%% @texfile{ %% filename="IsoJac.tex", %% version="1.1", %% date="2-9-95", %% cdate="19950902", %% filetype="LaTeX2e", %% journal="Comm. Math. Phys. 181, 631-645 (1996)", %% doi="10.1007/BF02101290", %% copyright="Springer". %% } \documentclass{amsart} %\newcommand{\href}[2]{ #2 } \usepackage{hyperref} \theoremstyle{plain} \newtheorem{exam}{Example}[section] \newtheorem{thm}[exam]{Theorem} \newtheorem{lem}[exam]{Lemma} \newtheorem{rem}[exam]{Remark} \newtheorem{hyp}[exam]{Hypothesis} \renewcommand{\Re}{{\rm Re}} \renewcommand{\Im}{{\rm Im}} \numberwithin{equation}{section} \title{On Isospectral Sets of Jacobi Operators} \author{F. Gesztesy} \address{Department of Mathematics\\ University of Missouri\\ Columbia, MO 65211, USA} \email{gesztesyf@missouri.edu} \author{M. Krishna} \address{Institute of Mathematical Sciences\\ Taramani, Madras 600 113, India} \email{krishna@imsc.ernet.in} \author{G. Teschl} \address{Department of Mathematics\\ University of Missouri\\ Columbia, MO 65211, USA} \curraddr{G. Teschl: Institut f\"ur Mathematik\\ Strudlhofgasse 4\\ 1090 Wien\\ Austria} \email{Gerald.Teschl@univie.ac.at} \urladdr{http://www.mat.univie.ac.at/\string~gerald/} \subjclass{Primary 47B39, 34B20; Secondary 34A55, 39A10} \keywords{Spectral theory, Jacobi operators, isospectral operators} \begin{document} \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} \maketitle \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} 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) & = a(n)^{-2} 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}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 ]|x0,\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}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}\in\!\!\!\!\! / \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}\in\!\!\!\!\! / \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. \begin{thebibliography}{99} \bibitem{Akh} N.I. Akhiezer, The Classical Moment Problem, Oliver and Boyd, Edinburgh, 1965. \bibitem{AK1} A.J. Antony and M. Krishna, Almost periodicity of some Jacobi matrices, Proc. Indian Acad. Sci. (Math. Sci.) {\bf 102}, 175--188 (1992). \bibitem{AK2} A.J. Antony and M. 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