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\newtheorem{thm}[exam]{Theorem} \newtheorem{lem}[exam]{Lemma}
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\title{On Isospectral Sets of Jacobi Operators}



\author{F. Gesztesy\thanks{E-mail:
mathfg@mizzou1.missouri.edu}\\
Department of Mathematics\\ University of
Missouri\\ Columbia, MO
65211, USA \and M. Krishna\thanks
{E-mail: krishna@imsc.ernet.in}\\
Institute of Mathematical Sciences\\ Taramani,
Madras 600 113, India
\and G. Teschl\thanks{E-mail:
gerald@iram.rwth-aachen.de}\\
Department of Mathematics\\ University of
Missouri\\ Columbia, MO
65211, USA}
\date{}

\begin{document}

\maketitle

\begin{abstract} We consider the inverse spectral
problem for a class
of reflectionless bounded Jacobi operators with
empty singularly
continuous spectra.  Our spectral hypotheses admit
countably many
accumulation points in the set of eigenvalues as well
as in the set of
boundary points of intervals of absolutely continuous
spectrum. The
corresponding iso\-spectral set of Jacobi operators
is explicitly
determined in terms of Dirichlet-type data.
\end{abstract}

\section{Introduction}

The principal aim of this paper is to study certain
bounded
self-adjoint Jacobi operators whose inverse spectral
theory and
isospectral class can be characterized explicitly.

Since the literature on inverse spectral theory for
Jacobi operators
(especially in the periodic and short-range scattering
case) is rather
extensive, we confine ourselves to a brief  account
of those results
which are close in spirit to our approach. In this
context, the use of
auxiliary spectral problems of the Dirichlet-type in
connection with
either the moment problem or the algebro-geometric
approach to
(quasi-)periodic finite-gap Jacobi operators, comes to
mind first. Dirichlet spectra and the moment
problem were first combined in the pioneering work by
Kac and van Moerbeke \cite{KvM1}, \cite{KvM2},
\cite{vM1}. The Jacobi inversion problem in connection
with Dirichlet divisors appear- \linebreak ed in
Date and Tanaka
\cite{DT} (see also \cite{T}) and simultaneously in
Dubrovin, \mbox{Matveev}, and Novikov
\cite{DN} with further developments in \cite{McvM},
\cite{vM2},
\cite{vMM}. (The algebro-geometric method is presented
in great detail in \cite{BBEIM}.) A complete
algebro-geometric treatment of Toda and Kac-van
Moerbeke hierarchies can be found in
\cite{BGHT}; the isospectral torus of quasi-periodic
Jacobi operators is explicitly realized in \cite{GT}.
The next step involved extensions to certain almost
periodic and random stationary Jacobi operators with
infinitely many gaps in their spectrum. Based on
fundamental contributions by Levitan \cite{Lev},
followed by Kotani and Krishna
\cite{KK} and Craig
\cite{Craig} in the case of Schr\"odinger operators,
Antony and Krishna \cite{AK1}, \cite{AK2} and
especially Sodin and Yuditski\u\i
\cite{SY1}, \cite{SY2} characterized the inverse
spectral problem for certain classes of almost periodic
Jacobi operators by solving an infinite dimensional
Jacobi inversion problem. In the random case Carmona
and Kotani \cite{CK} provided necessary and sufficient
conditions for a Herglotz function to be the
expectation of a half-line Weyl $m$-function for a
class of random stationary Jacobi operators. These
extensions use elements of harmonic analysis, in
particular, Herglotz properties of diagonal Green's
functions and their boundary behavior on the real line.

In this paper we consider a different class of bounded
Jacobi operators $H$ with infinitely many gaps in
their spectrum. More precisely, we assume that the
spectrum $\Sigma $ of $H$ satisfies
\begin{equation}\label{eq1.1} \Sigma ={\mathbb
R}\backslash
\bigcup_{j\in J_0\cup \{\infty \}}\rho
_j,\end{equation} where $J\subseteq
{\mathbb N}$, $J_0=J\cup \{ 0\}$,
\begin{equation}\begin{split}\label{eq1.2}
&\rho_0=(-\infty ,E_0),\quad \rho_\infty =
(E_\infty ,\infty),\\ & E_0\leq E_{2j-1}
<E_{2j}\leq E_\infty ,\quad \rho_j=(E_{2j-1},E_{2j}),
\quad j\in J,\\ &-\infty <E_0<E_\infty <\infty ,
\rho_j\cap \rho_k=\emptyset \text{ for } j\neq k
\end{split}\end{equation}
such that the set $\mathcal{A}$ of all
accumulation points of
$\{E_{2j-1}, E_{2j}\}_{j\in J}$ is countable, that
is,
\begin{equation}\label{eq1.3}
\mathcal{A}=\{E_{2j-1}, E_{2j}\}_{j\in J}'
\quad \text{is countable}.
\end{equation}
(Here $A'$
denotes the derived set of $A\subset {\mathbb R}$,
i.e., the set of all accumulation points of $A$.)

Hypotheses \eqref{eq1.1}--\eqref{eq1.3} include
situations such as the class of algebro-geometric
finite-gap Jacobi operators on one hand and Jacobi
operators with pure point spectrum with at most
countably many accumulation points on the other hand.
Our methods integrate the use of trace formulas and
Herglotz functions (as recently outlined by Gesztesy
and Simon \cite{GS1}, \cite{GS2}) and the moment
problem. In particular, the isospectral set of all
Jacobi operators with spectrum
$\Sigma $ satisfying \eqref{eq1.1}--\eqref{eq1.3} is
explicitly determined in Theorem~\ref{t4.3}.

\section{Preliminaries}

In this section we recall some of the basic facts on
Jacobi operators needed in Sections 3 and 4. Detailed
accounts of this material can be found, for instance,
in \cite{Ju}, Ch. VII,
\cite{CL}, Ch. III, \cite{GT}, Appendices A--D.

Let $\{a(m)>0\}_{m\in {\mathbb Z}}$, $\{b(m)\}_{m\in
{\mathbb Z}}\in
\ell^\infty_{\mathbb{R}}(\mathbb{Z})$ be bounded
real-valued \linebreak sequences and introduce the bounded
self-adjoint Jacobi operator $H$ in
$\ell^2(\mathbb{Z})$ by
\begin{equation}\label{eq2.1}(Hf)(m)=(\tau f)(m),\quad
f=\{f(m)\}_{m\in \mathbb{Z}}\in
\ell^2(\mathbb{Z}),\end{equation} with the difference
expression $\tau $ defined by
\begin{equation}\label{eq2.2} (\tau
f)(m)=a(m)f(m+1)+a(m-1)f(m-1)+b(m)f(m),\quad m\in
\mathbb{Z}.\end{equation} The Green's function
$G(z,m,n)$ associated with the resolvent $(H-z)^{-1}$
of $H$ then can be represented by
\begin{equation}\begin{split}\label
{eq2.3}G(z,n,n')&=(\delta
(n),(H-z)^{-1}\delta (n'))\\ &=
W(u_-(z),u_+(z))^{-1}\left\{
\begin{array}{ll}u_-(z,n)u_+(z,n'), & n\leq n'\\
u_+(z,n)u_-(z,n'),& n\geq n'\end{array}\right. ,\\
&\hspace*{5cm} z\in \mathbb{C}\backslash \sigma (H),\quad
n, n'\in \mathbb{Z}.\end{split}\end{equation} Here
$\delta (n)=\{\delta _{m,n}\}_{m\in {\mathbb Z}}$,
$\sigma (.)$ abbreviates the spectrum,
$u_\pm (z,.)$ are Weyl solutions satisfying
\begin{equation}\label{eq2.4}\tau u_\pm (z)=zu_\pm
(z),\quad u_\pm (z,.)\in \ell^2((m_0,\pm
\infty )\cap \mathbb{Z}),\quad m_0\in \mathbb{Z},\quad
z\in
\mathbb{C}\backslash (H),\end{equation} and
$W(f,g)(n)$ denotes the Wronskian
\begin{equation}\label{eq2.5}
W(f,g)(m)=a(m)[f(m)g(m+1)-f(m+1)g(m)],\quad m\in
\mathbb{Z}.
\end{equation}
Since $H$ is in the limit
point case at $\pm
\infty $, $u_\pm (z,.)$ are unique up to constant
multiples. They can be chosen to be holomorphic for
$z\in
\mathbb{C}\backslash\sigma_{ess}(H)$
($\sigma_{ess}(.)$ denoting the essential spectrum).

Next, denote by $H_{\pm ,n}$, $n\in \mathbb{Z}$ the
restrictions of $H$ to $\ell^2([n\pm 1, \pm
\infty )\cap \mathbb{Z} )$ with a Dirichlet boundary
condition at the point $n\in \mathbb{Z}$, that is,
\begin{equation}\label{eq2.6} (H_{\pm ,n}f)(m)=(\tau
f)(m),\quad f\in
\{ g\in \ell^2 ([n\pm 1,\pm \infty )\cap \mathbb{Z})|
g(n)=0 \} .\end{equation} The Weyl $m$-functions
$m_{\pm ,n}(z)$ associated with $H_{\pm ,n}$ are then
given by
\begin{equation}\begin{split}\label{eq2.7}m_{\pm
,n}(z) & = (\delta (n\pm 1),(H_{\pm ,n} -z)^{-1}\delta
(n\pm 1))\\  & = \left\{
\begin{array}{l} -u_+(z,n+1)/[a(n)u_+(z,n)]\\
-u_-(z,n-1)/[a(n-1)u_-(z,n)]\end{array}
\right. .\end{split}\end{equation} Introducing the
abbreviations,
\begin{align}\label{eq2.8} g(z,n) & = G(z,n,n),\\
\label{eq2.9} h(z,n) & = 2a(n)G(z,n,n+1)-1,\\
\label{eq2.10} M_+(z,n) & = a(n)^2m_{+,n}(z),\\
\label{eq2.11} M_-(z,n) & =
a(n-1)^2m_{-,n}(z)+z-b(n),\end{align} one infers
\begin{align}\label{eq2.12} g(z,n) & =
-[M_+(z,n)+M_-(z,n)]^{-1},\\
\label{eq2.13} g(z,n+1) & =
M_+(z,n)M_-(z,n)[M_+(z,n)+M_-(z,n)]^{-1},\\
\label{eq2.14} h(z,n) &=
[M_+(z,n)-M_-(z,n)][M_+(z,n)+M_-(z,n)]^{-1}.\end{align}
We recall that for all $n\in \mathbb{Z}$, $g(.,n)$ and
$M_\pm (.,n)$ are Herglotz functions (in contrast to
$h(.,n)$). Finally, if $\sigma_p(.)$ abbreviates the
point spectrum (i.e., the set of eigenvalues) one
obtains
\begin{align}\label{eq2.15} E&\in \sigma _p(H)\text{
if and only if }-\lim_{\epsilon
\downarrow 0}i\epsilon [g(E+i\epsilon
,n)+g(E+i\epsilon ,n+1)]>0,\\
\label{eq2.16}\mu &\in \sigma _p(H_{\pm ,n})\text{ if
and only if }-\lim_{\epsilon \downarrow 0}i\epsilon
M_\pm (\mu+i\epsilon ,n )>0.\end{align}

\section{The Direct Spectral Problem}

In this section we discuss the direct spectral problem
for a certain class of reflectionless bounded Jacobi
operators.

In order to set the stage we first recall that $g(z,n)$
admits an exponential Herglotz representation
\cite{AD} of the form
\begin{equation}\label{eq3.1} g(z,n)=|g(i,n)|\exp
\left\{\int_{\mathbb{R}}\left[\frac{1}{\lambda
-z}-\frac{\lambda}{1+\lambda^2}\right]\xi (\lambda
,n)d\lambda
\right\},\end{equation} where, for all $n\in
\mathbb{Z}$,
\begin{align}\label{eq3.2} &0\leq \xi (\lambda , n)\leq
1 \text{ for a.e. }\lambda \in
\mathbb{R},\\
\label{eq3.3}& \xi (\lambda ,n)=\lim_{\epsilon
\downarrow 0}\Im\{\ln [g(\lambda +i\epsilon
,n)]\}\text{ for a.e. } \lambda \in
\mathbb{R}.\end{align}
One can normalize $\xi (\lambda
,n)$ by demanding
\begin{equation}\label{eq3.4}
\xi (\lambda ,n)=0\text{
for } \lambda <\inf \{ \sigma (H)\}.\end{equation} Our
principal spectral hypothesis on $H$ then reads as
follows.

\begin{hyp}\label{H.4.1} (i). $H$ is a bounded
self-adjoint Jacobi operator. Hence its spectrum can be
written as
\begin{equation}\label{eq3.5} \sigma
(H)=\mathbb{R}\backslash
\bigcup_{j\in J_0\cup \{\infty
\}}\rho_j,\end{equation} where $J\subseteq
\mathbb{N}$, $J_0=J\cup \{0\}$,
\begin{equation}\begin{split}\label{eq3.6}
&\rho_0=(-\infty ,E_0),\quad \rho_\infty =(E_\infty ,
\infty ),\\  & E_0\leq E_{2j-1}<E_{2j}\leq
E_\infty, \quad \rho_j=(E_{2j-1},E_{2j}), \quad j\in J,
\\ & -\infty
<E_0<E_\infty <\infty ,\quad \rho_j\cap
\rho_k=\emptyset \text{ for } j\neq
k.\end{split}
\end{equation}
(ii). The set of all
accumulation points of the set $\{E_{2j-1}, E_{2j}\}_{j\in J}$
is assumed to be
countable and denoted by
\begin{equation}\label{eq3.7}
\mathcal{A}=\{\tilde{E}_j\}_{j\in \tilde{J}},\quad
\tilde{J}\subseteq
\mathbb{N}.
\end{equation}
(iii). For all $n\in
\mathbb{Z}$,
\begin{equation}\label{eq3.8} \xi (\lambda,
n)=\frac{1}{2}\text{ for a.e. } \lambda \in
\sigma_{ess}(H).\end{equation}\end{hyp}

We emphasize that the notation employed in
\eqref{eq3.6} implies that
$E_{2\ell}=E_{2k+1}$ for some $k\in J_0\cup \{\infty
\}$ whenever
$E_{2\ell}\in \sigma_d(H)$ ($\sigma _{d}(.)$ abbreviating
the discrete spectrum).

\begin{rem}\label{r3.2}
(i). Since by hypotheses \eqref{eq3.5} and
\eqref{eq3.6}, $\sigma (H)\subset [E_0,E_\infty ]$ is
bounded, the corresponding sequences $\{a(n)>0\}_{n\in
\mathbb{Z}}$ and
$\{b(n)\}_{n\in \mathbb{Z}}\subset \mathbb{R}$
associated with the difference expression
\eqref{eq2.2} are necessarily bounded.\\
(ii). Hypothesis (H.3.1)(i) implies that $\sigma (H)$ is a
countable union of closed intervals (which may
degenerate to points) of the type,
\begin{align}\label{eq3.9}
\sigma (H) &
=\left(\bigcup_{j\in J_0}\Sigma_j\right)\cup
\left(\bigcup_{j\in
\tilde{J}}\tilde{\Sigma}_j\right),\\
\intertext{where}
\label{eq3.10} \Sigma_j & =[E_{2j},E^{(r)}_{2j}],\quad
j\in J_0,\quad
\tilde{\Sigma
}_j=[\tilde{E}_j,\tilde{E}_j^{(r)}],\quad j\in
\tilde{J},\\
\intertext{with}
\label{eq3.11}x^{(r)} & = \inf \{E_n\in [E_0,E_\infty
]|x<E_n\}\text{ for } x\in [E_0,E_\infty].
\end{align}
\end{rem}

Reflectionless conditions such as \eqref{eq3.8} have
been used by a variety of authors for particular cases
such as almost periodic potentials (see, e.g.,
\cite{AK1}, \cite{AK2}, \cite{Craig}, \cite{DeS},
\cite{KK},
\cite{SY1}, \cite{SY2}) and scattering theoretic
situations (cf., e.g.,
\cite{DS}, \cite{GNP}). The following result further
illustrates (H.3.1)(iii).

\begin{lem}\label{l3.3} Suppose $H$ is a bounded Jacobi
operator and
$\Omega \subset \sigma (H)$. Then the following
conditions are equivalent.
\begin{enumerate}
\item[(i).] For all $n\in
\mathbb{Z}$, $\xi (\lambda ,n)=\frac{1}{2}$ for a.e.
$\lambda \in \Omega$.
\item[(ii).] For some $n_0\in \mathbb{Z}$, $n_1\in
\mathbb{Z}\backslash
\{n_0,n_0+1\}$,
$$ \xi (\lambda, n_0)=\xi (\lambda ,n_0+1)=\xi (\lambda
,n_1)=\frac{1}{2}\text{ for a.e }
\lambda \in \Omega.
$$
\item[(iii).] For some $n_0\in \mathbb{Z}$,
$$ M_+(\lambda +i0,n_0)=-\overline{M_-(\lambda
+i0,n_0)}\text{ for a.e. } \lambda \in \Omega.
$$
\end{enumerate}
\end{lem}

\begin{proof} Clearly (i) implies (ii). In order to prove
that (ii) implies (iii) we first recall that $M_\pm
(z,n_0)$, being Herglotz functions, have nontangential
\linebreak limits $z\to \lambda $ for a.e. $\lambda
\in \mathbb{R}$. Next, consider a particular
representation of $u_\pm (z,n,n_0)$ in \eqref{eq2.7}
normalized by $u_\pm (z,n_0,n_0)=1$, $z\in
\mathbb{C}\backslash \sigma (H_{\pm , n_0})$, given by
\begin{equation}\label{eq3.12}
u_\pm
(z,n,n_0)=c(z,n,n_0)\mp a(n_0)^{-1}M_\pm
(z,n_0)s(z,n,n_0).
\end{equation}
Here $c(z,n,n_0)$,
$s(z,n,n_0)$ are solutions of $\tau \psi (z)=z\psi
(z)$, $z\in\mathbb{C}$ defined by
\begin{equation}\label{eq3.13}
s(z,n_0,n_0)=c(z,n_0+1,n_0)=0,\quad
s(z,n_0+1,n_0)=c(z,n_0,n_0)=1,
\end{equation}
in particular, $c(z,n,n_0)$,
$s(z,n,n_0)$ are polynomials with respect to $z$ and
real-valued for $z\in \mathbb{R}$. The requirement $\xi
(\lambda ,m)=\frac{1}{2}$, that is,
$g(\lambda +i0,m)=\overline{-g(\lambda +i0,m)}$ for
a.e. $\lambda \in
\Omega $ then yields upon choosing $m=n_0$,
\begin{equation}\label{eq3.14}
\Re [M_+(\lambda
+i0,n_0)]=-\Re [M_-(\lambda +i0,n_0)]\quad
\text{for a.e. } \lambda\in \Omega
\end{equation}
and
\begin{align}\notag
&[M_+(\lambda
+i0,n_0)M_-(\lambda +i0,n_0)-\overline{M_+(\lambda
+i0,n_0)}\overline{M_-(\lambda +i0,n_0)}]s(\lambda
,m,n_0)^2\\ &\quad -\{\Im [M_+(\lambda +i0,n_0)]-\Im
[M_-(\lambda +i0,n_0)]\}2ia(n_0)c(\lambda
,m,n_0)s(\lambda ,m,n_0)\notag\\ \label{eq3.15}
&\quad=0 \quad \text{ for a.e. }\lambda \in \Omega .
\end{align}
Taking $m=n_0+1$ in \eqref{eq3.15} yields
\begin{align}\label{eq3.16}
&M_+(\lambda
+i0,n_0)M_-(\lambda +i0,n_0)-\overline{M_+(\lambda
+i0,n_0)}\overline{M_-(\lambda +i0,n_0)}\\ &\quad
=0=\Re[M_+(\lambda +i0,n_0)]\{\Im [M_+(\lambda
+i0,n_0)]-\Im [M_-(\lambda +i0,n_0)]\}\notag\\
&\hspace*{10cm} \text{ for a.e. } \lambda \in
\Omega \notag\end{align} since
$s(\lambda,n_0+1,n_0)=1$. Taking $m=n_1$ in
\eqref{eq3.15} finally proves
\begin{equation}\label{eq3.17} \Im [M_+(\lambda
+i0,n_0)]=\Im [M_-(\lambda +i0,n_0)] \quad \text{ for a.e. }
\lambda \in
\Omega\end{equation} and hence (iii) since $c(\lambda ,
n_1,n_0)s(\lambda ,n_1,n_0)\neq 0$ for a.e. $\lambda
\in
\mathbb{R}$. (iii) implies (i) by combining
\eqref{eq2.3},
\eqref{eq3.12}, and the real-valuedness of $c(\lambda
,n, n_0)$ and
$s(\lambda ,n,n_0)$ for $\lambda \in
\mathbb{R}$.\end{proof}

Next we turn to Dirichlet eigenvalues associated with
$\tau $ corresponding to a Dirichlet boundary
condition at $n\in \mathbb{Z}$. Associated with each
spectral gap $\rho_j$ we set
\begin{equation}\label{eq3.18} \mu _j(n)=\sup
\{\{E_{2j-1}\}\cup \{\lambda \in
\rho_j|g (\lambda ,n)<0\}\}\in
\overline{\rho_{j}},\quad j\in J.\end{equation} The
strict monotonicity of $g(\lambda ,n)$ with respect
to $\lambda \in \rho_j$, that is,
\begin{equation}\label{eq3.19}
\frac{d}{d\lambda }g
(\lambda ,n)=\sum_{m\in
\mathbb{Z}}G(\lambda ,n,m)^2>0,\quad \lambda \in
\rho_j,
\end{equation}
then yields
\begin{align}
\begin{array}{ll} g(\lambda, n)<0, \quad & \lambda  \in
(E_{2j-1},\mu_j(n)),\\
\label{eq3.20} g(\lambda , n) >0, & \lambda \in
(\mu_j(n), E_{2j}),\end{array}\quad j\in J.
\end{align}

A more detailed analysis of the exponential Herglotz
representation
\eqref{eq3.1} of $g (z,n)$ then yields

\begin{lem}\label{l3.4}
Assume (H.3.1)(i). Then
\begin{equation}
\begin{split}\label{eq3.21}
g (z,n) & = |g (i,n)|\exp
\left\{\int_{\mathbb{R}} \left[\frac{1}{\lambda
-z}-\frac{\lambda }{1+\lambda^2}\right]\xi (\lambda
,n)d\lambda \right\}\\ & =\frac{-1}{z-E_\infty }\exp
\left\{\int^{E_{\infty}}_{E_0}\frac{\xi (\lambda
,n)d\lambda }{\lambda -z}\right\}\\ &=
\frac{-1}{(z-E_0)^{1/2}(z-E_\infty )^{1/2}}
\prod_{j\in
J}\left[\frac{z-\mu_j(n)}{(z-E_{2j-1})^{1/2}
(z-E_{2j})^{1/2}}\right],
\end{split}\end{equation}
where the square root branch used is defined
by
\begin{equation}\label{eq3.22}
z^{1/2}=|z^{1/2}|\exp [i\arg(z)/2],\quad
-\pi <\arg (z)\leq \pi.
\end{equation}
In particular, denoting by
$\chi_\Omega (.)$ the characteristic function
of the set $\Omega
\subset \mathbb{R}$, one can represent
$\xi (\lambda,n)$ by
\begin{equation}
\begin{split}\label{eq3.23} \xi (\lambda ,n)
&= \frac{1}{2} \left[\chi _{(E_0,\infty )}
(\lambda )+\chi_{(E_\infty
,\infty )}(\lambda )\right]\\ & +
\frac{1}{2}\sum_{j\in J}\left[
\chi_{(E_{2j-1},\infty )}(\lambda )+
\chi_{(E_{2j},\infty )}(\lambda
)-2\chi_{(\mu_j(n),\infty )}
(\lambda )\right]\\ & =
\frac{1}{2}\chi_{(E_0,E_\infty)}(\lambda )+
\frac{1}{2}\sum_{j\in J}
\left[ \chi_{(E_{2j-1},\mu_j(n))}(\lambda
)-\chi_{(\mu_j(n),E_{2j})}(\lambda)\right]\\&+
\chi _{(E_\infty
,\infty)}(\lambda ) \quad \text{for a.e. }
\lambda\in\mathbb{R}.
\end{split}
\end{equation}
\end{lem}

For later purpose we observe that the Laurent
expression of $g(z,n)$
near $\frac{1}{z}=0$,
\begin{equation}\label{eq3.24}
g(z,n)=-\frac{1}{z}-\frac{b(n)}{z^2}+0(z^{-3}),
\end{equation}
combined with \eqref{eq3.21} implies the trace
formula
(cf.  \cite{GS1})
\begin{equation}\label{eq3.25}
b(n) =\frac{1}{2}(E_0+E_\infty
)+\frac{1}{2} \sum_{j\in J}[E_{2j-1}
+E_{2j}-2\mu_j(n)].
\end{equation}
Next, we denote for all $n\in \mathbb{Z}$,
\begin{align}\label{eq3.26}
\gamma_{\pm ,j}(n) & = -\lim_{\epsilon
\downarrow 0}i\epsilon M_\pm
(\mu_j(n)+i\epsilon,n)\geq 0,\quad j\in J,\\
\label{eq3.27}
\gamma_j(n) & =
\lim_{\epsilon \downarrow 0}i\epsilon
g(\mu_j(n)+i\epsilon
,n)^{-1}=\gamma_{+,j}(n)+\gamma_{-,j}(n)\geq 0,
\quad j\in J,\\
\label{eq3.28} \tilde{\gamma }_{\pm ,j}(n) &
=-\lim_{\epsilon
\downarrow 0}i\epsilon M_\pm (\tilde{E}_j+i\epsilon
,n)\geq 0,\quad j\in \tilde{J},\\ \label{eq3.29}
\tilde{\gamma}_j(n) &
= \lim_{\epsilon \downarrow 0}i\epsilon
g(\tilde{E}_j+i\epsilon
,n)^{-1}=\tilde{\gamma}_{+,j}(n)+\tilde{\gamma}_{-,j}
(n)\geq 0,\quad
j\in \tilde{J},
\end{align}
where we used the fact that by the Herglotz
property of $M_\pm (z,n)$, $-g(z,n)^{-1}$, the
limits in
\eqref{eq3.26}--\eqref{eq3.29} exist and take
on nonnegative values (cf.
\eqref{eq3.38} and
\eqref{eq3.39}). Associated with the limits
\eqref{eq3.26}--\eqref{eq3.29} are the following
ones $(n\in
\mathbb{Z})$,
\begin{align}\label{eq3.30}
\sigma _j (n) & =
\begin{cases} \lim\limits_{\epsilon \downarrow 0}
h(\mu _j(n)+i\epsilon
,n)=\frac{\gamma_{+,j}(n)-\gamma_{-,j}(n)}
{\gamma _j(n)}\in
[-1,1] & \text{if}\;  \gamma_j(n)>0\\ 2 &
\text{if}\;
\gamma_j(n)=0\end{cases},\\ \label{eq3.31}
\tilde{\sigma }_j(n) & =
\begin{cases} \lim\limits_{\epsilon \downarrow 0}
h(\tilde{E}_j+i\epsilon
,n)=\frac{\tilde{\gamma}_{+,j}(n)-
\tilde{\gamma}_{-,j}(n)}
{\tilde{\gamma}_j(n)}\in [-1,1] & \text{if}\;
\tilde{\gamma}_j(n)>0\\
2 & \text{if}\;  \tilde{\gamma }_j(n)=0\end{cases}.
\end{align}
The actual value of $\sigma_j(n)$ (resp.
$\tilde{\sigma }_j(n)$) if
$\gamma _j(n)=0$ (resp. $\tilde{\gamma}_j(n)=0$) in
\eqref{eq3.30}
(resp.~\eqref{eq3.31}) is immaterial.  For notational
convenience
later on, we chose a value outside the interval $[-1,1]$
in this case.

We note that
\begin{equation}\label{eq3.32}
\mu_j(n)\not\in
\mathcal{A}\text{ implies } \gamma_j(n)>0.
\end{equation}


Finally, we summarize the direct spectral problem in
the following

\begin{thm}\label{t3.5}
Assume (H.3.1) and let $n\in \mathbb{Z}$. Then
\begin{enumerate}
\item[(i).]
\begin{equation}\label{eq3.33}
\begin{split}
\sigma_p(H_{\pm,n})= & \left\{\mu_j(n)\in \overline{\rho_j}
| \sigma_j(n)\in^{(-1,1]}_{[-1,1)}\right\}_{j\in J}\\
\cup & \left\{\tilde{E}_j\in
\mathcal{A} | \tilde{\sigma}_j(n)\in^{(-1,1]}_{[-1,1)}
\right\}_{j\in \tilde{J}}.
\end{split}
\end{equation}
\item[(ii).] If $\mu_j(n)\in \sigma_p(H)$ and
$\overset{(\sim)}{\gamma}_{+,j}(n)>0$ (resp.
$\overset{(\sim)}{\gamma}_{-,j}(n)>0$) then
$\overset{(\sim)}{\gamma}_{-,j}(n)>0$ (resp.
$\tilde{\gamma
}_{+,j}(n)>0$), that is, if $\mu_j(n)\in
\sigma_p(H)$ then $\mu
_j(n)\in \sigma_p (H_{+,n})$ if and only if
$\mu_j(n)\in
\sigma_p(H_{-,n})$.
\item[(iii).] The following cases may occur:\\[2mm]
$\mu_j(n)\in \rho_j$ and  $\sigma_j(n)\in
\{-1,1\}$,\\ $\mu_j(n)\in \sigma_p(H)$ and
$\sigma_j(n)\in
[-1,1]$ implying that $\mu_j(n) \in
\sigma_p(H_{\pm,n})$,\\
$\mu_j(n) = \tilde{E}_k$ and  $\sigma_j(n)=
\tilde{\sigma}_k(n)$.\\[2mm]
Moreover, we have
\begin{align*} &\overset{(\sim
)}{\sigma} _j(n)\in [-1,1] & \hspace*{-10mm}\text{if}\quad
\lim_{\epsilon
\downarrow 0} i\epsilon g(\overset{(\sim )}
{E}_j+i\epsilon
,n)^{-1}>0,\\ &\overset{(\sim )}{\sigma }
_j(n)=2 & \hspace*{-10mm}\text{if}\quad
\lim_{\epsilon \downarrow 0} i\epsilon g(
\overset{(\sim )}{E}_j+i\epsilon
,n)^{-1}=0.\end{align*}
\item[(iv).]
\begin{align}
\begin{split}\label{eq3.34}\sigma_{ac}(H)
& = \sigma_{ac}(H_{\pm,n})\\ &=\overline{\{\lambda
\in [E_0,E_\infty ] | \xi (\lambda ,n_0)=1/2\}}^{ess}
 \quad \text{for some } n_0\in\mathbb{Z}\\
&=\overline{\bigcup_{\substack{j\in J\\E_{2j}\neq
E^{(r)}_{2j}}}\Sigma_j},\end{split}\\
\label{eq3.35}\sigma_{sc}(H) & = \sigma_{sc}(H_{\pm
,n})=\emptyset,
\end{align}
$\sigma _{ac}(H)$ being of uniform spectral
multiplicity two whereas $\sigma_p(H)$,\linebreak[4]
$\sigma_p(H_\pm,n)$, and $\sigma_{ac}(H_{\pm,n})$ are
all simple. In addition, if $d\nu_{\pm,n}$ denote the
measures associated with the Herglotz representations
of $M_\pm (z,n)$, that is,
\begin{align}\label{eq3.36}
M_+(z,n) & =
\int_{\mathbb{R}}\frac{d\nu_{+,n}(\lambda)}{\lambda-z},
& M_-(z,n) & =
z-b(n)+\int_{\mathbb{R}}\frac{d\nu_{-,n}(\lambda
)}{\lambda -z},
\end{align}
then
\begin{equation}\label{eq3.37}
d\nu_{+,n,ac} = d\nu_{-,n,ac} ,\qquad
d\nu_{\pm,n,sc} = \emptyset .
\end{equation}
$d\nu_{\pm,n}$ are both supported on infinite sets.
(Here
$\sigma_{ac}(.)$, $\sigma_{sc}(.)$ abbreviate
absolutely and singularly continuous spectra,
respectively, $d\nu=d\nu_{ac} \linebreak
+d\nu _{sc}+d\nu_{pp}$
denotes the usual Lebesgue decomposition of measures,
and $\overline{A}^{ess}$ denotes the essential closure of
$A\subset \mathbb{R}$ with respect to Lebesgue
measure, i.e., $\overline{A}^{ess}=\{\lambda \in
\mathbb{R} |m(A\cap(\lambda-\epsilon,\lambda+\epsilon
))>0$ for all
$\epsilon >0\}$, $m(.)$ denoting the Lebesgue
measure.)
\end{enumerate}
\end{thm}

\begin{proof} If $F(z)$ denotes a Herglotz function
with representation
\begin{equation}\label{eq3.38}
F(z)=cz+d+\int_{\mathbb{R}}\left[\frac{1}{\lambda-z}-
\frac{\lambda}{1+\lambda^2}\right] d\omega (\lambda ),
\quad c\geq 0,\quad d\in \mathbb{R},
\end{equation}
then
\begin{equation}\label{eq3.39}
\omega(\{\lambda_0\})=-\lim_{\epsilon
\downarrow
0}i\epsilon F(\lambda_0+i\epsilon ) \quad \text{for all }
 \lambda_0 \in
\mathbb{R}
\end{equation}
yields (i), taking into account
\eqref{eq3.26}--\eqref{eq3.29}.

In order to prove (ii) one can argue as follows.
$\overset{(\sim)}\gamma_{+,j}(n)>0$ implies $\mu_j(n)\in
\sigma_p(H_{+,n})$ by \eqref{eq2.16}, \eqref{eq3.26},
and
\eqref{eq3.27}
and $\overset{(\sim )}\gamma_j(n)>0$ yields
$$ \lim_{\epsilon\downarrow 0}\epsilon
g(\mu_j(n)+i\epsilon ,n)=0 $$ by \eqref{eq3.27} and
\eqref{eq3.29}.
Since $\mu_j(n)\in \sigma_p(H)$, \eqref{eq2.13},
\eqref{eq2.15},
\eqref{eq3.26}, and \eqref{eq3.27} yield
\begin{equation}\label{eq3.40}0<\overset{(\sim )}
{\gamma}_j(n) =
-\lim_{\epsilon \downarrow 0}i\epsilon g(\mu_j(n)
+i\epsilon
,n+1)=\frac{\overset{(\sim )}{\gamma}_{+,j}
\overset{(\sim
)}{\gamma}_{-,j}}{\overset{(\sim )}{\gamma}_{+,j}
+\overset{(\sim
)}{\gamma }_{-,j}}.\end{equation}
Hence
$\overset{(\sim)}{\gamma}_{-,j}>0$ and thus
$\mu_j(n)\in
\sigma_p(H_{-,n})$. Alternatively, one can \linebreak
invoke
the eigenfunction
$u_+(\mu_j(n),m)$ which then satisfies
 $u_+(\mu_j(n),n)=0$,
$u_+(\mu_j(n),.)\in\ell^2(\mathbb{Z})$ since
$\mu_j(n)\in
\sigma_p(H)\cap \sigma _p(H_{+,n})$. The limit
point property of $H$
at $\pm \infty $ then yields $u_-(\mu_j(n),.)
=Cu_+(\mu_j(n),.)$ for
some constant $C$ and again one concludes that
$\mu _j(n)\in
\sigma_p(H_{-,n})$.

(iii) is clear from \eqref{eq3.30} and \eqref{eq3.31}.

Next, define \begin{align}\label{eq3.41}
\Sigma_{\pm,n,sc} & =
\{\lambda \in [E_0,E_\infty ] |\lim_{\epsilon
\downarrow 0}\Im
[m_\pm (\lambda +i\epsilon
,n)]\;\text{exists and equals}\; +\infty\},\\
\begin{split}\label{eq3.42}\Sigma_{sc}&=\{\lambda
\in [E_0,E_\infty
] | \lim_{\epsilon \downarrow 0}\Im [g(\lambda
+i\epsilon
,n_0)+g(\lambda +i\epsilon
,n_0+1)]\\ & \hspace*{3cm}\text{exists and equals}\;
+\infty\}\;\text{for some}\; n_0\in \mathbb{Z}.
\end{split}\end{align}
Then $\Sigma_{\pm,n,sc}$ and $\Sigma_{sc}$ are
minimal supports (cf., e.g.,
\cite{Aron},
\cite{Gil}, \cite{GP}, \cite{Sim1}, \cite{Sim2}) of
$d\nu_{\pm ,n,sc}$ and
$d\nu^{tr}_{sc}$, where
$d\nu^{tr}=d\nu_{1,1}+d\nu_{2,2}$ abbreviates the
trace measure of the $2\times 2$ matrix-valued spectral
measure
$d\nu_{p,q}$, $1\leq p,q \leq 2$ of $H$ (derived
from $g(z,n_0)$,
$g(z,n_0+1)$, $h(z,n_0)$, cf.
\eqref{eq2.12}--\eqref{eq2.14}). By
\eqref{eq2.12} one has
\begin{equation}\label{eq3.43}
-g(z,n)^{-1}=M_+(z,n)+M_-(z,n)\end{equation} and by
the reflectionless property \eqref{eq3.8},
\begin{equation}\label{eq3.44}-\lim_{\epsilon
\downarrow 0}g(\lambda +i\epsilon
,n)^{-1}=2i\lim_{\epsilon
\downarrow 0}\Im [M_\pm (\lambda +i\epsilon
,n)]\;\text{for a.e.}\;  \lambda \in
\sigma_{ess}(H).\end{equation}
Consider
\begin{equation}\label{eq3.45}
\sigma(H)^0=(\bigcup_{j\in
J_0}\Sigma^0_j)\cup (\bigcup_{j\in
\tilde{J}}\tilde{\Sigma}_j^0),\quad
\Sigma_j^0= (E_{2j},E^{(r)}_{2j}),\quad
\tilde{\Sigma}_j^0=(\tilde{E}_j,
\tilde{E}_j^{(r)}),
\end{equation} where $A^0$ denotes the open interior
of $A\subseteq
\mathbb{R}$. Then the representation \eqref{eq3.21}
shows that
$\Sigma_{sc}\cap \sigma (H)^0=\emptyset$. But
$\sigma(H)\backslash
\sigma (H)^0$ is countable by Hypothesis (H.3.1)
and hence
$\sigma_{sc}(H)=\emptyset$.  \eqref{eq3.44} then
also yields
$\Sigma_{\pm ,sc}\cap \sigma (H)^0=\emptyset $ and
$\sigma_{sc}(H_{\pm,n})=\emptyset$, $d\nu_{\pm
,n,sc}=0$ since $\sigma (H_{\pm,n})\backslash
\sigma (H)^0$ is countable as well. Next, we recall
that $(n_0\in \mathbb{Z})$
\begin{align}\label{eq3.46}
\Sigma_{\pm ,ac} & = \{\lambda \in
[E_0,E_\infty ]|0<\lim_{\epsilon \downarrow 0}
\Im [M_\pm (\lambda
+i\epsilon
,n_0)]<\infty \text{ exists}\},\\
\label{eq3.47}\Sigma_{ac} & =
\{\lambda \in[E_0,E_\infty ]|0<\lim_{\epsilon
\downarrow
0}\Im[g(\lambda +i\epsilon ,n_0)]<\infty
\text{ exists}\}\end{align}
are minimal supports of $d\nu_{\pm ,n,ac}$ and
$d\nu_{ac}^{tr}$,
respectively.  By \eqref{eq3.44} one infers
$d\nu_{+,n,ac}=d\nu_{-,n,ac}$ and hence \eqref{eq3.37}.
\eqref{eq3.34} then follows from \eqref{eq3.8},
\eqref{eq3.44},
\eqref{eq3.46}, \eqref{eq3.47}, and Theorem~5.2 of
\cite{GS1} which states
\begin{equation}\label{eq3.48}
\sigma_{ac}(H) =
\overline{\{\lambda \in [E_0,E_\infty ]|0<\xi
(\lambda,n_0)<1\}}^{ess}.\end{equation} Finally, spectral
multiplicity
two on $\sigma_{ac}(H)$ is a consequence of \eqref{eq3.8}
and
\eqref{eq3.44}; $\sigma_p(H)$ is simple since
$H$ is in the limit
point case at $\pm \infty $, and half-line spectra
$\sigma(H_{\pm,n})$ are well-known to be simple.
$d\nu_{\pm ,n}$ are
both supported on infinite sets since $H_{\pm ,n}$ are
defined on the
discrete half-lines $\mathbb{Z}\cap (0,\pm\infty)$.
\end{proof}

That $H,H_{\pm,n}$ have purely absolutely continuous
spectra on
$\sigma(H)^0$ (cf.
\eqref{eq3.45}) also follows from Theorem~3.1 in
\cite{MS}.

\section{The Inverse Spectral Problem}

In this section we describe our principal new result on
the isospectral set of self-adjoint Jacobi operators
satisfying Hypothesis (H.3.1).

We start by introducing the following hypothesis.

\begin{hyp}\label{h4.1}
\begin{enumerate}\item[(i).] Let
\begin{equation}\label{eq4.1} \Sigma
=\mathbb{R}\backslash \bigcup_{j\in J_0\cup\{\infty
\}}\rho _j,\end{equation} where $J\subseteq
\mathbb{N}$,
$J_0=J\cup\{0\}$,
\begin{equation}\begin{split}\label{eq4.2}
&\rho_0=(-\infty ,E_0),\quad\rho_\infty =
(E_\infty ,\infty ),\\
&\rho_j=(E_{2j-1},E_{2j}),\quad E_0\leq
E_{2j-1}<E_{2j}\leq E_\infty , j\in J,\\
& -\infty
<E_0<E_\infty <\infty,\quad \rho_j\cap
\rho_k=\emptyset\;
\text{for}\; j\neq k.\end{split}\end{equation} By
$\Sigma_d$ we denote the set of isolated (discrete)
points of $\Sigma$.

\item[(ii).] The set $\mathcal{A}$ of all accumulation
points of the set
$\{E_{2j-1}, E_{2j}\}_{j\in J}$ is assumed to be
countable and denoted by
\begin{equation}\label{eq4.3} \mathcal{A}
=\{\tilde{E}_j\}_{j\in
\tilde{J}},\quad
\tilde{J}\subseteq \mathbb{N}.\end{equation}

\item[(iii).] We introduce the set $\{\mu_j\in
\overline{\rho_j}\}_{j\in J}$ and define $g(z)$ as in
\eqref{eq3.21}. In addition, we introduce
\begin{equation}\label{eq4.4}
\left\{ (\mu_j,\sigma_j)\in \overline{\rho_j}\times
[-1,1]\right\}_{j\in J},\end{equation} where
\begin{align}\label{eq4.5} &\sigma_j\in \{-1,1\}
\quad\text{if} \quad
\mu_j\in \rho_j,\\ \label{eq4.6} &\sigma_j\in(-1,1)
\quad\text{if}\quad \mu_j\in
\partial\rho_j\cap \Sigma_d.\text{ In this case }\\ &
\mu_j=\mu_k \text{ and } \sigma_j=\sigma_k \text{ for
some } j\neq k\in J.\notag\end{align} Next, we consider
\begin{equation}\label{eq4.7}
\{(\tilde{E}_j,\tilde{\sigma }_j)\in
\mathcal{A}\times \{[-1,1]\cup
\{2\}\}\}_{j\in \tilde{J}},
\end{equation}
where
\begin{equation}\begin{split}\label{eq4.8}
\tilde{\sigma}_j&\in
[-1,1]\text{ if }\lim_{\epsilon\downarrow 0}
i\epsilon g(\tilde{E}_j+i\epsilon )^{-1}>0,\\
\tilde{\sigma}_j & = 2\text{ if } \lim_{\epsilon
\downarrow 0} i \epsilon g(\tilde{E}_j+i\epsilon
)^{-1} =0.\end{split}
\end{equation}
Finally,
\begin{equation}\label{eq4.9}
\text{if }
\mu_j=\tilde{E}_k\text{ for some } j\in J, k\in
\tilde{J},\text{ then }
\sigma_j=\tilde{\sigma}_k.
\end{equation}

\item[(iv).] If $\Sigma =\overline{\Sigma_{d}}$, the
index sets
\begin{equation}\label{eq4.10}
J_\pm =\{j\in J|\sigma_j\in ^{(-1,1]}_{[-1,1)}\}\text{ are
infinite.}
\end{equation}
\end{enumerate}
\end{hyp}

\begin{rem}\label{l4.2} Conditions (i) and (ii) just
reintroduce the necessary notation from Hypothesis
(H.3.1).
\eqref{eq4.4}--\eqref{eq4.8} in condition (iii) takes
care of items (ii) and (iii) in Theorem~\ref{t3.5}. In
particular, the fact that two Dirichlet eigenvalues
must simultaneously hit a point in $\sigma_d(H)$ is
taken into account in \eqref{eq4.6}. If only a single
Dirichlet eigenvalue $\mu_j$ would hit a point $E^*\in
\sigma_d(H)$, then, since
$E^*$ necessarily occurs twice in the product
\eqref{eq3.21}, the term
$[z-\mu_j]/[(z-E^*)^{1/2}(z-E^*)^{1/2}]$
simply drops out and one would have ``lost'' $E^*$. In
other words, such a deformation of $\mu_j(n)$ would be
non\-isospectral. (A detailed account of such
(non)isospectral deformations will appear in
\cite{GST}.) Condition
\eqref{eq4.9} is a consistency requirement and condition
(iv) reflects the fact that we are working with
infinite matrix operators on the discrete half-lines
$\mathbb{Z}\cap (0,\pm \infty )$.
\end{rem}

Given Hypothesis (H.4.1) we define the set of Dirichlet
and accumulation data
\begin{equation}\label{eq4.11}
\mathcal{D}_\Sigma
=\{\{(\mu_j,\sigma_j)\in \overline{\rho_j}\times
[-1,1]\}_{j\in J},
\{\tilde{\sigma }_j\}_{j\in \tilde{J}}|\text{assuming
(H.4.1)}\}.
\end{equation}
The isospectral set of
self-adjoint reflectionless Jacobi operators $H$
satisfying (H.3.1) with $\sigma (H)=\Sigma$ is denoted
by
\begin{equation}\label{eq4.12}
I(\Sigma)=\{\text{Jacobi operators }H
\text{ in }\ell^2(\mathbb{Z})|\sigma(H)=\Sigma
\}.
\end{equation}

\begin{thm}\label{t4.3}
Suppose $\Sigma$ satisfies
(H.4.1). Then the map
\begin{equation}\begin{cases}\label{eq4.13} I(\Sigma)\to
\mathcal{D}_\Sigma\\ H\to \{\{(\mu_j^\circ
,\sigma^\circ_j)\}_{j\in J},
\{\tilde{\sigma}^\circ_j\}_{j\in
\tilde{J}}\},\end{cases}\end{equation} constructed in
Theorem~\ref{t3.5} is a bijection, where
\begin{align}\label{eq4.14}
\sigma(H) & = \Sigma ,\\ \nonumber
\label{eq4.15}\sigma _p(H_{\pm ,n_0}) & =
\{\mu^\circ_j\in
\overline{\rho_j}|\sigma_j^\circ \in
^{(-1,1]}_{[-1,1)}\}_{j\in J}\cup
\{\tilde{E}_j\in \mathcal{A}|\tilde{\sigma}_j^\circ \in
^{(-1,1]}_{[-1,1)}\}_{j\in \tilde{J}}\\
& \hspace*{5cm}\text{for some }n_0\in\mathbb{Z}.
\end{align}


\end{thm}

\begin{proof} We first show that the map
\eqref{eq4.13} is surjective. Fix a point
\begin{equation}\label{eq4.16}
\{\{(\mu^\circ_j,\sigma^\circ_j)\}_{j\in J},
\{\tilde{\sigma }^\circ_j\}_{j\in \tilde{J}}\}\in
\mathcal{D}_\Sigma .\end{equation} We shall construct
a unique Jacobi operator $H\in I(\Sigma )$ satisfying
\eqref{eq4.14} and
\eqref{eq4.15}. Given \eqref{eq4.16}, define
$g(z,n_0)$ as in
\eqref{eq3.21}. Let $\nu_{n_0}$ be the measure in the
Herglotz representation of $-g(z,n_0)^{-1}$, that is
\begin{equation}\label{eq4.17}
-g(z,n_0)^{-1}=z-b(n_0)+\int_{\mathbb{R}}
\frac{d\nu_{n_0}(\lambda
)}{\lambda -z},\end{equation} with
\begin{equation}\label{eq4.18}
b(n_0)=\frac{1}{2}(E_0+E_\infty
)+\frac{1}{2}\sum_{j\in J}[E_{2j-1}+E_{2j}-2\mu_
j^\circ].\end{equation} Next, we split up
$\nu_{n_0}=\nu_{+,n_0}+\nu_{-,n_0}$ as follows. Since
the pure point part of $\nu_{n_0}$ is supported on
$\{\mu_j^\circ\in \overline{\rho_j}\}$ we define
\begin{equation}\label{eq4.19} \nu_{\pm
,n_0}(\{\mu^\circ_j\})=\frac{1}{2}(1\pm
\sigma_j^\circ)\nu_{n_0}(\{\mu_j^\circ
\})\end{equation} and similarly,
\begin{equation}\label{eq4.20}
\nu_{\pm,n_0}(\{\tilde{E}_j\})=\frac{1}{2}(1\pm\tilde{\sigma
}_j^\circ )\nu_{n_0}(\{\tilde{E}_j\}).\end{equation}
(The split up of the pure point part in \eqref{eq4.19}
resembles the one in Theorem~3.6 of \cite{GS2} in the
case of Schr\"odinger operators with purely discrete
spectra.) The absolutely continuous part of
$\nu_{n_0}$ is then split up according to
Lemma~\ref{l3.3}, respectively \eqref{eq3.44}, by
\begin{equation}\label{eq4.21} \nu_{\pm
,n_0,ac}=\frac{1}{2}\nu_{n_0,ac}.\end{equation} We
note that
\begin{equation}\label{eq4.22} \nu_{n_0,sc}=\nu_{\pm
,n_0,sc}=0\end{equation} by the argument following
\eqref{eq3.45}. Next, define
\begin{equation}\label{eq4.23}
a(n_0)=\left[\int_{\mathbb{R}}d\nu_{+,n_0}(\lambda
)\right]^{1/2},\quad
a(n_0-1)=\left[\int_{\mathbb{R}}d\nu_{-,n_0}(\lambda
)\right]^{1/2}\end{equation} and consider the
probability measures
\begin{equation}\label{eq4.24}
\omega_{+,n_0}=a(n_0)^{-2}\nu_{+,n_0},\quad
\omega_{-,n_0}=a(n_0-1)^{-2}\nu_{-,n_0}\end{equation}
(which are both supported on infinite sets).
$\omega_{\pm ,n_0}$ enable one to compute
$H_{\pm ,n_0}$ by the moment approach as outlined, for
instance, in
\cite{Akh}, Ch. 4 and
\cite{Ju}, Ch. 7. One obtains,
\begin{equation}\begin{split}\label{eq4.25}
a(n)&= \int_{\mathbb{R}}\lambda s_\pm(\lambda
,n,n_0)s_{\pm}(\lambda,n+1,n_0)d\omega_{\pm ,n_0}(\lambda
), \quad \pm(n-n_0)\geq \begin{cases} 1 \\ 2 \end{cases},\\
b(n) & = \int_{\mathbb{R}}\lambda s_\pm(\lambda ,n,n_0)^2
d\omega_{\pm ,n_0}(\lambda ),\quad \pm(n- n_0)\geq 1,
\end{split}\end{equation}
where $s_{\pm}(\lambda, n, n_0)$, $\pm(n-n_0) \geq 1$ are
polynomials (of degree $\pm(n-n_0)$) orthonormal with respect
to $d\omega_{\pm, n_0}(\lambda)$. This determines
$H$ and
\eqref{eq4.15}. Introducing
\begin{equation}\begin{split}\label{eq4.26} M_+(z,n_0)
& =
a(n_0)^2\int_{\mathbb{R}}\frac{d\omega_{+,n_0}(\lambda
)}{\lambda -z},\\
M_-(z,n_0)&=z-b(n_0)+a(n_0-1)^2\int_{\mathbb{R}}
\frac{d\omega_{-,n_0}(\lambda )}{\lambda
-z},\end{split}\end{equation} one verifies
\eqref{eq4.14} using \eqref{eq2.12},
\eqref{eq2.14}, and \eqref{eq2.15}. It remains to show
that the map
\eqref{eq4.13} is injective. Suppose $H_1\in I(\Sigma
)$ and $H_2\in I(\Sigma )$ are both mapped to the same
point in \eqref{eq4.16}. Then one infers $\nu_{\pm
,1,n_0}=\nu_{\pm ,2,n_0}$ and
$b_1(n_0)=b_2(n_0)$ (where, in obvious notation,
$\nu_{\pm, j,n_0}$ and $b_j$ refer to
$H_j,j=1,2$) and hence $H_1=H_2$.\end{proof}

We conclude with a simple example illustrating an
explicit construction to the effect that an
accumulation point of eigenvalues of $H$ may or may not
be an eigenvalue of $H$.

\begin{exam}\label{ex4.4}
Suppose $H$ satisfies
(H.3.1), $H$ has pure point spectrum only, and
$\mathcal{A} \neq \emptyset$. Let
$\tilde{E}_{j_0}\in \mathcal{A}$ and define
\begin{align}\label{eq4.27}
\tilde{\gamma }_{j_0} &
=\lim_{\epsilon
\downarrow 0} i\epsilon g (\tilde{E}_{j_0}+i\epsilon
,0)^{-1}\\
\intertext{and}
\label{eq4.28} g_\delta (z,0) & = -[-g(z,0)^{-1}-(\delta
-\tilde{\gamma}_{j_0})(z-\tilde{E}_{j_0})^{-1}]^{-1},\quad
\delta
\geq 0.
\end{align}
Then
$\tilde{\gamma }\geq 0$ and $g _\delta $ is a Herglotz
function corresponding to a pure point measure in its
representation of the type \eqref{eq3.38}. Computing
the zeros
$\mu_{\delta,j}$ of $g_\delta (z,0)$ and choosing
$\sigma_{\delta,j}$,
$\tilde{\sigma }_{\delta,j}\in [-1,1]\times \{2\}$
according to (H.4.1) yields a corresponding Jacobi
operator $H_\delta $ by Theorem~\ref{t4.3}. Since
\begin{equation}\label{eq4.29} \lim_{\epsilon
\downarrow 0}i\epsilon g_\delta
(\tilde{E}_{j_0}+i\epsilon ,0)^{-1}=\delta,
\end{equation}
one obtains the following case distinctions.
\begin{equation*}
\begin{split} \text{(i).} &
\quad\delta =0, \text{ then }
\tilde{E}_{j_0}\nin
\sigma_p(H_{\delta,\pm}). \\
\text{(ii).} & \quad\delta  >0,\: \tilde{\sigma
}_{\delta ,j_0}\in
\{\pm 1\},\quad
\text{then } \tilde{E}_{j_0}\in \sigma_p(H_{\delta
,\tilde{\sigma }_{\delta,j_0}}),\quad
\tilde{E}_{j_0}\nin \sigma_p(H_\delta). \\
\text{(iii).} & \quad\delta >0, \:\tilde{\sigma }_{\delta
,j_0}\in (-1,1),\quad \text{then }
\tilde{E}_{j_0}\in \sigma_p(H_{\delta ,\pm })\cap
\sigma _p(H_\delta ).\end{split}\end{equation*} Case
(i) is clear (in this case $\tilde{E}_{j_0}$ may or may
not belong to $\sigma_p(H)$). Case (ii) follows from
Theorem~\ref{t3.5} (ii). In case (iii) one has
$\lim_{\epsilon
\downarrow 0}\epsilon g_\delta
(\tilde{E}_{j_0}+i\epsilon ,0)=0$ but
$-\lim_{\epsilon
\downarrow 0}i\epsilon g_\delta
(\tilde{E}_{j_0}+i\epsilon ,1)>0$ in analogy to
\eqref{eq3.40}. $\tilde{E}_{j_0}\in \sigma_p(H_\delta
)$ then follows from
\eqref{eq2.15}.\end{exam}

\section*{Acknowledgements.}

F.G. would like to thank Barry Simon for numerous
discussions and joint work on inverse spectral problems
which helped to shore up the foundations for this paper.
M.K. wishes to thank Walter Craig for discussions and
the Departments of Mathematics at Brown University and
the University of Missouri-Columbia for an invitation
which made this work possible.

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