j}(n)$ are given by 
\begin{equation}
m_{\pm,j}(n) = \spr{\delta_{n\pm 1}}{(H_{\pm,n})^j\delta_{n\pm 1}}, \quad
j\in\N
\end{equation}
and satisfy
\bea \nn 
&& m_{\pm,0} =1, \quad m_{\pm,1} = b^\pm,\\
&& m_{\pm,j+1} = b^\pm m_{\pm,j} + \genfrac{}{}{0pt}{}{(a^+)^2}{(a^{--})^2}
\sum_{\ell=0}^{j-1} m_{\pm,j -\ell -1} m_{\pm,\ell}^+, \quad j \in\N.
\eea
\el

\br
As in Remark~\ref{remsigna} we have that (\ref{asephi}) converges for
$|z| > \| H_{\pm,n}\|$ if $H_{\pm,n}$ is bounded and $m_\pm(z,n)$ depend
only on $a^2$.
\er



\section{Inverse spectral theory}
\setcounter{equation}{0}
\setcounter{thm}{0}



In this section we present a simple recursive method of reconstructing the
sequences $a^2, b$ when the Weyl matrix
\bea \nn
M(z,n) &=& \left( \ba{cc} G(z,n,n) & G(z,n+1,n) \\ G(z,n,n+1) & G(z,n+1,n+1) \ea
\right) - \frac{1}{2a(n)} \left( \ba{cc} 0 &1 \\ 1 & 0\ea \right)\\ 
&=& \left( \ba{cc} g(z,n) & \frac{h(z,n)}{2a(n)} \\ \frac{h(z,n)}{2a(n)} & g(z,n+1)
\ea \right), \qquad z \in \C\bs\sig(H) 
\eea
is known for one fixed $n\in\Z$. As a consequence, we are led to several
uniqueness results.

{}From the previous section we know
\bea
g(z,n) &=& - \frac{1}{z} - \frac{b(n)}{z^2} + O(\frac{1}{z^3}),\\
h(z,n) &=& -1 - \frac{2 a(n)^2}{z^2} + O(\frac{1}{z^3}).
\eea
Here and in the remainder of this paper all $O(\frac{1}{z^\ell})$ terms apply for
$|z|\to\infty$, $|\im(z)| \ge\eps>0$. Hence
\bea
b(n) &=& -\lim_{z \to \I\infty} z(1 + z g(z,n)), \\
a(n)^2 &=& -\frac{1}{2}\lim_{z \to \I\infty} z^2 (1+h(z,n)).
\eea
Moreover, we have the useful identities (use (\ref{gzn}) and (\ref{hzn}))
\begin{equation} \label{detgf}
4a(n)^2 g(z,n) g(z,n+1) = h(z,n)^2 -1
\end{equation}
and
\begin{equation}
h(z,n+1) +h(z,n) = 2(z-b(n+1)) g(z,n+1), 
\end{equation}
which show that $g(z,n)$ and $h(z,n)$ together with $a(n)^2$ and
$b(n)$ can be determined recursively if, say, $g(z,n_0)$ and $h(z,n_0)$ are
given.

In addition, we infer that $a(n)^2$, $g(z,n)$, $g(z,n+1)$ determine
$h(z,n)$ up to one sign,
\begin{equation}
h(z,n) = \Big(1 + 4a(n)^2 g(z,n) g(z,n+1)\Big)^{1/2}
\end{equation}
since $h(z,n)$ is holomorphic with respect to $z\in\C \bs \sig(H)$ and
$\ol{h(z,n)}=h(\ol{z},n)$. However, this sign can be determined from the
asymptotic behavior $h(z,n)= -1 + O(z^{-2})$.

Hence we have reproved the well-known result that $M(z,n_0)$
determines the sequences $a^2,b$.  In fact, we have proved the slightly stronger
result:

\bth \label{thmunibas}
One of the following set of data\\
(i) $g(.,n_0)$ and $h(.,n_0)$\\
(ii) $g(.,n_0+1)$ and $h(.,n_0)$\\
(iii) $g(.,n_0)$, $g(.,n_0+1)$, and $a(n_0)^2$\\
for one fixed $n_0\in\Z$ uniquely determines the sequences $a^2$ and $b$.
\eth

\br
(i) We want to emphasize that the diagonal elements $g(z,n_0)$ and $g(z,n_0+1)$
alone  plus $a(n_0)^2$ are sufficient to reconstruct $a(n)^2, b(n)$. This is in
contradistinction to the case of one-dimensional Schr\"odinger operators, where
the diagonal elements of the Weyl matrix determine the potential only up to
reflection. It is not clear to me whether this different behavior of Jacobi
operators has been previously noted in the literature.

The reader might wonder how the Weyl matrix of the operator $H_R$ associated
with the (at $n_0$) reflected coefficients $a_R, b_R$ (i.e., $a_R(n_0-k-1) =
a(n_0+k)$, $b_R(n_0-k) = b(n_0+k)$, $k\in \Z$) look like. Since reflection at
$n_0$ exchanges $m_\pm(z,n_0)$ (i.e., $m_{R,\pm}(z,n_0) = m_\mp(z,n_0)$) we
infer
\bea
g_R(z,n_0) &=& g(z,n_0),\\
h_R(z,n_0) &=& - h(z,n_0) + 2(z-b(n_0)) g(z,n_0),\\ \nn
g_R(z,n_0+1) &=& \frac{a(n_0)^2}{a(n_0-1)^2} g(z,n_0+1) +
\frac{z-b(n_0)}{a(n_0-1)^2} \Big( h(z,n_0)\\
&& {} +(z-b(n_0)) g(z,n_0)\Big),
\eea
in obvious notation.\\
(ii) Remark~\ref{remsigna}(ii) shows that the sign of $a(n)$ cannot be determined
from either $g(z,n_0)$, $h(z,n_0)$, or $g(z,n_0+1)$.\\
(iii). Clearly, if $H$ is $l.c.$ at $\pm\infty$ the corresponding boundary
condition is determined by $M(z,n)$ as well.\\
(iv). Equation (\ref{detgf}) is equivalent to $\det M(z,n)= -1/(2a(n))^2$.
The analogous equation for the Schr\"odinger case was first used by
Rofe-Beketov in connection with inverse problems (see \cite{lv}, Section~7.3).
\er

The off diagonal Green function can be recovered as follows
\begin{equation}
G(z,n,n+k) = g(z,n) \prod_{j=n}^{n+k-1} \frac{1+h(z,j)}{2a(j)g(z,j)},
\quad k>0,
\end{equation}
and we remark
\bea \nn
&& a(n)^2 g(z,n+1) - a(n-1)^2 g(z,n-1) +(z-b(n))^2 g(z,n)\\
&& \qquad = (z-b(n)) h(z,n). 
\eea

A similar procedure works for $H_+$. The asymptotic expansion
\begin{equation}
m_+(z,n) = -\frac{1}{z} -\frac{b(n+1)}{z^2} - \frac{a(n+1)^2 +
b(n+1)^2}{z^3} + O(z^{-4})
\end{equation}
shows that $a(n+1)^2,b(n+1)$ can be recovered from $m_+(z,n)$.
In addition, (\ref{riccwm}) shows that $m_+(z,n_0)$ determines
$a(n)^2,b(n),m_+(z,n)$, $n>n_0$. Similarly, (by reflection) $m_-(z,n_0)$
determines $a(n-1)^2,b(n),m_-(z,n-1)$,
$n<n_0$. Hence both $m_\pm(z,n_0)$ determine $a(n)^2,b(n)$ except
for $a(n_0-1)^2,a(n_0)^2,b(n_0)$. However, introducing $\ti{m}_\pm(z,n)=
\mp u_\pm(z,n+1) / (a(n)u_\pm(z,n))$ and considering
\begin{equation} \label{conmtim}
\ti{m}_+(z,n) = m_+(z,n), \quad \ti{m}_-(z,n) = \frac{z - b(n) +
a(n-1)^{-2} m_-(z,n)}{a(n)^2}
\end{equation}
we see that $\ti{m}_-(z,n_0)$ determines $a(n_0-1)^2, a(n_0)^2, b(n_0)$ and
$m_-(z,n_0)$. Summarizing:

\bth \label{thmugamb}
The quantities $\ti{m}_\pm(z,n_0)$ uniquely determine $a(n)^2, b(n)$ for all
$n\in\Z$. Moreover, we have
\bea \nn
g(z,n) = \frac{-a(n)^{-2}}{\ti{m}_+(z,n) + \ti{m}_-(z,n)}, \quad
g(z,n+1) = \frac{\ti{m}_+(z,n) \ti{m}_-(z,n)}{\ti{m}_+(z,n) +
\ti{m}_-(z,n)},\\
h(z,n) = \frac{\ti{m}_+(z,n) - \ti{m}_-(z,n)}{\ti{m}_+(z,n) +
\ti{m}_-(z,n)},
\eea
and conversely
\begin{equation} \label{contimgh}
\ti{m}_\pm(z,n) = \frac{1\pm h(z,n)}{2a(n)^2g(z,n)} =
-\frac{2 g(z,n+1)}{1\mp h(z,n)}.
\end{equation}
\eth

Next we recall the function $\gam^\beta(z,n)$ introduced in (\ref{defgambet})
with asymptotic expansion
\begin{equation} \label{asymgabe}
\gam^\beta(z,n) = -\frac{\beta}{a(n)} - \frac{1+\beta^2}{z} 
- \frac{b(n+1) + 2\beta a(n) + \beta^2b(n)}{z^2}+ O(\frac{1}{z^3}).
\end{equation}
Our goal is to prove

\bth \label{thmunigabot}
Let $\beta_{1,2} \in \R\cup\{\infty\}$ with $\beta_1 \ne \beta_2$
be given. Then $\gam^{\beta_j}(.,n_0)$, $j=1,2$ for one fixed $n_0\in\Z$
uniquely determines $a(n)^2, b(n)$ for all $n\in\Z$ (set
$\gam^\infty(z,n)=g(z,n)$) unless $(\beta_1,\beta_2) = (0,\infty), (\infty,0)$.
In the latter case $a(n_0)^2$ is needed in addition.
More explicitly, we have
\bea \label{solsyg}
g(z,n) &=& \frac{\gam^{\beta_1}(z,n) +
\gam^{\beta_2}(z,n) + 2 R(z)}{(\beta_2-\beta_1)^2},\\ \label{solsygp}
g(z,n+1) &=& \frac{\beta_2^2 \gam^{\beta_1}(z,n) + \beta_1^2
\gam^{\beta_2}(z,n) + 2 \beta_1\beta_2 R(z)}{(\beta_2-\beta_1)^2},\\
\label{solsyh} h(z,n) &=& \frac{\beta_2 \gam^{\beta_1}(z,n) + \beta_1
\gam^{\beta_2}(z,n) + (\beta_1+\beta_2) R(z)}{(-2a(n))^{-1}
(\beta_2-\beta_1)^2},
\eea
where $R(z)$ is the branch of
\begin{equation} \label{solsyr}
R(z)= \left( \frac{(\beta_2-\beta_1)^2}{4a(n)^2} + \gam^{\beta_1}(z,n)
\gam^{\beta_2}(z,n) \right)^{1/2} = \frac{\beta_1+\beta_2}{2a(n)} +
O(\frac{1}{z}),
\end{equation}
which is holomorphic for $z\in\C\bs\R$ and has asymptotic behavior as 
indicated. If one of the numbers $\beta_{1,2}$ equals $\infty$, one has to replace
all formulas by their limit using $g(z,n)=\lim\limits_{\beta\to\infty} \beta^{-2}
\gam^\beta(z,n)$.
\eth

\bpf
Clearly, if $(\beta_1,\beta_2) \neq (0,\infty), (\infty,0)$ we can determine
$a(n)$ from (\ref{asymgabe}). Hence by Theorem~\ref{thmunibas} it suffices to
show (\ref{solsyg}) -- (\ref{solsyh}). Since (\ref{solsyg}) follows from
(\ref{detgf}) and the other two, it remains to establish (\ref{solsygp}) and
(\ref{solsyh}). This will follow if we prove that the system
\begin{equation} \label{systgph}
(g^+)^2 +2\frac{\beta_j}{2a(n)} h g^+ + \frac{\beta_j^2}{4a(n)^2} (h^2-1)
= g^+ \gam^{\beta_j}(z,n), \quad j=1,2
\end{equation}
has a unique solution $(g^+,h)=(g(z,n+1),h(z,n))$ for $|z|$ large enough,
$|\im(z)|\ge\eps$, which is holomorphic with respect to $z$ and satisfies the
asymptotic requirements from above. We first consider the case $\beta_j \ne
0,\infty$. Changing to new variables $(x_1,x_2)$,
$x_j = (2a(n)/ \beta_j) g^+ + h$, our system reads
\begin{equation}
x_j^2-1 = \frac{\beta_1\beta_2}{\beta_j^2}
\frac{2a(n)\gam^{\beta_j}(z,n)}{\beta_2-\beta_1}
(x_1-x_2), \quad j=1,2.
\end{equation}
Picking $|z|$ large enough we can assume $\gam^{\beta_j}(z,n)\ne 0$
and the solution set of the new system is given by the intersection of
two parabolas. In particular, (\ref{systgph}) has at most four solutions.
Two of them are clearly $g^+= 0, \quad h=\pm 1$.
But they do not have the correct asymptotic behavior and hence are of no
interest to us. The remaining two solutions are given by (\ref{solsygp})
and (\ref{solsyh}) with the branch of $R(z)$ arbitrarily.
However, we only get correct asymptotics ($g^+= -z^{-1} + O(z^{-2})$ resp.\ 
$h= -1 + O(z^{-2})$) if we fix the branch as in (\ref{solsyr}).
This shows that $g(z,n+1), h(z,n)$ can be reconstructed from
$\gam^{\beta_j}$, $j=1,2$ and we are done. The remaining cases
can be treated similarly.
\epf

\bk
Suppose $H$ has purely discrete spectrum. Then $a(n_0)$, $\sig(H)$ plus
$\beta_j$, $\sig(H^{\beta_j}_{n_0})$, $j=1,2$ for two values $\beta_1 \ne
\beta_2$ uniquely determine the coefficients $a(n)^2,b(n)$ (and the boundary 
condition at $\pm\infty$ if any).
\ek

\bpf
Since $H$ has purely discrete spectrum the same is true for $H^\beta_{n_0}$.
Hence $\gam^\beta(z,n_0)$ is meromorphic with poles at the eigenvalues of
$H$ and zeros at the eigenvalues of $H^\beta_{n_0}$ following from
(\ref{defgambet}) (if eigenvalues of $H$ and $H^\beta_{n_0}$ coincide we have a
double zero in numerator of (\ref{defgambet}) and a single zero in the
denominator). Thus we know when $\gam^\beta(z,n_0)$ changes sign implying that
we know the exponential Herglotz measure of $\gam^\beta(z,n_0)$ (cf.
(\ref{heex})). The remaining constant $c$ in (\ref{heex}) follows from the
asymptotic behavior (see also (\ref{Gambethergr})). Hence we can reconstruct
$\gam^\beta(z,n_0)$ from
$a(n_0)$, $\sig(H)$ and $\beta$, $\sig(H^\beta_{n_0})$ completing the proof.
\epf

Finally, let us turn to half line operators $H^\beta_+=H^\beta_{+,0}$ (cf.\
Remark~\ref{rembc}). Since the dependence one $a(0)$ can be removed by scaling
$\beta$, we assume without restriction $a(0)=1$ for the remainder of this section.
We will now prove the following generalization of a result by Fu and Hochstadt
\cite{fh} (where the special case $\beta_1=0, \beta_2=\infty$ is proved under
somewhat more restrictive conditions).

\bth
Suppose the spectrum of $H^\beta_+$ is purely discrete for one
$\beta\in\R\cup\{\infty\}$ (and hence for all $\beta$) and let
$\beta_j$, $j=1,2$ be  two different values which have opposite signs if
$0<|\beta_j|<\infty$. Then $\beta_j$ plus $\sig(H^{\beta_j}_+)$, $j=1,2$
uniquely determine the coefficients $a(n)^2,b(n)$ (and the boundary condition at
$+\infty$ if any).
\eth

\bpf
Without restriction we suppose $\beta_2 \ne 0$ and $\beta_1\ne\infty$.
Then
\begin{equation}
F(z) = \frac{\beta_2(m_+(z) + \beta_1)}{m_+(z) + \beta_2} =
\frac{-1}{\frac{-1}{\beta_2} - \frac{1}{m_+(z)}} +
\frac{\beta_1 \beta_2}{m_+(z) + \beta_2}
\end{equation}
is a meromorphic Herglotz function since $m_+(z)=m_+(z,0)$ is. Moreover, since
$m_+(z) = \frac{u_+(z,1)}{u_+(z,0)}$ (where $u_+(z,0)$ has to be defined as
$-a(1) u_+(z,2) +(z-b(1)) u(z,1)$; recall our convention $a(0)=1$), we infer
that the zeros of $F(z)$ are given by the eigenvalues of $H^{\beta_1}_+$ and
the poles by the eigenvalues of $H^{\beta_2}_+$. Thus we know the exponential 
Herglotz measure $\xi(\lam)$ of $F(z)$ (cf.\ (\ref{heex})). The remaining constant
$c$ in (\ref{heex}) can be determined from the asymptotic behavior
$F(z) = \beta_1 - (1-\beta_1 \beta_2^{-1}) z^{-1} + O(z^{-2})$. Thus $F(z)$
is known and solving $F(z)$ for $m_+(z)$ finishes the proof.
\epf



\section{General trace formulas and $\xi$ functions}
\setcounter{equation}{0}
\setcounter{thm}{0}


In this section we will investigate trace formulas for Jacobi operators
$H$. We will essentially follow the philosophy of \cite{gsxi}, \cite{ghsztr}
and use the exponential Herglotz representation (\ref{heex}) rather than
(\ref{heru}). This will produce generalizations of the formula (\ref{tfper}).

To avoid the Abelian limits of \cite{gsxi} we will first consider
the case where $H$ (and thus $a,b$) is bounded. We abbreviate
\begin{equation}
E_0 = \inf \sig(H), \qquad E_\infty = \sup \sig(H),
\end{equation}
and note that $G(\lam,n,n) >0$ for $\lam < E_0$, which follows from
$(H - \lam) > 0$ (implying $(H - \lam)^{-1} > 0$). Similarly,
$G(\lam,n,n) < 0$ for $\lam > E_\infty$, following from
$(H - \lam) < 0$. Our main tool will be the following exponential
representation of the Herglotz function $g(z,n)=G(z,n,n)$
(cf.\ Theorem~\ref{thmhexp})
\begin{equation} \label{expherg}
g(z,n) = |g(\I,n)| \exp \left( \int_\R \Big( \frac{1}{\lam-z} -
\frac{\lam}{1+\lam^2} \Big) \xi(\lam,n) d\lam \right), \quad z\in
\C \bs \sig(H),
\end{equation}
where the $\xi$ function $\xi(\lam,n)$ is defined by
\begin{equation} \label{defxif}
\xi(\lam,n) = \frac{1}{\pi} \lim_{\eps \downarrow 0} \arg g(\lam +
\I \eps,n), \qquad \arg(.) \in (-\pi,\pi].
\end{equation}
In addition, $\xi(\lam,n)$ (which is only defined a.e.) satisfies
$0 \le \xi(\lam,n) \le 1$,
\begin{equation} \label{xifproper}
\int_\R \frac{\xi(\lam,n)}{1+\lam^2} d\lam = \arg g(\I,n), 
\text{ and } \xi(\lam,n)= \left\{ \ba{c@{\text{ for
}}l} 0 & z<E_0 \\ 1 & z>E_\infty \ea \right. .
\end{equation}
Using (\ref{xifproper}) together with the asymptotic behavior of $g(.,n)$ we
infer
\begin{equation} \label{exgxi}
g(z,n) = \frac{1}{E_\infty -z} \exp \left( \int_{E_0}^{E_\infty}
\frac{\xi(\lam,n) d\lam}{\lam-z} \right).
\end{equation}

\bth
Suppose $H$ is bounded and let $\xi(\lam,n)$ be defined as above.
Then we have the following trace formula
\begin{equation} \label{gentracef}
b^{(\ell)}(n) = E_\infty^\ell - \ell \int_{E_0}^{E_\infty} \lam^{\ell-1}
\xi(\lam,n) d\lam,
\end{equation}
where
\bea \nn
b^{(1)}(n) &=& b(n),\\ \label{expcoeflng}
b^{(\ell)}(n) &=& \ell \, g_\ell(n) - \sum_{j=1}^{\ell-1}
g_{\ell-j}(n) b^{(j)}(n), \quad \ell >1.
\eea
\eth

\bpf
The claim follows after expanding both sides of
\begin{equation}
\ln \Big( (E_\infty -z) g(z,n) \Big) = \int_{E_0}^{E_\infty} \frac{\xi(\lam,n)
d\lam}{\lam-z}
\end{equation}
and comparing coefficients using the following connections between the series
of $g(z)$ and $\ln (1+g(z))$ (cf., e.g., \cite{olv}). Let $g(z)$ have the
asymptotic expansion
\begin{equation}
g(z) = \sum_{\ell=1}^\infty \frac{g_\ell}{z^\ell}
\end{equation}
as $z \to \infty$. Then we have
\begin{equation}
\ln(1+g(z)) = \sum_{\ell=1}^\infty \frac{c_\ell}{z^\ell},
\end{equation}
where
\begin{equation}
c_1 = g_1, \quad c_\ell = g_\ell - \sum_{j=1}^{\ell-1} \frac{j}{\ell}
g_{\ell-j} c_j, \quad \ell \ge 2.
\end{equation}
\epf

We remark that the special case $\ell=1$ of equation (\ref{gentracef})
\begin{equation} \label{gentracefo}
b(n) = E_\infty - \int_{E_0}^{E_\infty} \xi(\lam,n) d\lam = \frac{E_0 + E_\infty}{2}
+ \frac{1}{2} \int_{E_0}^{E_\infty} (1-2\xi(\lam,n) ) d\lam
\end{equation}
has first been given in \cite{gsxi}.

Next we turn to unbounded operators. In order to avoid Abelian limits
here as well, we resort to a little trick. This will also show how our
investigations tie in with the theory of Krein \cite{kr} and
rank one perturbations (see also \cite{gsxi}, Appendix A, \cite{gsro},
\cite{siro}). Consider
\begin{equation}
H_{n,\theta} = H + \theta \spr{\delta_n}{.} \delta_n,\quad \theta\ge 0.
\end{equation}
Then, as in \cite{gsxi}, Appendix A, one computes
\begin{equation}
\tr\Big( (H-z)^{-1} - (H_{n,\theta}-z)^{-1}\Big) =
\frac{d}{dz} \ln(1 + \theta g(z,n)) = \int_\R
\frac{\xi_\theta(\lam,n)}{(\lam-z)^2} d\lam,
\end{equation}
where
\begin{equation} \label{hxiherg}
1 + \theta g(z,n) = \exp \Big( \int_\R
\frac{\xi_\theta(\lam,n)}{\lam-z} d\lam \Big), \quad
\xi_\theta(\lam,n) = \frac{1}{\pi} \lim_{\eps \downarrow 0} \arg 
\Big( 1+ \theta g(\lam+\I\eps,n)\Big).
\end{equation}
By Theorem \ref{thmhexp} (iii) all moments of $\xi_\theta(\lam,n)d\lam$
are finite and $\int_\R \xi_\theta(\lam,n) d\lam = \theta$.

Taking logarithms in (\ref{hxiherg}) and expanding yields as before

\bth
Let $\xi_\theta(\lam,n)$ be defined as above. Then we have
\begin{equation} \label{tracelambcell}
b_\theta^{(\ell)}(n) = (\ell+1) \int_\R \lam^\ell \xi_\theta(\lam,n) d\lam,
\end{equation}
with
\begin{equation}
b_\theta^{(0)}(n)= \theta, \quad b_\theta^{(\ell)}(n) = \theta (\ell+1) g_\ell(n) +
\theta \sum_{j=1}^\ell g_{\ell-j}(n) b_\theta^{(j-1)}(n),
\quad \ell\in\N.
\end{equation}
\eth

Again, in the special case $\ell=1$ we obtain
\begin{equation}
b(n) = \frac{1}{\theta} \int_\R \lam \xi_\theta(\lam,n) d\lam - \frac{\theta}{2}.
\end{equation}
In addition, we remark that letting the coupling constant $\theta$ tend
to $\infty$ implies $H_{n,\theta} \to H_n^\infty$ in a suitable sense
(i.e., norm resolvent sense on $\{f\in \lz | \spr{\delta_n}{f}=0\}$,
cf.\ \cite{gsro}). Similarly, $H_{n_0}^\beta$ can be obtained as the limit of
the operator $H + \theta \spr{\delta_n^\beta}{.} \delta_n^\beta$ as
$\theta\to\infty$.

Clearly, the same procedure can be applied to (cf.\ Theorem~\ref{thmhexp}
(i), (iii))
\begin{equation} \label{Gambethergr}
\gam^\beta(z,n) = -\frac{\beta}{a(n)} \exp \Big( \int_\R
\frac{\xi^\beta(\lam,n) d\lam}{\lam-z}
\Big), \quad z\in \C \bs \sig(H^\beta_n), \beta \in \R \bs \{0\},
\end{equation}
where
\begin{equation}
\xi^\beta(\lam,n) = \frac{1}{\pi} \lim_{\eps \downarrow 0}\arg\Big(
\gam^\beta(\lam + \I \eps,n) \Big) - \delta^\beta, \quad
\delta^\beta = \left\{ \ba{cl} 0, &
\beta a(n) <0 \\ 1, & \beta a(n) >0\ea \right. 
\end{equation}
and $0\le\sgn(-a(n)\beta) \, \xi^\beta(\lam,n) \le 1$.
This yields as before

\bth
Let $\xi^\beta(\lam,n)$ be defined as above. Then we have
\begin{equation} \label{gentrfal}
b^{\beta,(\ell)}(n) = (\ell+1) \frac{\beta}{a(n)} \int_\R \lam^\ell
\xi^\beta(\lam,n) d\lam, \quad \ell \in \N
\end{equation}
where
\bea \nn
b^{\beta,(0)}(n) &=& 1 +\beta^2, \\
b^{\beta,(\ell)}(n) &=& (\ell+1) \gam^\beta_\ell(n)
- \frac{\beta}{a(n)} \sum_{j=1}^\ell 
\gam^\beta_{\ell-j}(n) b^{\beta,(j-1)}(n), \quad \ell \in\N.
\eea
\eth

Again specializing for $\ell=0$ in (\ref{gentrfal}) we obtain
\begin{equation}
a(n) = \frac{1}{\beta + \beta^{-1}} \int_\R \xi^\beta(\lam,n) d\lam.
\end{equation}

Finally, we want to find out when $\xi^{\beta_j}(\lam,n_0)$,
$j=1,2$, for one fixed $n_0$ determines $a(n), b(n)$, $n\in\Z$.
Since $\xi^{\beta}(.,n_0)$, $\beta\in\R$ and $a(n_0)$ determines
$\gam^\beta(z,n_0)$ by (\ref{Gambethergr}) we conclude from
Theorem~\ref{thmunigabot}

\bk
Let $\beta_{1,2} \in \R\cup\{\infty\}$ be given. Then $(\beta_j,
\xi^{\beta_j}(.,n_0))$, $j=1,2$, and $a(n_0)$ for one fixed $n_0\in\Z$
uniquely determines $a(n)^2, b(n)$ for all $n\in\Z$.
\ek



\section{Reflectionless operators}
\setcounter{equation}{0}
\setcounter{thm}{0}



Reflectionless operators have attracted a considerable amount of interest
recently in connection with inverse spectral theory \cite{akr}, \cite{gkt},
\cite{sy1}, \cite{sy2} and  completely integrable lattices \cite{bght},
\cite{vm}. In this section we show that the trace formulas of the previous
section become particularly transparent in this case.

We will assume that $H$ is a bounded self-adjoint Jacobi operator.
Hence its spectrum can be written as the complement of a countable union
of disjoint open intervals, that is,
\begin{equation} \label{defsetsig}
\sig(H)= \R\bs\bigcup_{j\in J_0\cup \{\infty\}} \rho_j,
\end{equation}
where $J\subseteq\N$, $J_0=J\cup \{0\}$,
\bea \nn
&\rho_0=(-\infty ,E_0),\quad \rho_\infty =(E_\infty,\infty ),&\\
\label{defrohj} & E_0\leq E_{2j-1}<E_{2j}\le E_\infty,
\quad \rho_j=(E_{2j-1},E_{2j}),\quad j\in J,&\\ \nn
& -\infty<E_0<E_\infty <\infty ,\quad \rho_j\cap
\rho_k=\emptyset \text{ for } j\ne k.&
\eea
In addition, we will require that $H$ is reflectionless, that is,
for all $n\in\Z$,
\begin{equation} \label{refl}
\xi(\lam,n)=\frac{1}{2} \text{ for a.e. } \lam \in\sig_{ess}(H).
\end{equation}
By \cite{gkt}, Lemma~3.3 the requirement (\ref{refl}) is
equivalent to one of the following\\
(i). For some $n_0\in \Z$, $n_1\in\Z\bs\{n_0,n_0+1\}$,
$$
\xi(\lam,n_0)=\xi(\lam ,n_0+1)=\xi(\lam,n_1)=\frac{1}{2}
\text{ for a.e }\lam\in\sig_{ess}(H).
$$
(ii). For some $n_0\in \Z$,
$$
\ti{m}_+(\lam +\I 0,n_0)= \ol{\ti{m}_-(\lam+\I 0,n_0)}\text{ for a.e. }
\lam\in\sig_{ess}(H),
$$
where $\ti{m}_-(\lam+\I 0,n_0)$ abbreviates $\lim_{\eps\downarrow 0}
\ti{m}_-(\lam+\I \eps,n_0)$.

The last equation implies
\begin{equation} \label{rfllupm}
u_+(\lam +\I 0,n)= \ol{u_-(\lam+\I 0,n)}\text{ for a.e. }
\lam\in\sig_{ess}(H)
\end{equation}
for $u_\pm(z,n)= c(z,n,n_0) + a(n_0) \ti{m}_\pm(z,n_0) s(z,n,n_0)$, where
$c,s$ are the solutions of $\tau u = z u$ corresponding to the initial
conditions $c(z,n_0,n_0)=s(z,n_0+1,n_0)=1$, $s(z,n_0,n_0)=c(z,n_0+1,n_0)=0$.

The name reflectionless will become clear in the next section.
There the above conditions will turn out to be equivalent to
the vanishing of the {\em reflection coefficients} $R_\pm(z)$
(cf.\ (\ref{scatmat})). For instance periodic operators, operators
with purely discrete spectrum, and stationary solutions of the Toda
hierarchy are special cases of reflectionless operators.

Next we turn to Dirichlet eigenvalues associated with $\tau$ corresponding
to a Dirichlet boundary condition at $n\in \Z$. Associated with each
spectral gap $\rho_j$ we set
\begin{equation} \label{defdie}
\mu_j(n)=\sup \{E_{2j-1}\}\cup \{\lam\in\rho_j | g(\lam,n)<0\}\in
\ol{\rho_j},\quad j\in J.
\end{equation}
The numbers $\mu_j(n)$ are called Dirichlet eigenvalues of $H$ since
we have
\begin{equation}
\sig(H_n^\infty) = \sig_{ess}(H) \cup \{\mu_j(n) \}_{j\in J}.
\end{equation}
However, we want to point out that $\mu_j(n)$ is not necessarily an
eigenvalue of $H_n^\infty$ unless $\mu_j(n)\not\in\sig_{ess}(H)$.

The strict monotonicity of $g(\lam,n)$ with respect
to $\lam \in \rho_j$, that is,
\begin{equation}
\frac{d}{d\lam}g(\lam,n)= \spr{\delta_n}{(H-\lam)^{-2}\delta_n} =
\sum_{m\in\Z} G(\lam ,n,m)^2>0, \quad \lam\in\rho_j,
\end{equation}
then yields
\begin{equation}
\ba{ll} g(\lam,n)<0, \quad & \lam\in(E_{2j-1},\mu_j(n)),\\
g(\lam,n)>0, & \lam\in(\mu_j(n),E_{2j}),\ea\quad j\in J.
\end{equation}
Thus we conclude $\xi(\lam,n)=1$, $\lam\in(E_{2j-1},\mu_j(n))$ and
$\xi(\lam,n)=0$, $\lam\in(\mu_j(n),E_{2j})$, $j\in J$.
Using this information to evaluate the exponential Herglotz representation
of $g(z,n)$ then implies (\cite{gkt}, Lemma~1.1)
\begin{equation} \label{exphggrl}
g(z,n) = \frac{-1}{\sqrt{z-E_0}\sqrt{z-E_\infty}}
\prod_{j\in J}\frac{z-\mu_j(n)}{\sqrt{z-E_{2j-1}} \sqrt{z-E_{2j}}},
\end{equation}
where the square root branch used is defined as $\sqrt{z}=|\sqrt{z}|
\exp(\I\arg(z)/2)$, $-\pi < z \le \pi$. In addition,
denoting by $\chi_\Omega(.)$ the characteristic function
of the set $\Omega\subset\R$, one can represent $\xi(\lam,n)$ by
\bea \nn
\xi(\lam,n) &=& \frac{1}{2} \Big(\chi_{(E_0,\infty)}(\lam)+
\chi_{(E_\infty,\infty )}(\lam)\Big)\\ \nn
&& {}+ \frac{1}{2}\sum_{j\in J}\Big(\chi_{(E_{2j-1},\infty)}(\lam)+
\chi_{(E_{2j},\infty)}(\lam)-2\chi_{(\mu_j(n),\infty)}(\lam )\Big)\\ \nn
&=& \frac{1}{2}\chi_{(E_0,E_\infty)}(\lam)+ \frac{1}{2}\sum_{j\in J}
\Big( \chi_{(E_{2j-1},\mu_j(n))}(\lam)-\chi_{(\mu_j(n),E_{2j})}(\lam)
\Big)\\ \label{xifrefl}
&& {}+ \chi_{(E_\infty,\infty)}(\lam) \quad \text{for a.e. }\lam\in\R.
\eea
Evaluation of (\ref{gentracef}) shows
\begin{equation} \label{bellrl}
b^{(\ell)}(n) = \frac{1}{2}\Big (E_0^\ell + E_\infty^\ell+
\sum_{j\in J} (E_{2j-1}^\ell+E_{2j}^\ell-2\mu_j(n)^\ell) \Big)
\end{equation}
and in the special case $\ell=1$
\begin{equation}
b(n) = \frac{1}{2}\Big( E_0+E_\infty + \sum_{j\in J} (E_{2j-1}
+E_{2j}-2\mu_j(n)) \Big).
\end{equation}
The formulas for $\ell=1,2$ were first  given in \cite{akr}, Theorem 5.2.

Next, we want to address the problem of expressing $a(n)^2$ as a function
of $E_j$ and $\mu_j(n)$. This endeavor turns out to be impossible
unless we introduce additional data. This will be done first by defining
\begin{equation}
\{ \ti{\mu}_j(n)\}_{j \in \ti{J}} = \{ \mu_j(n)\}_{j \in J} \cup \sig_p(H^\infty_n),
\quad \ti{J} \subseteq\N
\end{equation}
and $\ti{E}_0=E_0$, $\ti{E}_\infty = E_\infty$,
\begin{equation}
\ti{E}_{2j-1} = \sup \{E\in\sig(H) | E < \ti{\mu}_j(n) \}, \quad
\ti{E}_{2j} = \inf \{E\in\sig(H) | \ti{\mu}_j(n) < E \}.
\end{equation}
A few remarks are in order:
\br
(i). We note that $\ti{\mu}_j = \mu_k$ implies $\ti{E}_{2j-1}  = E_{2k-1}$,
$\ti{E}_{2j} = E_{2k}$ and $\ti{E}_{2j-1}  < \ti{E}_{2j}$ implies
$\ti{\mu}_j(n) =\mu_k(n)$ for some $k\in J$. Indeed, if $\ti{E}_{2j-1}  <
\ti{E}_{2j}$ we infer $\lim_{\lam\to\ti{\mu}_j(n),\, \lam\in(\ti{E}_{2j-1},
\ti{E}_{2j})} g(\lam,n) =0$ and hence $\ti{\mu}_j(n)=\mu_k(n)$ for some
$k\in J$ by monotonicity of $g(.,n)$ in spectral gaps. In other words,
computing all previous formulas with $\mu_j(n), E_j$ replaced by
$\ti{\mu}_j(n), \ti{E}_j$ leaves them unchanged since the new
factors drop out.\\
(ii). Our notation concerning $\ti{E}_j$ is imprecise since the
list of numbers $[\ti{E}_j]_{j\in\ti{J}}$ might, in general, depend on $n$.
Suppose for instance, that $\ti{\mu}_j(n)$ is also an eigenvalue of $H$
such that $\ti{E}_{2j-1}=\ti{\mu}_j(n)=\ti{E}_{2j}$. Then the pair
$\ti{E}_{2j-1},\ti{E}_{2j}$ shows up in the list corresponding to $n$ but not in the
one corresponding to $n+1$ since the eigenfunction for $\ti{\mu}_j(n)$ cannot
vanish at two consecutive points.
\er

Moreover, following \cite{gkt}, we introduce the numbers
\begin{equation}
\ti{R}_j(n) = \lim\limits_{\eps\downarrow 0} \I\eps
g(\ti{\mu}_j(n)+\I\eps,n)^{-1}\ge 0,
\end{equation}
and
\begin{equation}
\ti{\sig}_j(n) = \left\{\ba{cl} \lim\limits_{\eps\downarrow 0}
h(\ti{\mu}_j(n)+\I\eps,n) & \mbox{if}\; \ti{R}_j(n)>0\\ 2 & \mbox{if}\; 
\ti{R}_j(n)=0
\ea\right. .
\end{equation}
The actual value of $\ti{\sig }_j(n)$ if $\ti{R}_j(n)=0$ is immaterial and is
chosen in accordance with \cite{gkt}. The above limits exist if
$\ti{\mu}_j \in \sig(H^\infty_n)$ (i.e., if $\ti{R}_j(n)>0$) and $\ti{\sig }_j(n)$ is
either $\pm1$ (depending on whether $\ti{\mu}_j$ is an eigenvalue of
$H_{\pm,n}$) or in $(-1,+1)$ (if $\ti{\mu}_j$ is an eigenvalue of both
$H_{\pm,n}$ and hence also of $H$). For more details see \cite{gkt}.

The numbers $\ti{R}_j(n)$ can be evaluated using (\ref{exphggrl})
\begin{equation}
\ti{R}_j(n) = \frac{\sqrt{\ti{\mu}_j(n)-E_0}\sqrt{\ti{\mu}_j(n)-E_\infty}
\sqrt{\ti{\mu}_j(n)-E_{2j-1}} \sqrt{\ti{\mu}_j(n)-E_{2j}}}{\prod_{k\in
J\bs\{j\}}\frac{\ti{\mu}_j(n)-\mu_k(n)}{\sqrt{\ti{\mu}_j(n)-E_{2k-1}}
\sqrt{\ti{\mu}_j(n)-E_{2k}}}}.
\end{equation}
If $\ti{\mu}_j=\mu_k=E_{2k}=E_{2j-1}$ for some $k$ (resp.
$\ti{\mu}_j=\mu_k=E_{2k-1}=E_{2j}$) the vanishing factors
$\ti{\mu}_j-\mu_k$ in the denominator and $\ti{\mu}_j-E_{2j}$ (resp.
$\ti{\mu}_j-E_{2j-1}$) in the numerator have to be omitted. In particular,
we want to point out that $\ti{R}_j(n)$ depend on $E_j, \mu_j$ only.

In addition, we require that the singularly continuous spectrum of
$H^\infty_n$ is empty (the absolutely continuous spectrum being taken
care of by the reflectionless condition). Then it is shown
in \cite{gkt} that the spectral data $ E_j$, $j\in J \cup
\{0,\infty\}$ plus $\mu_j(n_0)$, $j\in J$ plus  $\ti{\sig}_j(n_0)$ , $j \in \ti{J}$
for one fixed $n_0\in\Z$ are minimal and uniquely determine $a(n)^2, b(n)$. 
(To be precise, the class of operators considered here is slightly larger than the
one in \cite{gkt}, however, the same proof applies.) Moreover, necessary and
sufficient conditions for given spectral data to be the spectral data of some Jacobi
operator were derived. Here we want to focus on the reconstruction of $a(n)^2,
b(n)$ from given spectral data as above and present an {\em explicit} expression
of $a(n)^2, b(n)$ in terms of the spectral data.

Our point of departure will be the formulas (use (\ref{conmtim}) and
(\ref{contimgh}))
\bea \nn
a(n)^2 m_+(z,n) \pm a(n-1)^2 m_-(z,n) &=& \mp z \pm b(n) - 
\left\{\ba{l} \frac{1}{g(z,n)}\\ \rule{0pt}{5mm} \frac{h(z,n)}{g(z,n)}\ea\right. \\
\label{tfaphi} &=& - \sum_{j=0}^\infty \frac{c_{\pm,j}(n)}{z^{j+1}},
\eea
where the coefficients $c_{\pm,j}(n)$ are to be determined.
Arguing similarly as for (\ref{tfnonex}) one obtains
\begin{equation} 
c_{\pm,\ell}(n)  = \int_\R \lam^\ell \Big( a(n)^2 d\rho_{+,n}(\lam) \pm a(n-1)^2
d\rho_{-,n}(\lam)\Big), \quad \ell\in\N_0,
\end{equation}
where $d\rho_{\pm,n}(\lam)$ are the spectral measures of $H_{\pm,n}$
associated with the vector $\delta_{n\pm1}$. 

The evaluation of this integral will now be done for the minus sign. Due  to the
reflectionless condition, the integral over the (absolutely) continuous spectrum is
zero (there is no singularly continuous part by assumption) and it remains to
evaluate the pure point part. To do this it suffices to know the jumps of the
measure which are given by the residues of the corresponding Herglotz function.
Evaluating the residues (using (\ref{tfaphi}) plus the notation from above) shows
\begin{equation} \label{cmtsd}
c_{-,\ell}(n)  = \sum_{j\in\ti{J}} \ti{\sig}_j(n) \ti{R}_j(n) \ti{\mu}_j(n)^\ell,
\quad \ell\in\N_0.
\end{equation}
Clearly it suffices to sum over all $\ti{\mu}_j(n)\in\sig_p(H^\infty_n)$
since for all other terms we have $\ti{R}_j(n)=0$.

Next we turn to the coefficients $c_{+,\ell}(n)$. They can be determined
from (cf.\ (\ref{exgxi}))
\begin{equation}
\frac{1}{g(z,n)}  = -z\, \exp\Big( - \sum_{\ell=1}^\infty \frac{b^{(\ell)}(n)}{\ell z^\ell}
\Big),
\end{equation}
which implies
\bea \nn
c_{+,-2}(n)&=&1,\\ \label{cptsd}
c_{+,\ell-2}(n)&=& \frac{1}{\ell} \sum_{j=1}^\ell c_{+,\ell-j-2}(n)
b^{(j)}(n),\quad \ell\in\N.
\eea
Thus $c_{+,\ell}(n)$ are expressed in terms of $E_j$, $\mu_j(n)$.
Here $c_{+,-2}(n)$ and $c_{+,-1}(n)$ have been introduced for
notational convenience only.

In particular, combining the case $\ell=0$ with our previous results we obtain
\begin{equation} \label{exaemu}
a(n - {\textstyle\genfrac{}{}{0pt}{}{0}{1}})^2 = \frac{b^{(2)}(n) - b(n)}{4} \pm
\sum_{j\in\ti{J}} \frac{\ti{\sig}_j(n)}{2} \ti{R}_j(n).
\end{equation}
Similarly, for $\ell=1$,
\bea\nn
b(n\pm1) &=& \frac{1}{a(n-\genfrac{}{}{0pt}{}{0}{1})^2} \Big( \frac{2
b^{(3)}(n) - 3 b(n) b^{(2)}(n) + b(n)^3}{12}\\ &&{}
\pm \sum_{j\in\ti{J}} \frac{\ti{\sig}_j(n)}{2} \ti{R}_j(n) \ti{\mu}_j(n)\Big).
\eea
However, these formulas are only the tip of the iceberg. Combining
\begin{equation}
c_{\pm,\ell}(n) = a(n)^2 m_{+,\ell}(n) \pm a(n-1)^2 m_{-,\ell}(n)
 \end{equation}
with some basic facts from the moment problem we obtain our main result:

\bth
Let $H$ be a given bounded reflectionless Jacobi operator. Suppose the
singularly continuous spectrum of $H^\infty_n$ is empty
and the spectral data corresponding to $H$ (as above) are given for one fixed
$n\in\Z$. Then the sequences $a^2,b$ can be expressed explicitly in terms of the
spectral data as follows
\bea \label{eqamom}
a(n\pm k - {\textstyle\genfrac{}{}{0pt}{}{0}{1}})^2 &=&
\frac{C_{\pm,n}(k+1)C_{\pm,n}(k-1)}{C_{\pm,n}(k)^2},\\ \label{eqbmom}
b(n\pm k) &=& \frac{D_{\pm,n}(k)}{C_{\pm,n}(k)} -
\frac{D_{\pm,n}(k-1)}{C_{\pm,n}(k-1)}, \quad k\in\N,
\eea
where $C_{\pm,n}(0)=1$, $D_{\pm,n}(0)=0$,
\begin{equation}
C_{\pm,n}(k) = \det \left(\ba{cccc}
m_{\pm,0}(n) & m_{\pm,1}(n) & \cdots &m_{\pm,k-1}(n)\\
m_{\pm,1}(n) & m_{\pm,2}(n) & \cdots &m_{\pm,k}(n)\\
\vdots & \vdots & \ddots & \vdots\\
m_{\pm,k-1}(n) & m_{\pm,k}(n) & \cdots &m_{\pm,2k-2}(n)\ea\right),
\end{equation}
\begin{equation}
D_{\pm,n}(k) = \det \left(\ba{ccccc}
m_{\pm,0}(n) & m_{\pm,1}(n) & \cdots & m_{\pm,k-2}(n) & m_{\pm,k}(n)\\
m_{\pm,1}(n) & m_{\pm,2}(n) & \cdots & m_{\pm,k-1}(n) & m_{\pm,k+1}(n)\\
\vdots & \vdots & \ddots & \vdots& \vdots\\
m_{\pm,k-1}(n) & m_{\pm,k}(n) & \cdots & m_{\pm,2k-3}(n)& m_{\pm,2k-1}(n)
\ea\right),
\end{equation}
and $m_{\pm,\ell}(n)=\frac{c_{+,\ell}(n)\pm c_{-,\ell}(n)}{2a(n-
\genfrac{}{}{0pt}{}{0}{1})}$. The quantities $a(n)^2$, $a(n-1)^2$, and
$c_{\pm,\ell}(n)$ have to be expressed in terms of the spectral data using
(\ref{exaemu}), (\ref{cptsd}), (\ref{cmtsd}) and (\ref{bellrl}).
\eth

\bpf
It remains to show the expressions (\ref{eqamom}) and (\ref{eqbmom}) for $a(n)$
and $b(n)$ in terms of the moments $M_{\pm,\ell}(n_0)$, $\ell\in\N$.
Both can be found in \cite{ak} (first equation on page 5). However, the
equation for $b(n)$ here differs from the one in \cite{ak} since we have
performed the integration (see \cite{gtjocil}, Section~2.5 for details).
\epf

In the special case of periodic Jacobi operators, the formula (\ref{exaemu}) was
first given in \cite{bght}.  In addition, we get a discrete version of Borg's
theorem.

\bk
Let $H$ be a reflectionless Jacobi operator with spectrum consisting of only one
band, that is $\sig(H) = [E_0,E_\infty]$. Then the sequences $a(n)^2, b(n)$ are
necessarily constant
\begin{equation}
a(n)^2 = \frac{(E_\infty-E_0)^2}{16}, \quad b(n) = \frac{E_0+E_\infty}{2}.
\end{equation}
\ek

The special case where $H$ is periodic seems due to \cite{flln} (Proposition 2
on p.\ 451). The formula for $b(n)$ also follows directly from
(\ref{gentracefo}).

\br
(i). If $J$ is finite, that is, $H$ has only finitely many spectral gaps, then
$\{ \ti{\mu}_j(n) \}_{j\in \ti{J}} =  \{ \mu_j(n) \}_{j\in J}$ and we can forget
about the additional $\mu$'s.\\
(ii). The reader might wonder whether a similar procedure for one-dimensional
Schr\"odinger operators $H=-\frac{d^2}{dx^2} + V(x)$ is possible. This is in fact the
case but under more restrictive conditions on $V(x)$. Without going into technical
details we remark that in the continuous case the asymptotic expansions of the
Weyl $m$-functions contain the information of all derivatives of $V$ at the base
point. Hence if $V$ is assumed real analytic (e.g., finite gap) it can be expressed in
terms of its derivatives using Taylor's formula.\\
(iii). Concerning general Jacobi operators we note that Theorem~\ref{thmunigabot}
indicates that $a(n_0)^2$, $\gam^{\beta_j}_\ell(n_0)$, $j=1,2$, $\ell\in\N$ is
solvable for $a(n)^2, b(n)$ as well.
\er

Finally, we turn to general eigenvalues associated with $H_n^\beta$.
Associated with each
spectral gap $\rho_j$ we set
\begin{equation}
\lam^\beta_j(n)=\sup \{E_{2j-1}\}\cup \{\lam\in\rho_j | \gam^\beta(\lam,n)<0\}
\in\ol{\rho_j},\quad j\in J.
\end{equation}

The strict monotonicity of $\gam^\beta(\lam,n)$ with respect
to $\lam \in \rho_j, j\in J_0\cup\{\infty\}$, that is,
\begin{equation}
\frac{d}{d\lam}\gam^\beta(\lam,n)= (1+\beta^2) \spr{\delta^\beta_n}{(H-\lam)^{-2}
\delta^\beta_n}, \quad \lam\in\rho_j,
\end{equation}
then yields
\begin{equation}
\ba{ll} \gam^\beta(\lam,n)<0, \quad & \lam\in(E_{2j-1},\lam^\beta_j(n)),\\
\gam^\beta(\lam,n) >0, & \lam\in(\lam^\beta_j(n),E_{2j}),\ea\quad j\in J.
\end{equation}
Since $\gam^\beta(\lam,n)$ is positive (resp.\ negative) for $a(n)\beta>0$
(resp.\ $a(n)\beta<0$) as $\lam\to\infty$ (resp.\ $\lam\to-\infty$), there
must be an additional zero $\lam^\beta_\infty$ for $\lam\ge E_\infty$ (resp.\
$\lam\le E_0$). Summarizing, $\xi^\beta(\lam,n)$ is given by
\bea \nn
\xi^\beta(\lam,n) &=& \frac{1}{2}\chi_{(E_0,E_\infty)}(\lam) +
\frac{1}{2}\sum_{j\in J} \Big( \chi_{(E_{2j-1},\lam^\beta_j(n))}(\lam) -
\chi_{(\lam^\beta_j(n),E_{2j})}(\lam) \Big)\\ && {}+
\chi_{(E_\infty,\lam^\beta_\infty)}(\lam),
\quad a(n)\beta>0
\eea
and
\bea \nn
\xi^\beta(\lam,n) &=& -\frac{1}{2}\chi_{(E_0,E_\infty)}(\lam) +
\frac{1}{2}\sum_{j\in J} \Big( \chi_{(E_{2j-1},\lam^\beta_j(n))}(\lam) -
\chi_{(\lam^\beta_j(n),E_{2j})}(\lam)
\Big)\\ && {}- \chi_{(\lam^\beta_\infty,E_0)}(\lam),
\quad a(n)\beta<0.
\eea
Thus we have for $\beta\ne 0,\infty$,
\begin{equation}
\gam^\beta(z,n) = \frac{z-\lam^\beta_\infty(n)}{\sqrt{z-E_0}\sqrt{z-E_\infty}}
\prod_{j\in J}\frac{z-\lam^\beta_j(n)}{\sqrt{z-E_{2j-1}}
\sqrt{z-E_{2j}}},
\end{equation}
and we remark that the numbers $\lam^\beta_j(n)$ are related to the spectrum
of $H_n^\beta$ as follows
\begin{equation}
\sig(H_n^\beta) = \sig_{ess}(H) \cup \{\lam^\beta_j(n) \}_{j\in J \cup
\{ \infty\}}.
\end{equation}
Again we point out that $\lam^\beta_j(n)$ is not necessarily an
eigenvalue of $H_n^\beta$ unless $\lam^\beta_j(n)\not\in\sig_{ess}(H)$.

Evaluation of (\ref{gentracef}) shows
\bea \nn
b^{\beta,(\ell)}(n) &=& \frac{-\beta}{2a(n)}\Big (E_0^{\ell+1} +
E_\infty^{\ell+1}- 2\lam^\beta_\infty(n)^{\ell+1}\\
&& {}+ \sum_{j\in J} (E_{2j-1}^{\ell+1}+E_{2j}^{\ell+1} -
2\lam^\beta_j(n)^{\ell+1}) \Big)
\eea
and in the special case $\ell=0$,
\begin{equation}
a(n) = \frac{1}{2(\beta+\beta^{-1})}\Big (E_0 + E_\infty -2
\lam^\beta_\infty(n) +\sum_{j\in J} (E_{2j-1}+E_{2j}-2\lam^\beta_j(n)) \Big).
\end{equation}





\section{Scattering theory}
\setcounter{equation}{0}
\setcounter{thm}{0}



One important class of Jacobi operators are periodic ones. In this
section we want to consider scattering theory with periodic background
operators and apply the results of Section~5. Even though this
problem arises naturally if one considers an infinite harmonic crystal (with
$N$ atoms in the base cell) with impurities, not too many articles are
available on this problem (cf., e.g., \cite{gerass}, \cite{kl}). The case
with constant background (i.e., only one atom in the base cell) is treated,
for instance in \cite{dinv1}, \cite{gu}. For a comprehensive treatment in the
case of Schr\"odinger operators with fairly arbitrary backgrounds we refer the
reader to \cite{gnp} and the references therein.

We first recall some basic facts from the theory of
periodic operators (cf., e.g., \cite{bght}, Appendix~B, \cite{kr1},
\cite{vm}). Let $H_p$ be a Jacobi operator associated with
periodic sequences $a_p\ne0,b_p$, that is,
\begin{equation}
a_p(n+N) =a_p(n), \qquad b_p(n+N) = b_p(n),
\end{equation}  
for some fixed $N \in\N$. The spectrum of $H_p$ is purely absolutely
continuous and consists of a finite number of gaps, that is,
\begin{equation}
\sig(H) = \bigcup_{j=1}^N [E_{p,2j-2},E_{p,2j-1}], \quad
E_{p,0} <\cdots< E_{p,2N-1}
\end{equation}
Moreover, Floquet theory implies the existence of solutions $u_{p,\pm}(z,.)$ of
$\tau_p u = z u$, $z\in\C$ ($\tau_p$ the difference expression corresponding
to $H_p$) satisfying
\begin{equation}
u_{p,\pm}(z,n+N) = m^\pm(z) u_{p,\pm}(z,n)
\end{equation}
and hence
\begin{equation}
u_{p,\pm}(z,n) = p_\pm(z,n) \exp(\pm\I q(z) n),\quad p_\pm(z,n)=p_\pm(z,n+N),
\end{equation}
where $m^\pm(z)=\exp(\pm\I q(z) N) \in \C$ are called Floquet multipliers and
$q(z)$ is called Floquet momentum ($m^\pm(z)$ is not related to the
Weyl $m$-function $m_\pm(z,n)$). $m^\pm(z)$ satisfy
$m^+(z) m^-(z)=1$, $m^\pm(z)^2 =1$ for $z \in \{ E_{p,j} \}_{j=0}^{2N-1}$,
$|m^\pm(z)|=1$ for $z \in \sig(H_p)$, and $|m^+(z)|<1$ for $z \in
\C\bs\sig(H_p)$. (This says in particular, that $u_{p,\pm}(z,.)$ are
bounded for $z\in\sig(H_p)$ and linearly independent for $z \in \C\bs\{ E_j
\}_{j=0}^{2N-1}$.) Requiring $m^\pm(\lam) = \lim_{\eps\downarrow 0}
m^\pm(\lam +\I\eps)$, $\lam\in\sig(H_p)$ determines $m^\pm(z)$ uniquely.

We are going to investigate scattering theory for the pair $(H,H_p)$,
where $H$ is a Jacobi operator satisfying
\begin{equation} \label{decay}
\sum_{n\in\Z} |n(a(n) - a_p(n))| < \infty, \quad
\sum_{n\in\Z} |n(b(n) - b_p(n))| < \infty.
\end{equation}
By \cite{gtosc}, Theorem~5.1 the requirement (\ref{decay}) implies
that the essential spectrum of $H$ is equal to $\sig(H_p)$ and purely
absolutely continuous. Moreover, the point spectrum of $H$ is finite
and confined to the spectral gaps of $H_p$, that is, $\sig_p(H) \subset
\R\bs\sig(H_p)$.

As in the proof of \cite{gtosc}, Theorem~5.1 one can use the sum equation
\begin{equation} \label{voltseq}
u_\pm(z,n) = \frac{a_p(n-\genfrac{}{}{0pt}{}{0}{1})}{a(n-\genfrac{}{}{0pt}{}{0}{1})}
 u_{p,\pm}(z,n) \mp \sum_{m=\genfrac{}{}{0pt}{}{n+1}{-\infty}
}^{\genfrac{}{}{0pt}{}{\infty}{n-1} }
\frac{a_p(n-\genfrac{}{}{0pt}{}{0}{1})}{a(n-\genfrac{}{}{0pt}{}{0}{1})}
K(z,n,m) u_\pm(z,m),
\end{equation}
where
\bea \nn
K(z,n,m) &=& \frac{ ((\tau-\tau_p) u_{p,-}(z))(m) u_{p,+}(z,n) 
- u_{p,-}(z,n) ((\tau-\tau_p) u_{p,+}(z))(m)
}{W_p(u_{p,-}(z),u_{p,+}(z))}\\ \nn
&=& \frac{s_p(\lam,n,m+1)}{a_p(m+1)}(a(m) - a_p(m))
+ \frac{s_p(\lam,n,m)}{a_p(m)}(b(m) - b_p(m))\\
&& {}+ \frac{s_p(\lam,n,m-1)}{a_p(m-1)}(a(m-1) - a_p(m-1))
\eea
($W_p(.,..)$ denotes the Wronskian formed with $a_p$ rather than $a$)
to show the existence of solutions $u_\pm(z,.)$ of $\tau u = z u$
satisfying
\begin{equation} \label{asupm}
\lim_{n\to\pm\infty} \exp(\mp\im(q(z))n)|u_\pm(z,n) - u_{p,\pm}(z,n)| =0,
\quad z\in\C.
\end{equation}

Since we are most of the time interested in the case $z\in\sig(H_p)$ we
shall normalize $u_{p,\pm}(\lam,0)=1$ for $\lam\in\sig(H_p)$. In what follows we
will freely use the notation and results found in \cite{bght}, Appendix~B.
In particular, note that we have $\ol{u_{p,\pm}(\lam)}=u_{p,\mp}(\lam)$,
where the bar denotes complex conjugation. Since one
computes
\begin{equation} \label{wruolu}
W(u_\pm(\lam),\ol{u_\pm(\lam)}) = W_p(u_{p,\pm}(\lam),u_{p,\mp}(\lam)) =
\mp\frac{2\I\sin(q(\lam)N)}{s_p(\lam,N)}, \quad \lam\in\sig(H_p)
\end{equation}
($s_p(\lam,n)$ is the solution of $\tau_p u = z u$ corresponding to the initial
condition $s(\lam,0)=0$, $s_p(\lam,1)=1$) we conclude that $u_\pm(\lam)$,
$\ol{u_\pm(\lam)}$ are linearly independent for $\lam$ in the interior
of $\sig(H_p)$ (if two bands collide at $E$, numerator and denominator of
(\ref{wruolu}) both approach zero when $\lam \to E$ and have a nonzero
limit). Hence we might set
\begin{equation}
u_\pm(\lam,n) = \alpha(\lam) \ol{u_\mp(\lam,n)} +
\beta_\mp(\lam) u_\mp(\lam,n), \quad \lam\in\sig(H_p),
\end{equation}
where
\bea
\alpha(\lam) &=& \frac{W(u_\mp(\lam), u_\pm(\lam))}{W(u_\mp(\lam),
\ol{u_\mp(\lam)})} = \frac{s_p(\lam,N)}{2\I\sin(q(\lam)N)}
W(u_-(\lam),u_+(\lam)),\\
\beta_\pm(z) &=& \frac{W(u_\mp(\lam)),\ol{u_\pm(\lam)}
}{W(u_\pm(\lam), \ol{u_\pm(\lam)})} = \pm
\frac{s_p(\lam,N)}{2\I\sin(q(\lam)N)} W(u_\mp(\lam),\ol{u_\pm(\lam)}).
\eea
The function $\alpha(\lam)$ can be defined for all $\lam\in\C\bs \{ E_{p,j}\}$.
Note that we have
\begin{equation} \label{propalbe}
|\alpha(\lam)|^2 = 1 + |\beta_\pm(\lam)|^2 \mbox{ and }
\ol{\beta_\pm(\lam)} = -\beta_\mp(\lam).
\end{equation}
Using (\ref{voltseq}) one can also show
\begin{equation}
W(u_-(\lam),u_+(\lam)) = W_p(u_{p,-}(\lam),u_{p,+}(\lam)) +
\sum_{n\in\Z} u_\pm(\lam,n) ((\tau-\tau_p)u_{p,\mp}(\lam))(n)
\end{equation}
and
\begin{equation}
W(u_\mp(\lam),\ol{u_\pm(\lam)}) = \mp
\sum_{n\in\Z} u_\pm(\lam,n) ((\tau-\tau_p)u_{p,\pm}(\lam))(n).
\end{equation}
We now define the scattering matrix
\begin{equation} \label{scatmat}
S(\lam) = \left( \ba{cc} T(\lam) & R_-(\lam) \\ R_+(\lam) & T(\lam) \ea \right),
\quad \lam\in\sig(H_p)
\end{equation}
of the pair $(H,H_p)$, where $T(\lam) = \alpha(\lam)^{-1}$ and
$R_\pm(\lam) = \alpha(\lam)^{-1} \beta_\pm(\lam)$. The matrix $S(\lam)$ is
easily seen to be unitary since by (\ref{propalbe}) $|T(\lam)|^2 + |R_\pm(\lam)|^2
=1$ and $T(\lam)\ol{R_+(\lam)} = -\ol{T(\lam)} R_-(\lam)$.

The quantities $T(\lam)$ and $R_\pm(\lam)$ are called transmission and
reflection coefficients respectively. The following equation further
explains this notation:
\begin{equation}
T(\lam) u_\pm(\lam,n) = \left\{\ba{ll}
T(\lam) u_{p,\pm}(\lam,n), & n\to\pm\infty\\ \\
u_{p,\pm}(\lam,n) + R_\mp(\lam) u_{p,\mp}(\lam,n), & n\to\mp\infty
\ea\right., \lam\in\sig(H_p).
\end{equation}
Clearly (\ref{rfllupm}) implies $R_\pm(\lam)=0$, explaining the term {\em
reflectionless} in the previous section. The quantities $T(\lam)$ and
$R_\pm(\lam)$ can be expressed in terms of
$\ti{m}_\pm(z)= \ti{m}_\pm(z,0)$ as follows
\bea
T(\lam) &=& \frac{\ol{u_\pm(\lam,0)}}{u_\mp(\lam,0)} \:
\frac{2\I\im(\ti{m}_\pm(\lam + \I 0))}{\ti{m}_-(\lam + \I 0) + \ti{m}_+(\lam +
\I 0)},\\
R_\pm(\lam) &=& - \frac{\ol{u_\pm(\lam,0)}}{u_\pm(\lam,0)} \:
\frac{\ti{m}_\mp(\lam + \I 0) + \ol{\ti{m}_\pm(\lam +
\I 0)}}{\ti{m}_-(\lam + \I 0) + \ti{m}_+(\lam +
\I 0)}, \quad \lam\in\sig(H_p).
\eea
In addition, one verifies
\bea\nn
g(\lam + \I 0,n) &=& \frac{u_-(\lam,n) u_+(\lam,n)}{W(u_-(\lam),u_+(\lam))}
= T(\lam) \frac{s_p(\lam,N)}{2\I\sin(q(\lam)N)} u_-(\lam,n) u_+(\lam,n)\\
&=& \frac{s_p(\lam,N)}{2\I\sin(q(\lam)N)} |u_\pm(\lam,n)|^2 \Big(
1 + R_\pm(\lam) \frac{u_\pm(\lam,n)}{\ol{u_\pm(\lam,n)}}\Big),\:\:
\lam\in\sig(H_p).
\eea

Construct the list $( E_j )_{j=0}^{2M+1}$ by taking all $E_{p,j}$ plus
two copies of each eigenvalue of $H$. We can assume
$E_0 \le E_1 < E_2 \le \cdots < E_{2M} \le E_{2M+1}$ and equality holds
if and only if $E_{2j}=E_{2j+1}$ is an eigenvalue of $H$.
Define the Dirichlet eigenvalues $\mu_j(n)$ associated with each
spectral gap $(E_{2j+1},E_{2j+2})$ as in (\ref{defdie}).
Then we infer 
\bea \nn
\xi(\lam,n) &=& \frac{1}{2}\chi_{(E_0,E_\infty)}(\lam)+ \frac{1}{2}\sum_{j=1}^M
\Big( \chi_{(E_{2j-1},\mu_j(n))}(\lam)-\chi_{(\mu_j(n),E_{2j})}(\lam) \Big)\\
&& {}+ \chi_{(E_\infty,\infty)}(\lam) + \frac{1}{\pi}\arg\Big(1 + R_\pm(\lam)
\frac{u_\pm(\lam,n)}{\ol{u_\pm(\lam,n)}}\Big) \chi_{\sig(H_p)}(\lam)
\eea
since we have
\begin{equation}
\xi(\lam,n)= \frac{1}{2} + \frac{1}{\pi}\arg\Big(1 + R_\pm(\lam)
\frac{u_\pm(\lam,n)}{\ol{u_\pm(\lam,n)}}\Big), \: \lam\in\sig(H_p).
\end{equation}
Hence we obtain from (\ref{gentracef})
\bea \nn
b^{(\ell)}(n) &=& \frac{1}{2} \sum_{j=0}^{2M+1} E_j^\ell -
\sum_{j=1}^{M-1} \mu_j(n)^\ell\\
&&{}+ \frac{\ell}{\pi}\int_{\sig(H_p)} \lam^{\ell-1} \arg\Big(1 + R_\pm(\lam)
\frac{u_\pm(\lam,n)}{\ol{u_\pm(\lam,n)}}\Big) d\lam,
\eea
and in the special case $\ell=1$
\bea \nn
b(n) &=& \frac{1}{2} \sum_{j=0}^{2M+1} E_j - \sum_{j=1}^{M-1} 
\mu_j(n)\\ \label{trfscbr}
&&{}+ \frac{1}{\pi}\int_{\sig(H_p)} \arg\Big(1 + R_\pm(\lam)
\frac{u_\pm(\lam,n)}{\ol{u_\pm(\lam,n)}}\Big) d\lam.
\eea
The analog of (\ref{trfscbr}) in the case of Schr\"odinger operators with constant
background and no eigenvalues was first derived in \cite{detr}. The general case
for Schr\"odinger operators can be found in \cite{ghsas}. For further trace
formulas in the constant background case, in particular in connection with the Toda
lattice, we refer the reader to \cite{caopt}, \cite{conl}.

\br
If $R_\pm(\lam)=0$ then $H$ can be obtained from $H_p$ by inserting the
corresponding number of eigenvalues using the double commutation method
provided in \cite{gtjc} since this transformation is easily seen to preserve  the
reflectionless property.
\er


\section*{Acknowledgments}

I thank the referee for making several valuable suggestions.



\begin{appendix}


\section{Herglotz functions}
\setcounter{equation}{0}
\setcounter{thm}{0}




The results stated in this section can be found in \cite{ad} (see also
\cite{ar}).

We set $\C_\pm=\{z \in\C |\ \pm\im(z)>0\}$. A function $F:\C_+\to\C_+$ is called
a Herglotz function (sometimes also Pick or Nevanlinna--Pick function),
if $F$ is analytic in $\C_+$. For convenience one usually defines $F$
on $\C_-$ by $F(\ol{z})=\ol{F(z)}$.

Herglotz functions can be characterized by 
\bth \label{hergl} 
$F$ is a Herglotz function if and only if
\begin{equation} \label{heru}
F(z)=a+b\,z+\int_\R \Big( \frac{1}{\lam-z}-\frac{\lam}{1+\lam^2} \Big)
\,d\rho(\lam), \quad z\in\C_+,
\end{equation}
where $a=\re\big(F(\I)\big)\in\R,\  b\ge 0$, and $ \rho $ is a measure 
on $ \R $ which satisfies $\int_\R (1+\lam^2)^{-1} d\rho(\lam) < \infty$.
\eth

Let $\ln(z)$ be defined such that $\ln(z)=\ln|z|+\I\arg(z)$,
$-\pi < \arg(z)\le \pi$. Then $\ln(z)$ is holomorphic and
$\im \big(\ln(z)\big) >0$ for $z\in\C_+$. Hence $\ln(z)$ is a Herglotz
function.

The sum of two Herglotz functions is again a Herglotz function, similarly the 
composition of two Herglotz functions is Herglotz. In particular, if $F(z)$ is a
Herglotz function, the same holds for $\ln\big( F(z) \big)$ and $-\frac{1}{F(z)}$.
Thus, using the  representation (\ref{heru}) for $\ln\big( F(z) \big)$, we get
another representation for $F(z)$.
\bth \label{thmhexp}
(i). $F$ is a Herglotz function if and only if it has the representation 
\begin{equation} \label{heex}
F(z) = \exp\Big\{c+\int_{\R} \Big( \frac{1}{\lam-z}-\frac{\lam}{1+\lam^2} 
\Big) \, \xi(\lam)\, d\lam \Big\}, \quad z\in \C_+, 
\end{equation}
where $c=\ln |F(\I)| \in\R$, $\xi\in L^1(\R, (1+\lam^2)^{-1}d\lam)$
real-valued and $\xi$ is not identically zero. Moreover, 
\begin{equation}
\xi(\lam) = \frac{1}{\pi}\lim_{\eps\downarrow 0}\im \Big( \ln\big(
F(\lam+\I\eps)\big) \Big)= \frac{1}{\pi}\lim_{\eps\downarrow
0}\arg\big(F(\lam+\I\eps) \big) 
\end{equation}
for a.e. $\lam\in\R$, and $0 \le \xi(\lam) \le 1$ for a.e. $\lam\in\R$.
Here $-\pi<\arg(F(\lam+\I\eps))\le\pi$ according to the definition of
$\ln(z)$.\\
(ii). Fix $n \in \N$ and set $\xi_+(\lam)= \xi(\lam)$, $\xi_-(\lam)=
1-\xi(\lam)$. Then
\begin{equation}
\int_\R |\lam|^n \xi_\pm(\lam) d\lam< \infty
\end{equation}
if and only if
\begin{equation}
\int_\R |\lam|^n \, d\rho(\lam) < \infty \quad\text{and}\quad
\lim_{z \to\I\infty} \pm F(z) = \pm a \mp \int_{\R}
\frac{\lam d\rho(\lam)}{1+\lam^2} > 0.
\end{equation}
(iii). We have
\begin{equation}
F(z) = \pm 1 + \int_{\R} \frac{d\rho(\lam)}{\lam-z} \quad\text{with}\quad
\int_{\R} d\rho(\lam) < \infty
\end{equation}
if and only if
\begin{equation}
F(z) = \pm\exp\Big( \pm\int_{\R} \xi_\pm(\lam)
\frac{d\lam}{\lam-z}\Big) \quad\text{with}\quad\xi_\pm \in L^1(\R)
\end{equation}
($\xi_\pm$ from above). In this case
\begin{equation}
\int_\R d\rho(\lam) = \int_\R \xi_\pm(\lam) d\lam.
\end{equation}
\eth


\end{appendix}

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\end{document}
