%% @texfile{ %% filename="diss.tex", %% version="1.0", %% date="12.12.95", %% cdate="19951212", %% filetype="LaTeX", %% journal="PhD Thesis, University of Missouri, 1995", %% copyright="Copyright (C) G.Teschl". %% } \documentclass[12pt]{report} \usepackage{latexsym} \author{Gerald Teschl} \title{Spectral Theory for Jacobi Operators} \setlength{\oddsidemargin}{.4in} \setlength{\evensidemargin}{.4in} \setlength{\textheight}{8.6in} \setlength{\topmargin}{0.2in} \setlength{\textwidth}{6in} \setlength{\footskip}{0.4in} %\renewcommand{\baselinestretch}{2} \renewcommand{\contentsname}{Table of Contents} %%%%%%%%%THEOREMS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\bth}{\begin{thm}} \newcommand{\eth}{\end{thm}} \newcommand{\bl}{\begin{lem}} \newcommand{\el}{\end{lem}} \newcommand{\bk}{\begin{coro}} \newcommand{\ek}{\end{coro}} \newcommand{\bh}{\begin{hypo}} \newcommand{\eh}{\end{hypo}} \newcommand{\br}{\begin{rem}} \newcommand{\er}{\end{rem}} \newcommand{\bpf}{Proof. } \newcommand{\epf}{\nolinebreak{\hspace*{1mm}\hfill$\Box$} \vspace{2mm}} %%%%%%%%%%%%%%FONTS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \font\twelveeuf=eufm10 scaled 1200 \font\nineeuf=eufm9 \font\seveneuf=eufm7 \newfam\euffam \textfont\euffam=\twelveeuf \scriptfont\euffam=\nineeuf \scriptscriptfont\euffam=\seveneuf \def\frak#1{\fam\euffam\relax#1} \font\twelvemsb=msbm10 scaled 1200 \font\ninemsb=msbm9 \font\sevenmsb=msbm7 \newfam\msbfam \textfont\msbfam=\twelvemsb \scriptfont\msbfam=\ninemsb \scriptscriptfont\msbfam=\sevenmsb \def\Bbb#1{\fam\msbfam\relax#1} \newfont{\Symb}{msbm10 scaled 1728} %%%%%%%%%%%%%%%GREEK%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\lam}{\lambda} \newcommand{\sig}{\sigma} \newcommand{\eps}{\varepsilon} \newcommand{\vphi}{\varphi} \newcommand{\gam}{\gamma} %%%%%%%%%%%%%%%%%%ABBRS%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\ti}{\tilde} \newcommand{\bs}{\backslash} \newcommand{\mzer}{\backslash \{ 0\}} \newcommand{\nn}{\nonumber} \newcommand{\mb}{\mbox} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}} \newcommand{\D}{\displaystyle} \newcommand{\ol}{\overline} \newcommand{\ul}{\underline} \newcommand{\R}{{\Bbb R}} \newcommand{\N}{{\Bbb N}} \newcommand{\Z}{{\Bbb Z}} \newcommand{\C}{{\Bbb C}} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\I}{{\rm i}} \newcommand{\mod}{\,{\rm mod}\, } \newcommand{\id}{{\Bbb I}} \newcommand{\Ran}{{\rm Ran}} \newcommand{\tr}{{\rm tr}} \newcommand{\sgn}{{\rm sgn}} \newcommand{\hr}{{\frak H}} \newcommand{\db}{{\frak D}} \newcommand{\lz}{\ell^2(\Z)} \newcommand{\lf}{\ell(\Z)} \newcommand{\spr}[2]{\langle #1 , #2 \rangle} \newcommand{\intp}[1]{[\![ #1 ]\!]} \newcommand{\gele}{\ba{c} \vspace*{-8mm} > \\ \vspace*{3mm} < \ea} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\keywords{Jacobi operators, spectral theory, oscillation theory, commutation methods} %\subjclass{Primary 36A10, 39A70; Secondary 34B24, 34L05} \begin{document} \newtheorem{thm}{Theorem}[chapter] \newtheorem{lem}[thm]{Lemma} \newtheorem{coro}[thm]{Corollary} \newtheorem{hypo}[thm]{Hypothesis {\bf H$\!\!\!$ .}\hspace*{-0.47ex}} \newtheorem{rem}[thm]{Remark} \begin{titlepage} \begin{center} \vspace*{2.3cm} {\Large SPECTRAL THEORY FOR JACOBI OPERATORS}\\[12mm] \rule{10cm}{.01in}\\[5mm] A Dissertation\\[1mm] presented to\\[1mm] the Faculty of the Graduate School\\[1mm] University of Missouri-Columbia\\[5mm] \rule{10cm}{.01in}\\[5mm] In Partial Fulfillment\\[1mm] of the Requirements for the Degree\\[1mm] Doctor of Philosophy\\[5mm] \rule{10cm}{.01in}\\[3mm] by \\ GERALD TESCHL\\[3mm] Professor Fritz Gesztesy, Dissertation Supervisor \\[5mm] DECEMBER 1995 \end{center} \end{titlepage} \renewcommand{\thepage}{\roman{page}} %\setcounter{page}{1} \thispagestyle{plain} \addcontentsline{toc}{section}{Acknowledgements} \vspace*{0.5in} \begin{center} ACKNOWLEDGEMENTS \end{center} I would like to thank my advisor Fritz Gesztesy for two fruitful years of collaboration and for his excellent supervision. Special thanks are also due to Karl Unterkofler and all students of Fritz: Ronnie Dickson, Miroslaw Mystkowski, Ray Nowell, Ratnam Ratnaseelan, Walter Renger, Don Steiger, Wilhelm Sticka, Susanne Timischl, and Mehmet Unal. Moreover, I would like to acknowledge the hospitality of the Department of Theoretical Physics, Technical University of Graz, Austria during the summer months of 1994 and 1995. Last but not least I am indebted to my parents Edeltrude Teschl and Karl Prenner and particularly to Susanne Timischl for always being there when I needed someone. \newpage \thispagestyle{plain} \addcontentsline{toc}{section}{Abstract} \vspace*{0.5in} \begin{center} {\large SPECTRAL THEORY FOR JACOBI OPERATORS}\\[3mm] Gerald Teschl\\ Professor Fritz Gesztesy, Dissertation Supervisor\\[2mm] \large ABSTRACT \end{center} %\begin{abstract} The present thesis discusses various aspects of spectral theory for Jacobi operators. The first chapter reviews Weyl-Titchmarsh theory for these operators and provides the necessary background for the following chapters. In the second chapter we provide a comprehensive treatment of oscillation theory for Jacobi operators with separated boundary conditions. Moreover, we present a reformulation of oscillation theory in terms of Wronskians of solutions, thereby extending the range of applicability for this theory. Furthermore, these results are applied to establish the finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Jacobi operators. In the third chapter we offer two methods of inserting eigenvalues into spectral gaps of a given background Jacobi operator: The single commutation method which introduces eigenvalues into the lowest spectral gap of a given semi-bounded background Jacobi operator and the double commutation method which inserts eigenvalues into arbitrary spectral gaps. Moreover, we prove unitary equivalence of the commuted operators, restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues, with the original background operator. Finally, we show how to iterate the above methods. Concrete applications include an explicit realization of the isospectral torus for algebro-geometric finite-gap Jacobi operators and the $N$-soliton solutions of the Toda and Kac-van Moerbeke lattice equations with respect to arbitrary background solutions. %\end{abstract} \newpage \tableofcontents \newpage \renewcommand{\thepage}{\arabic{page}} \setcounter{page}{1} \chapter[Weyl-Titchmarsh Theory]{Weyl-Titchmarsh Theory for Jacobi Operators} \section{General Background} First of all we need to fix some notation. For $I \subseteq \Z$ we denote by $\ell(I)$ the set of $\C$-valued sequences $\{ f(n) \}_{n \in I}$. For $M,N \in \Z \cup \{ \pm\infty \}$ we abbreviate $\ell(M,N) = \ell(\{n\in \Z | M < n < N\})$ (sometimes we will also write $\ell(N,-\infty)$ instead of $\ell(-\infty,N)$). $\ell^2(I)$ is the Hilbert space of all square summable sequences with scalar product and norm defined as \begin{equation} \spr{f}{g} = \sum_{n \in I} \ol{f(n)} g(n), \quad \| f \| = \sqrt{\spr{f}{f}}, \quad f,g \in \ell^2(I). \end{equation} Furthermore, $\ell_0(I)$ denotes the set of sequences with only finitely-many values being nonzero, $\ell^1(I)$ the set of summable sequences, $\ell^\infty(I)$ the set of bounded sequences, and $\ell^2_\pm(\Z)$ denotes the set of sequences in $\ell(\Z)$ which are $\ell^2$ near $\pm\infty$. For brevity we focus in the following on the case $I=\Z$. To set the stage, we shall consider operators on $\lz$ associated with the difference expression \begin{equation} (\tau f)(n) = a(n) f(n+1) + a(n-1) f(n-1) -b(n) f(n), \end{equation} where $a,b \in \ell(\Z)$ satisfy the following hypothesis. \bh \label{hypoab} Suppose $a,b \in \ell(\Z)$ satisfy \begin{equation} a(n) \in \R \mzer, \quad b(n) \in \R, \quad n \in \Z. \end{equation} \eh If $\tau$ is limit point ($l.p.$) at both $\pm\infty$ (cf., e.g., \cite{at}, \cite{be}) then $\tau$ gives rise to a unique self-adjoint operator $H$ when defined maximally. Otherwise we need to fix a boundary condition at each endpoint where $\tau$ is limit circle ($l.c.$). Throughout this thesis we denote by $u_\pm(z,.)$, $z \in \C$ nontrivial solutions of $\tau u = z u$ which satisfy the boundary condition at $\pm\infty$ (if any) and $u_\pm(z,.) \in \ell^2_\pm(\Z)$, respectively. $u_\pm(z,.)$ might not exist for $z \in \R$ (cf.\ Lemma~\ref{upmholz}) but if it exists it is unique up to a constant multiple. Picking $z_0 \in \C \bs \R$ we can characterize $H$ by \begin{equation} \ba{lccl} H :& \db(H) & \to & \lz \\ & f &\mapsto& \tau f \ea , \end{equation} where the domain of $H$ is explicitly given by \begin{equation} \db(H) = \{ f \in \lz | \ba[t]{l} \tau f \in \lz, \: \lim_{n \to +\infty} W_n(u_+(z_0),f) = 0, \\ \lim_{n \to -\infty} W_n(u_-(z_0),f) = 0 \} \ea \end{equation} and \begin{equation} \label{wr} W_n(f,g) = a(n) \Big( f(n)g(n+1) - f(n+1)g(n) \Big) \end{equation} denotes the (modified) Wronskian. By $\sig(.)$, $\sig_p(.)$, and $\sig_{ess}(.)$ we denote the spectrum, point spectrum (i.e., the set of eigenvalues), and essential spectrum of an operator, respectively. A simple calculation yields Green's formula for $f,g \in \lf$ \begin{equation} \label{gf} \sum_{j=m}^n \Big( f (\tau g) - g \tau f \Big)(j) = W_n(f,g) - W_{m-1}(f,g). \end{equation} A glance at (\ref{gf}) shows that the modified Wronskian of two solutions of \begin{equation} \label{gl} \tau u = z u \end{equation} is constant and nonzero if and only if they are linearly independent. If we choose $f=u(z)$, $g=\ol{u(z)}$ in (\ref{gf}), where $u(z)$ is a solution of (\ref{gl}) with $z\in \C \bs \R$, we obtain \bea \label{weylkl} [u(z)]_n = [u(z)]_{m-1} - \sum_{j=m}^n |u(z,j)|^2, \eea where $[.]_n$ denotes the Weyl bracket \bea [u(z)]_n = \frac{W_n(u(z),\ol{u(z)})}{2 \I \im(z)} = a(n) \frac{\im (u(z,n) \ol{u(z,n+1)})}{\im(z)}, \quad n \in \Z. \eea Taking limits in (\ref{gf}) shows that $W_{\pm\infty}(f,g) = \lim_{n \to \pm\infty} W_n(f,g)$ exists if $f,g,\tau f$, and $\tau g$ are square summable near $\pm\infty$. The following sections generalize some well-known facts about Sturm--Liouville operators (to be found, e.g., in \cite{cole},\cite{com},\cite{9},\cite{wd}) to Jacobi operators. The presented material is essentially taken from \cite{ak},\cite{at},\cite{be},\cite{cl}. \section{Weyl $m$-functions} \label{secwm} Let $c_\alpha(z,.),s_\alpha(z,.)$ be the fundamental system of (\ref{gl}) corresponding to the initial conditions \bea \label{funsy} \ba{l@{\qquad}l} s_\alpha(z,0) = -\sin(\alpha), & s_\alpha(z,1) = \cos(\alpha), \\ c_\alpha(z,0) = \D \frac{\cos(\alpha)}{a(0)}, & c_\alpha(z,1) = \D \frac{\sin(\alpha)}{a(0)} \ea \eea such that \bea W(c_\alpha(z),s_\alpha(z)) = 1. \eea Next pick $\lam_1 \in \R$ and define the following rational function with respect to $z$, \bea \label{mfun} m_N(z,\alpha) = \frac{W_N(s_\alpha(\lam_1), c_\alpha(z))}{W_N(s_\alpha(\lam_1), s_\alpha(z))}, \qquad N \in \Z \bs \{ 0 \}, \eea which has poles at the zeros $\lam_j(N) \in \R$, $\lam_1(N) \equiv \lam_1$ of $W_N(s_\alpha(\lam_1), s_\alpha(.)) = 0$. The fact that one can rewrite $m_N(z,\alpha)$ with $\lam_1$ replaced by $\lam_j(N)$ together with \bea \lim_{z \to \lam_j(N)} W_N(s_\alpha(\lam_j(N)), c_\alpha(z)) &=& -1, \\ \label{limw} \lim_{z \to \lam_j(N)} \frac{W_N(s_\alpha(\lam_j(N)), s_\alpha(z)) }{z-\lam_j(N)} &=& W_N(s_\alpha(\lam_j(N)), \frac{d}{dz} s_\alpha(\lam_j(N)) ) \eea imply that all poles of $m_N(z,\alpha)$ are simple. Using (\ref{gf}) to evaluate (\ref{limw}) one infers that $\mp 1$ times the residue at $\lam_j(N)$ is given by \bea \gam_j(\alpha,N) = \Big( \sum_{n={1 \atop N+1}}^{N \atop 0} s_\alpha(\lam_j(N),n)^2 \Big)^{-1}, \quad N \gele 0. \eea The $\gam_j(\alpha,N)$ are called norming constants. Hence one gets \bea m_N(z,\alpha) = \sum_j \frac{\mp\gam_j(\alpha,N)}{z - \lam_j(N)} + \left\{ \ba{l} \frac{\pm\tan(\alpha)^{\pm1}}{a(0)}, \: \alpha \in {[0,\pi) \atop (0,\pi]} \\ \frac{\pm z - b({1 \atop 0})}{a(0)^2}, \: \alpha = {\pi \atop 0}\ea \right. , \quad N \gele 0. \eea (We note that $\lam_j(N)$ depend on $\alpha$ for $j>1$.) Furthermore, the function \bea u_N(z,n) = c_\alpha(z,n) - m_N(z,\alpha) s_\alpha(z,n) \eea satisfies \bea \sum_{n={1 \atop N+1}}^{N \atop 0} |u_N(z,n)|^2 = \pm \frac{\im(m_N(z,\alpha))}{\im(z)}, \quad N \gele 0, \eea i.e., $\pm m_N(z,\alpha)$ are Herglotz functions for $N \gele 0$. Next we want to investigate the limits $N \to \pm\infty$. Fix $z \in \C \bs \R$. Then, as in the Sturm-Liouville case, the function $m_N(z,\alpha)$ (for different values of $\lam_1 \in \R$) lies on a circle given by \bea \{ m \in \C | [c_\alpha(z) - m s_\alpha(z)]_N =0 \}. \eea Since $[.]_N$ is decreasing in $N$ for $N>0$, the circle corresponding to $N+1$ lies inside the circle corresponding to $N$. Similarly for $N<0$. Hence these circles either tend to a limit point or a limit circle, depending on whether \bea \sum^{\pm\infty} |s_\alpha(z,n)|^2 = \infty, \quad\mb{or}\quad \sum^{\pm\infty} |s_\alpha(z,n)|^2 < \infty. \eea Accordingly, one says that $\tau$ is limit point ($l.p.$) respectively limit circle ($l.c.$) at $\pm\infty$. One can show that this definition is independent of $z \in \C \bs \R$. Thus the pointwise convergence of $m_N(z,\alpha)$ is clear in the $l.p.$ case. In the $l.c.$ case both Wronskians in (\ref{mfun}) converge and we may set \bea \ti{m}_\pm(z,\alpha) = \lim_{N \to \pm\infty} m_N(z,\alpha). \eea \begin{rem} (i).\ $\ti{m}_\pm(z,0)$ are not the usual Weyl $m$-functions defined in the literature. For a connection with the standard Weyl $m$-functions $m_\pm(z)$ see (\ref{mpl}), (\ref{mmi}). We have chosen to introduce $\ti{m}_\pm(z,\alpha)$ in order to simplify our notation in various places.\\ (ii).\ This explicit construction of converging sequences, even in the $l.c.$ case, also works for Sturm-Liouville operators and seems to be novel to the best of our knowledge. Previously one usually proved the existence of such sequences using Helly's selection theorem (cf., e.g., \cite{cole}). \end{rem} Moreover, the above sequences are locally bounded in $z$ (fix an $N$ and take all circles corresponding to a (sufficiently small) neighborhood of any point $z$ and note that all following circles lie inside the ones corresponding to $N$) and by Vitali's theorem (\cite{tit}, p.\ 168) they converge uniformly on every compact set in $\C_\pm = \{z \in \C | \pm\im(z)>0 \}$, implying that $\pm\ti{m}_\pm(z,\alpha)$ are again Herglotz functions. Upon setting \bea u_\pm(z,n) = c_\alpha(z,n) - \ti{m}_\pm(z,\alpha) s_\alpha(z,n) \eea we get a function which is square summable near $\pm\infty$ \bea \sum_{n= {1 \atop -\infty}}^{\infty \atop 0} |u_\pm(z,n)|^2 = \pm \frac{\im(\ti{m}_\pm(z,\alpha))}{\im(z)}, \quad N \gele 0. \eea In addition, \bea W_{\pm\infty}(s_\alpha(\lam_1),u_\pm(z))=0, \eea if $\tau$ is $l.c.$ at $\pm\infty$. We remark that (independently of the $l.c.$ and $l.p.$ case at $\pm\infty$) \bea \ti{m}_\pm(z) = \ti{m}_\pm(z,0) = \frac{-u_\pm(z,1)}{a(0)u_\pm(z,0)} \eea and that $\ti{m}_\pm(z,\alpha)$ can be expressed in terms of $\ti{m}_\pm(z,\beta)$ (use that $u_\pm$ is unique up to a constant) by \bea \hspace*{1cm} \ti{m}_\pm(z,\alpha) = \frac{1}{a(0)} \frac{a(0) \cos(\beta-\alpha) \ti{m}_\pm(z,\beta) - \sin(\beta-\alpha)}{a(0) \sin(\beta-\alpha) \ti{m}_\pm(z,\beta) + \cos(\beta-\alpha)}. \eea \section[Weyl-Titchmarsh Theory on $\N$]{Weyl-Titchmarsh Theory on {\Symb \N}} \label{secwtn} Let $H_+$ be a given self-adjoint operator associated with $\tau$ on $\N$ and a Dirichlet boundary condition at $n=0$. Abbreviate $s(z,n)=s_0(z,n)$ and let $u_+(z,n)$, $z \in \C \backslash \sig(H_+)$ be a solution of (\ref{gl}) which is square summable near $\infty$ and fulfills the boundary condition at $\infty$ (if any). The resolvent of $H_+$ then reads \bea ((H_+ - z)^{-1} f)(n) &=& \sum_{m \in \N} G_+(z,m,n) f(m), \quad z \in \C \backslash \sig(H_+), \eea where \bea G_+(z,m,n) = \frac{1}{W(s(z), u_+(z))} \left\{ \ba{l@{,\quad}l} s(z,n) u_+(z,m) & m \ge n \\ s(z,m) u_+(z,n) & m \le n \ea \right. . \eea Since $s(z,n)$ is a polynomial in $z$ we infer by induction \bea \label{relhl} s(H_+,n) \delta_{1} = \delta_{n}, \qquad \delta_n(k) = \left\{ \ba{c@{\quad}l} 1, & k=n \\ 0, & k \ne n \ea \right. , \eea implying that $\delta_{1}$ is a cyclic vector for $H_+$. If $E_+(.)$ denotes the family of spectral projections corresponding to $H_+$ we introduce the measure \bea d\rho_+(.) = d\spr{\delta_1}{E_+(.) \delta_1}. \eea Equation (\ref{relhl}) now shows that the polynomials $s(z,n), \: n \in \N$ are orthogonal with respect to this measure, i.e., \bea \spr{s(j)}{s(k)} = \int\limits_{-\infty}^\infty s(\lam,j) s(\lam,k) \, d\rho_+(\lam) = \delta_j(k) , \eea implying \bea \label{polab} a(n) = \spr{s(n+1)}{\lam s(n)}, \quad b(n)= -\spr{s(n)}{\lam s(n)}, \:\: n \in \N. \eea Now consider the following transformation $U$ from the set $\ell_0(\N)$ onto the set of polynomials \bea (Uf)(\lam) &=& \sum_{n=1}^\infty f(n) s(\lam,n), \\ \label{unitary2} (U^{-1} F)(n) &=& \int_\R s(\lam,n) F(\lam) d\rho_+(\lam). \eea A simple calculation for $F(\lam) = (Uf)(\lam)$ shows that \bea \sum_{n=1}^\infty |f(n)|^2 = \int_\R |F(\lam)|^2 d\rho_+(\lam). \eea Thus $U$ extends to a unitary transformation \bea \tilde{U}: \ell^2(\N) \to L^2(\R,d\rho_+) \eea (since the set of polynomials is dense in $L^2(\R,d\rho_+)$, \cite{be}, Theorem VII.1.7) which maps the operator $H_+$ to the multiplication operator by $\lam$, \bea \tilde{U} H_+ \tilde{U}^{-1} = \tilde{H}, \eea where \bea \tilde{H} F(\lam) = \lam F(\lam), \quad \db(\tilde{H})= \{F \in L^2(\R,d\rho_+) | \lam F(\lam) \in L^2(\R,d\rho_+)\}. \eea This is easily verified for $f \in \ell_0(\N)$. If $\tau$ is $l.p.$ at $\infty$ note that $\ell_0(\N)$ is a core for $H_+$ and if $\tau$ is $l.c.$\ at $\infty$ note that $d\rho_+$ is a pure point measure and that eigenfunctions are mapped onto eigenfunctions (all finite linear combinations of eigenfunctions form again a core). This implies that the spectrum of $H_+$ can be characterized as follows. Let the Lebesgue decomposition of $d\rho_+$ be given by \bea d\rho_+ = d\rho_{+,p} + d\rho_{+,ac} + d\rho_{+,sc}, \eea then we have ($\rho_+(\lam) = \int_{(-\infty,\lam]} d\rho_+$) \bea \sig(H_+) &=& \{\lam \in \R |\mb{$\lam$ is a growth point of $\rho_+$}\},\\ \sig_{p}(H_+) &=& \{\lam \in \R |\mb{$\lam$ is a growth point of $\rho_{+,p}$}\},\\ \sig_{ac}(H_+) &=& \{\lam \in \R |\mb{$\lam$ is a growth point of $\rho_{+,ac}$}\},\\ \sig_{sc}(H_+) &=& \{\lam \in \R |\mb{$\lam$ is a growth point of $\rho_{+,sc}$}\}. \eea The Stieltjes transform of the spectral function $\rho_+$ is called the Weyl $m$-function \bea m_+(z) = \int_\R \frac{d\rho_+(\lam)}{z-\lam}, \qquad z \in \C \bs \R. \eea Conversely, the spectral function $\rho_+$ can be recovered from $m_+(z)$ by the Stieltjes inversion formula \bea \rho_+(\lam) = \frac{-1}{\pi} \lim_{\delta \downarrow 0} \lim_{\eps \downarrow 0} \int\limits_{-\infty}^{\lam+\delta} \im(m_+(\nu + \I \eps)) d\nu. \eea We have normalized $\rho_+$ such that it is right continuous and satisfies $\lim\limits_{\lam \to -\infty} \rho_+(\lam) = 0$. One infers \bea \label{mpl} m_+(z) = G_+(z,1,1) = \frac{-u_+(1)}{a(0) u_+(0)} = \ti{m}_+(z), \eea and we remark that the local compact convergence of $m_N(z,0)$ to $\ti{m}_+(z) = m_+(z)$ implies the convergence of the associated spectral functions at every point of continuity (\cite{ad}, p.\ 332). The second Weyl $m$-function is usually defined as \bea \label{mmi} m_-(z) = G_-(z,-1,-1) = \frac{-u_-(-1)}{a(-1) u_-(0)} = -\frac{ z+b(0) + a(0)^2 \ti{m}_-(z)}{a(-1)^2}. \eea $m_\pm(z)$, like $\pm\ti{m}_\pm(z)$, are Herglotz functions. \section[Weyl--Titchmarsh Theory on $\Z$]{Weyl--Titchmarsh Theory on {\Symb \Z}} \label{secwtz} In Section~\ref{secwtn} we have dealt with the half-line $\N$. In this section we extend these results to all of $\Z$. Let $H$ be a given self-adjoint operator associated with $\tau$. Let $u_\pm(z,n)$ be a solution of (\ref{gl}) which is square summable near $\pm\infty$ (provided such a solution exists) and fulfills the boundary condition at $\pm\infty$ if any. The resolvent of $H$ then reads \bea ((H - z)^{-1} f)(n) &=& \sum_{m \in \Z} G(z,m,n) f(m), \quad z \in \rho(H), \eea where \bea G(z,m,n) = \frac{1}{W(u_-(z),u_+(z))} \left\{ \ba{l@{\quad}l} u_-(z,n) u_+(z,m), & m \ge n \\ u_-(z,m) u_+(z,n), & m \le n \ea \right. . \eea Consider the vector-valued polynomials \bea \ul{S}(z,n) = \Big( s(z,n), c(z,n) \Big), \eea where $s(z,n)$, $c(z,n)$ are solutions of (\ref{gl}) satisfying the initial conditions \bea \ba{c@{\qquad}c} s(z,0) = 0, & s(z,1) = 1,\\ c(z,0) = 1, & c(z,1) = 0. \ea \eea The analog of (\ref{relhl}) reads \bea \label{relwl} s(H,n) \delta_1 + c(H,n) \delta_0 = \delta_n. \eea This is obvious for $n=0,1$ and the rest follows from induction upon applying $H$ to (\ref{relwl}). If $E(.)$ denotes the spectral resolution of the identity corresponding to $H$ we introduce the measures \bea d\rho_{j,k}(.) = d\spr{\delta_j}{E(.) \delta_k}, \eea and the (hermitian) matrix-valued measure \bea d\rho = \left( \ba{cc} d\rho_{1,1} & d\rho_{1,2} \\ d\rho_{2,1} & d\rho_{2,2} \ea \right). \eea By (\ref{relwl}) the vector-valued polynomials are orthogonal with respect to $d\rho$ \bea \nn \spr{\ul{S}(m)}{\ul{S}(n)} &=& \sum_{j,k=1}^2 \int_\R S_j(\lam,m) \; S_k(\lam,n) d\rho_{j,k}(\lam)\\ &\equiv& \int_\R \ul{S}(\lam,m) d\rho(\lam) \; \ul{S}(\lam,n) = \delta_n(m). \eea The analogous formulas to (\ref{polab}) then read \bea a(n) = \spr{\ul{S}(n+1)}{\lam \ul{S}(n)}, \quad b(n) = \spr{\ul{S}(n)} {\lam \ul{S}(n)}, \:\: n \in \Z. \eea Next we consider the following transformation $U$ from the set $\ell_0(\Z)$ onto the set of vector-valued polynomials \bea (Uf)(\lam) &=& \sum_{n\in \Z} f(n) \ul{S}(\lam,n), \\ (U^{-1} \ul{F})(n) &=& \int_\R \ul{S}(\lam,n) d\rho(\lam) \ul{F}(\lam). \eea Again a simple calculation for $\ul{F}(\lam) = (Uf)(\lam)$ shows that \bea \sum_{n \in \Z} |f(n)|^2 = \int_\R \ol{\ul{F}(\lam)} d\rho(z) \ul{F}(\lam). \eea Thus $U$ extends to a unitary transformation \bea \tilde{U}: \ell^2(\Z) \to L^2(\R,d\rho) \eea which maps the operator $H$ to the multiplication operator by $\lam$, \bea \tilde{U} H \tilde{U}^{-1} = \tilde{H}, \eea where \bea \tilde{H} \ul{F}(\lam) = z \ul{F}(\lam), \quad \db(\tilde{H})= \{\ul{F} \in L^2(\R,d\rho) | \lam \ul{F}(\lam) \in L^2(\R,d\rho)\}. \eea In order to characterize the spectrum of $H$ one only needs to consider the trace $d\rho^t$ of $d\rho$ \bea d\rho^t = d\rho_{1,1} + d\rho_{2,2}. \eea Let the Lebesgue decomposition of $d\rho^t$ be given by \bea d\rho^t = d\rho^t_{p} + d\rho^t_{ac} + d\rho^t_{sc}, \eea then we have ($\rho^t(\lam) =\int_{(-\infty,\lam]} d\rho^t$, etc.) \bea \sig(H) &=& \{\lam \in \R |\mb{$\lam$ is a growth point of $\rho^t$}\},\\ \sig_{p}(H) &=& \{\lam \in \R |\mb{$\lam$ is a growth point of $\rho^t_{p}$}\},\\ \sig_{ac}(H) &=& \{\lam \in \R |\mb{$\lam$ is a growth point of $\rho^t_{ac}$}\},\\ \sig_{sc}(H) &=& \{\lam \in \R |\mb{$\lam$ is a growth point of $\rho^t_{sc}$}\}. \eea The Weyl-matrix $M(z)$ is defined as \bea M(z) = \int\limits_{-\infty}^\infty \frac{d\rho(\lam)}{z-\lam}, \qquad z \in \C \bs \R. \eea Explicit evaluation yields \bea \nn && M(z) = \left( \ba{cc} G(z,0,0) & G(z,1,0) \\ G(z,0,1) & G(z,1,1) \ea \right)\\ && = \frac{a(0)^{-2}}{\ti{m}_-(z) - \ti{m}_+(z)} \left( \ba{cc} 1 & -a(0) \ti{m}_+(z) \\ -a(0) \ti{m}_+(z) & a(0)^2 \ti{m}_+(z) \ti{m}_-(z) \ea \right). \eea Finally, assuming $\rho$ to be right continuous and normalizing $\rho(-\infty)=0$ one obtains \bea \rho_{j,k}(\lam) = \frac{-1}{\pi} \lim_{\delta \downarrow 0} \lim_{\eps \downarrow 0} \int\limits_{-\infty}^{\lam+\delta} \im(M_{j,k}(\nu + \I \eps)) d\nu, \quad 1 \le j,k \le 2. \eea \section{Some Useful Lemmas} This section provides some useful results needed later on. Denote by $s(z,n)$ and $c(z,n)$ the solutions of $\tau u = z u$ corresponding to the initial conditions $s(z,0)=c(z,1)=0$, $s(z,1)=c(z,0)=1$. \bl \label{upmholz} Let $\lam_0<\lam_1$ be such that $[\lam_0,\lam_1] \cap \sig_{ess}(H_+) = \emptyset$. Then there exists a solution $u_+(z,.)\in\ell^2_+(\Z)$ of $\tau u =z u$ satisfying the boundary condition of $H$ at $+\infty$ (if any) which is holomorphic with respect to $z$ for $z \in \C \bs ((-\infty,\lam_0] \cup [\lam_1,\infty))$. In addition, we can assume $u_+(z,.) \not\equiv 0$ and $\ol{u_+(z,.)} = u_+(\ol{z},.)$. Similarly, $[\lam_0,\lam_1] \cap \sig_{ess}(H_-) = \emptyset$ implies the existence of a solution $u_-(z,.)\in\ell_-(\Z)$ fulfilling the boundary condition of $H$ at $-\infty$ (if any) and, as a function of $z$, satisfies the same conditions as $u_+(z,.)$. \el \bpf Explicitly, we can set \begin{equation} u_\pm(z,n) = \Big(\prod_{\mu \in \sig(H_+) \cap [\lam_0,\lam_1]} \hspace*{-3mm} (z-\mu) \Big) \Big( a(0)^{-1}c(z,n) - \ti{m}_\pm(z) s(z,n) \Big). \end{equation} \epf \bl \label{upmpos} Suppose $a(n)<0$ and let $\lam < \inf\sig(H)$. Then we can assume \begin{equation} u_\pm(\lam,n) >0, \quad n \in \Z, \end{equation} \begin{equation} n \, s(\lam,n) >0, \quad n \in \Z \mzer. \end{equation} They solutions $u_\pm(\lam,.)$ are called principal solutions of $(H-\lam)u=0$ near $\pm\infty$ in \cite{har}. \el \bpf From $(H-\lam)>0$ one infers $(H_{+,n} - \lam)>0$ and hence \begin{equation} \label{posauou} 0 < \spr{\delta_{n+1}}{(H_{+,n} -\lam)^{-1}\delta_{n+1}} = \frac{u_+(\lam,n+1)}{-a(n) u_+(\lam,n)} \end{equation} showing that $u_+(\lam)$ can be chosen to be positive. Furthermore, for $n>0$ we obtain \begin{equation} 0 < \spr{\delta_n}{(H_+ -\lam)^{-1}\delta_n} = \frac{u_+(\lam,n)s(\lam,n)}{-a(0) u_+(\lam,0)} \end{equation} implying $s(\lam,n)>0$ for $n>0$. Similarly one proves the remaining results. \epf Let $u_\pm(z,n)$ are solutions of $\tau u =z u$ as in Lemma \ref{upmholz}. Then Green's formula (\ref{gf}) implies \begin{equation} W_n(u_+(z),u_+(\ti{z})) = (z-\ti{z}) \sum_{j=n+1}^\infty u_+(z,j) u_+(\ti{z},j) \end{equation} and furthermore, \bea \nn W_n(u_+(z),\dot{u}_+(z)) &=& \lim_{\ti{z} \to z} W_n(u_+(z),\frac{u_+(z)-u_+(\ti{z})}{z-\ti{z}})\\ &=& \sum_{j=n+1}^\infty u_+(z,j)^2. \eea Here the dot denotes the derivative with respect to $z$. An analogous result holds for $u_-(z,n)$. Interchanging limit and summation can be justified using (cf.~Remark~\ref{rembc}) \begin{equation} u_+(\ti{z},j) = const(\ti{z}) (H^\beta_{+,n-1} - \ti{z})^{-1} \delta_n(j) \quad\mb{ for } j \le n \end{equation} (with $\beta$ such that $z \not\in \sig(H^\beta_{+,n-1})$) and the first resolvent identity. Summarizing (compare \cite{at}, Theorem 4.2.2): \bl \label{wuudot} Let $u_\pm(z,n)$ be solutions of $\tau u =z u$ as in Lemma \ref{upmholz}. Then we have \begin{equation} W_n(u_\pm(z),\dot{u}_\pm(z)) = \left\{ \ba{l} -\sum\limits_{j=n+1}^\infty u_+(z,j)^2 \\ \sum\limits_{j=-\infty}^n u_-(z,j)^2 \ea \right. . \end{equation} \el \chapter{Oscillation Theory} \section{Introduction} In 1836 Sturm originated the investigations of oscillation properties of solutions of second-order differential and difference equations \cite{stu}. Since then numerous extensions have been made. Especially, around 1948, Hartman and others have shown the following in a series of papers (\cite{har1}, \cite{har2}, \cite{har3}). For a given Sturm--Liouville operator $H$ on $L^2(0,\infty)$, the dimension of the spectral projection $P_{(-\infty, \lam)}(H)$ equals the number of zeros of certain solutions of $H u = \lam u$. Moreover, the dimension of $P_{(\lam_1, \lam_2)}(H)$ can be obtained by considering the difference of the number of zeros inside a finite interval $(0,x)$ of two solutions corresponding to their respective spectral parameters $\lam_1$ and $\lam_2$, and performing a limit $x \to \infty$. Only recently it was shown in \cite{gst} by F.~Gesztesy, B.~Simon, and myself that these limits can be avoided by using a renormalized version of oscillation theory, that is, counting zeros of Wronskians of solutions instead. This naturally raises the question whether similar results hold for second-order difference equations. Despite a variety of literature on this subject (cf., e.g., \cite{at}, \cite{bo}, \cite{do}, \cite{ft}, \cite{crit}, \cite{glz}, Sections 14 and 37, \cite{har}, \cite{hl}, \cite{hp}, \cite{hkp}, \cite{kp}, \cite{pat1}, \cite{pat2} and the references therein) only a few things concerning the connections between oscillation properties of solutions and spectra of the corresponding operators appear to be known. In particular, the analogs of the aforementioned theorems seem to be unknown. Moreover, even the analog of the well-known fact that the $n$-th eigenfunction of a Sturm-Liouville operator (below the essential spectrum) has $n-1$ nodes is only known in the special case of finite Jacobi operators (i.e., finite tri-diagonal matrices) \cite{at}, Theorem~4.3.5, \cite{ft}. The present thesis aims at filling these gaps and provides a complete solution to these problems. Now, we want to give the reader an intuitive idea of how oscillation theory works. In the sequel a solution of $\tau u = \lam u$, $\lam \in \R$ will always mean a real-valued, non-zero solution. We first need to define what we mean by a node of a real-valued sequence $u \in \ell(\Z)$. A point $n \in \Z$, is called a node of $u$ if either \begin{equation} u(n) =0 \quad \mbox{or} \quad a(n)u(n)u(n+1)>0. \end{equation} In the special case $a(n)<0$, $n \in \Z$ a node of $u$ is precisely a sign flip of $u$ as one would expect. In the general case, however, one has to take the sign of $a(n)$ into account. For simplicity we shall assume $a(n)<0$ (cf.\ Lemma~\ref{lemaeps}) and $a,b$ bounded (implying $H$ bounded) for the remainder of this section. By Lemma~\ref{upmholz} $u_-(\lam,.)$ can be assumed to be continuous with respect to $\lam$ as long as $\lam$ is below the essential spectrum of $H$. In addition, $u_-(\lam,.)$ can be assumed positive for $\lam$ below the spectrum of $H$ and hence has no nodes in this case. Increasing $\lam$ one needs to observe three things: (i) Nodes of $u_-(\lam)$ move to the right (by (\ref{thetaminc})) without colliding; (ii) $u_-(\lam)$ cannot pick up nodes locally (by (\ref{sigch})); (iii) $u_-(\lam)$ cannot lose nodes at $-\infty$. By (i) and (ii) we infer that $u_-(\lam)$ can only pick up nodes at $+\infty$. Intuitively this happens if $u_-(\lam) \in \lz$ (or equivalently, if $\lam$ an eigenvalue of $H$) and hence $\lim_{n\to\infty} u_-(\lam,n) =0$. Summarizing, $u_-(\lam)$ has no nodes below the spectrum of $H$ and picks up one additional node whenever $\lam$ is an eigenvalue of $H$. Since no nodes get lost we are lead to (cf.\ Theorem~\ref{thmbelesssph}) \begin{equation} \dim\Ran\, P_{(-\infty,\lam)}(H) = \#(u_-(\lam)), \end{equation} where $\#(u)$ denotes the total number of nodes of $u$ and $P_\Omega(H)$ is the spectral projection of $H$ corresponding to the Borel set $\Omega \subseteq \R$. As a corollary we conclude, as already anticipated, that the $n$-th eigenfunction (below the essential spectrum) has $n-1$ nodes. To obtain the number of eigenvalues between two given values $\lam_1$ and $\lam_2$ it seems natural to consider $\#(u_-(\lam_2)) - \#(u_-(\lam_1))$. This gives nothing new below the essential spectrum and otherwise we have $\#(u)=\infty$ for any solution of $\tau u = \lam u$ with $\lam$ above the infimum of the essential spectrum. Hence, a naive use of oscillation theory in the latter case yields $\infty-\infty$. There are two ways to overcome this problem. The first, due to \cite{har2} in the case of differential operators, uses a limiting procedure which only works for half-line operators and can be found in Theorem~\ref{thmhartman}. The second, due to \cite{gst} in the case of differential operators, uses the fact that the nodes of the Wronskian of two solutions $u_1, u_2$ corresponding to $\lam_1, \lam_2$, respectively, essentially counts the additional nodes of $u_2$ with respect to $u_1$ (cf.\ Corollary~\ref{cornodw}). In this sense the Wronskian comes with a built-in renormalization. Moreover, the nodes of Wronskians behave similar to the nodes of solutions and satisfy the above properties (i), (ii), and (iii) as well. Hence, similar techniques apply. To give rigorous proofs for the indicated results we first introduce and investigate Pr\"ufer variables in Section~\ref{secpruf}. They will be our main tool in Section~\ref{secosc} and Section~\ref{secrnosc} where our major theorems are derived. Section~\ref{secapp} uses the results of Section~\ref{secosc} and \ref{secrnosc} to investigate the spectra of short-range perturbations of periodic Jacobi operators. \section{Pr\"ufer Variables} \label{secpruf} For the rest of this chapter we assume for convenience \bh \label{hypoaposb} Suppose \begin{equation} a,b \in \ell(\Z), \qquad a(n)< 0, b(n) \in \R. \end{equation} \eh We remark that the case $a(n) \ne 0$ can be reduced to the case $a(n)>0$ or $a(n)<0$ (cf., e.g., \cite{eil}, p.\ 141). In fact one has \begin{lem} \label{lemaeps} Assume (H.\ref{hypoab}) and let $H$ be a Jacobi operator associated with the difference expression (\ref{diffex}). Introduce $a_\eps$ by \bea a_\eps(n) = \eps(n)a(n), \; \eps(n) \in\{+1, -1\}, \quad n\in\Z \eea and the unitary operator $U_\eps$ by \bea U_\eps =\{ \tilde{\eps}(n) \delta_{m,n} \}_{m,n\in\Z}, \quad \tilde{\eps}(n) \in\{+1,-1\}, \; \tilde{\eps}(n) \tilde{\eps}(n+1) = \eps(n). \eea Then $H_\eps$ defined as \bea H_\eps = U_\eps^{-1} H U_\eps, \eea is associated with the difference expression \bea (\tau_\eps f)(n) = a_\eps(n) f(n+1) + a_\eps(n-1) f(n-1) - b_\eps(n) f(n). \eea In particular, $H_\eps$ is unitarily equivalent to $H$. \end{lem} In addition, by a solution of $\tau u = \lam u$, $\lam \in \R$ we will always mean a real-valued solution not vanishing identically. Given a solution $u(\lam,.)$ of $\tau u =\lam u$, $\lam\in\R$ we introduce Pr\"ufer variables $\rho_u(\lam,.)$, $\theta_u(\lam,.)$ via \bea \label{pruef1} u(\lam,n) &=& \rho_u(\lam,n) \sin \theta_u(\lam,n), \\ \label{pruef2} u(\lam,n+1) &=& \rho_u(\lam,n) \cos \theta_u(\lam,n). \eea Notice that the Pr\"ufer angle $\theta_u(\lam,.n)$ is only defined up to an additive integer multiple of $2\pi$ (which depends on $n$). Inserting (\ref{pruef1}), (\ref{pruef2}) into $(\tau - \lam) u=0$ yields \begin{equation} \label{ricphi} a(n) \cot \theta_u(\lam,n) + a(n-1) \tan \theta_u(\lam,n-1) = b(n) + \lam, \end{equation} \begin{equation} \label{eqrho} \rho_u(\lam,n) \sin \theta_u(\lam,n) = \rho_u(\lam,n-1) \cos \theta_u(\lam,n-1). \end{equation} Equation (\ref{ricphi}) is a discrete Riccati equation (cf.\ \cite{hp}) for $\cot\theta_u(n)$ and (\ref{eqrho}) can be solved if $\theta_u(n)$ is known provided it is replaced by \begin{equation} a(n) \rho_u(\lam,n) = a(n-1) \rho_u(\lam,n-1) =0 \end{equation} if $\sin \theta_u(\lam,n) = \cos \theta_u(\lam,n-1) =0$ (use $\tau u =\lam u$ and (\ref{sigch}) below). The Wronskian of two solutions $u_{1,2}(\lam_{1,2},n)$ reads \begin{equation} W_n(u_1(\lam_1), u_2(\lam_2)) = a(n) \rho_{u_1}(\lam_1,n) \rho_{u_2}(\lam_2,n) \sin(\theta_{u_1}(\lam_1,n) - \theta_{u_2}(\lam_2,n)). \end{equation} The next lemma considers nodes of solutions and their Wronskians more closely. \bl Let $u_{1,2}$ be solutions of $\tau u_{1,2} =\lam_{1,2} u_{1,2}$ corresponding to $\lam_1 \ne \lam_2$, respectively. Then \begin{equation} \label{sigch} u_1(n) =0 \quad\Rightarrow\quad u_1(n-1)u_1(n+1)<0. \end{equation} Moreover, suppose $W_n(u_1,u_2) = 0$ but $W_{n-1}(u_1,u_2) W_{n+1}(u_1,u_2) \ne 0$, then \begin{equation} W_{n-1}(u_1,u_2) W_{n+1}(u_1,u_2) < 0. \end{equation} Otherwise, if $W_n(u_1,u_2) = W_{n+1}(u_1,u_2) =0$, then necessarily \begin{equation} u_1(n+1)=u_2(n+1)=0, \quad\mbox{and}\quad W_{n-1}(u_1,u_2) W_{n+2}(u_1,u_2)< 0. \end{equation} \el \bpf The fact $u(n)=0$ implies $u_1(n-1)u_1(n+1) \ne 0$ (otherwise $u_1$ vanishes identically) and $a(n) u_1(n+1) = -a(n-1) u_1(n-1)$ (from $\tau u =\lam u$) shows $u_1(n-1) u_1(n+1)<0$. Next, $W_n(u_1,u_2)=0$ is equivalent to $u_1(n) = c u_2(n)$, $u_1(n+1) = c u_2(n+1)$ for some $c \ne 0$ and from (\ref{gf}) we infer \begin{equation} W_{n+1}(u_1,u_2) - W_n(u_1,u_2) = (\lam_2-\lam_1) u_1(n+1) u_2(n+1). \end{equation} Applying the above formula gives \begin{equation} W_{n-1}(u_1,u_2) W_{n+1}(u_1,u_2) = - c^2 (\lam_2-\lam_1)^2 u_1(n)^2 u_1(n+1)^2 \end{equation} proving the first claim. If $W_n(u_1,u_2)$, $W_{n+1}(u_1,u_2)$ are both zero we must have $u_1(n+1)=u_2(n+1)=0$ and as before $W_{n-1}(u_1,u_2) W_{n+1}(u_1,u_2) = - (\lam_2-\lam_1)^2 u_1(n-1) u_1(n+2) u_2(n-1) u_2(n+2)$. Hence the claim follows from the first part. \epf We can make the Pr\"ufer angel $\theta_u(\lam,.)$ unique by fixing, for instance, $\theta_u(\lam,0)$ and requiring \begin{equation} \label{normalth} \intp{\theta_u(\lam,n)/ \pi} \le \intp{\theta_u(\lam,n+1)/ \pi} \le \intp{\theta_u(\lam,n)/ \pi} +1, \end{equation} where \begin{equation} \intp{x} = \sup \{n \in \Z \,|\, n \sin(\Delta(\lam_1)) = \cos(k\pi)\sin(k\pi + \Delta(\lam_1)) = R^{-1} u(\lam_1,n+1)^2>0$, contradicting (i). Case (ii) is similar. Case (iii) implies $\delta(\lam_1)=\pi/2$ and hence $1=\sin(k\pi+\pi/2)\cos(k\pi) = R^{-1} u(\lam_1,n) u(\lam_1,n+2)$ contradicting (\ref{sigch}). Again, case (iv) is similar. \epf Let us call a point $n \in \Z$ a node of a solution $u$ if either $u(n)=0$ or $a(n)u(n)u(n+1)>0$. Then, $\intp{\theta_u(n)/ \pi} = \intp{\theta_u(n+1)/ \pi}$ implies no node at $n$. Conversely, if $\intp{\theta_u(n+1)/ \pi} = \intp{\theta_u(n)/ \pi} +1$, then $n$ is a node by (\ref{sigch}). Denote by $\#(u)$ the total number of nodes of $u$ and by $\#_{(m,n)}(u)$ the number of nodes of $u$ between $m$ and $n$. More precisely, we shall say that a node $n_0$ of $u$ lies between $m$ and $n$ if either $m0$ for $m0$ for $m\le j\le n$. Moreover, $u_1(m) \le 0$, $u_1(n+1)<0$ and $u_2(n+1) \ge 0$ provided $m,n$ are finite. By Green's formula (\ref{gf}) \begin{equation} 0 \le (\lam_2 - \lam_1) \sum_{j=m+1}^n u_1(j) u_2(j) = W_n(u_1,u_2) - W_m(u_1,u_2). \end{equation} Evaluating the Wronskians shows $W_n(u_1,u_2) < 0$, $W_m(u_1,u_2) > 0$ which is a contradiction. It remains to prove the last part. We will use induction on $k$. The case $k=1$ is trivial and $k=2$ has already been proven. Denote the nodes of $u_2$ lower or equal than $m_{k+1}$ by $n_k > n_{k-1} > \cdots$. If $n_k > m_k$ we are done since there are $k-1$ nodes $n$ such that $m_1 \le n \le m_k$ by induction hypothesis. Otherwise we can find $k_0$, $0 \le k_0 \le k$ such that $m_j=n_j$ for $1+k_0 \le j \le k$. If $k_0=0$ we are clearly done and we can suppose $k_0 \ge 1$. By induction hypothesis it suffices to show that there are $k-k_0$ nodes $n$ of $u_2$ with $m_{k_0} \le n \le m_{k+1}$. By assumption $m_j=n_j$, $1+k_0 \le j \le k$ are the only nodes $n$ of $u_2$ such that $m_{k_0} \le n \le m_{k+1}$. Abbreviate $m=m_{k_0}$, $n= m_{k+1}$ and assume without restriction $u_1(m+1)>0$, $u_2(m)>0$. Since the nodes of $u_1$ and $u_2$ coincide we infer $0<\sum_{j=m+1}^n u_1(j) u_2(j)$ and we can proceed as in the first part to obtain a contradiction. \epf We call $\tau$ oscillatory if one solution of $\tau u =0$ has an infinite number of nodes. In addition, we call $\tau$ oscillatory at $\pm\infty$ if one solution of $\tau u =0$ has an infinite number of nodes near $\pm\infty$. We remark that if one solution of $(\tau - \lam)u =0$ has infinitely many nodes so has any other (corresponding to the same $\lam$) by (\ref{diffnb}). Furthermore, $\tau - \lam_1$ oscillatory implies $\tau - \lam_2$ oscillatory for all $\lam_2>\lam_1$ (again by (\ref{diffnb})). Now we turn to the special solution $s(\lam,n)$ characterized via the initial conditions $s(\lam,0)=0$, $s(\lam,1)=1$. As in Lemma~\ref{wuudot} we infer \bea W_n(s(\lam),\dot{s}(\lam)) &=& \sum_{j=n+1}^0 s(\lam,j)^2, \quad n<-1, \\ W_n(s(\lam),\dot{s}(\lam)) &=& \sum_{j=1}^n s(\lam,j)^2, \quad n\ge 1. \eea Here the dot denotes the derivative with respect to $\lam$. Notice also $W_{-1}(s(\lam),\dot{s}(\lam)) = W_0(s(\lam),\dot{s}(\lam)) =0$. Evaluating the above equation using Pr\"ufer variables shows \bea \label{thetaspos} \dot{\theta}_s(\lam,n) &=& \frac{\sum_{j=1}^n s(\lam,j)^2}{ -a(n) \rho_s(\lam,n)^2} > 0, \quad n \ge 1,\\ \dot{\theta}_s(\lam,n) &=& \frac{\sum_{j=n+1}^0 s(\lam,j)^2}{ a(n) \rho_s(\lam,n)^2} < 0, \quad n<-1. \eea Notice, again that $\dot{\theta}_s(\lam,-1)=\dot{\theta}_s(\lam,0)=0$. Equation (\ref{thetaspos}) implies that nodes of $s(\lam,n)$ for $n \in \N$ move monotonically to the left without colliding (cf., \cite{at} Theorem~4.3.4). In addition, since $s(\lam,n)$ cannot pick up nodes locally by (\ref{sigch}), all nodes must enter at $\infty$ and since $\dot{\theta}_s(\lam,0) =0$ they are trapped inside $(0,\infty)$. We shall normalize $\theta_s(\lam,0)=0$ and hence $\theta_s(\lam,-1)=-\pi/2$. Since $s(\lam,n)$ is a polynomial in $\lam$ we easily infer $s(\lam,n) \gele 0$ for fixed $n\gele 0$ and $\lam$ sufficiently small. This implies \begin{equation} \label{thetassmlam} -\pi < \theta_s(\lam,n) < -\pi/2, \: n < -1, \quad 0 < \theta_s(\lam,n) < \pi, \: n \ge 1, \end{equation} for fixed $n$ and $\lam$ sufficiently small. Moreover, dividing (\ref{ricphi}) by $\lam$ and letting $\lam \to -\infty$ using (\ref{thetassmlam}) shows \begin{equation} \lim_{\lam \to \pm\infty} \frac{\cot(\theta_s(\lam,n))^{\pm1}}{\lam} = \frac{1}{a(n)}, \quad n \, {\ge +1 \atop < -1 } \end{equation} and hence \begin{equation} \label{limthetas} \theta_s(\lam,n) = -\frac{\pi}{2}-\frac{a(n)}{\lam} + o(\frac{1}{\lam}), \: n<-1, \quad \theta_s(\lam,n) = \frac{a(n)}{\lam} + o(\frac{1}{\lam}), \: n \ge 1, \end{equation} as $\lam \to -\infty$. Analogously, let $u_\pm(\lam,n)$ be solutions of $\tau u =\lam u$ as in Lemma \ref{upmholz}. Then Lemma \ref{wuudot} implies \bea \dot{\theta}_+(\lam,n) &=& \frac{\sum_{j=n+1}^\infty u_+(\lam,j)^2}{ a(n)\rho_+(\lam,n)^2} < 0,\\ \label{thetaminc} \dot{\theta}_-(\lam,n) &=& \frac{\sum_{j=-\infty}^n u_-(\lam,j)^2}{ -a(n)\rho_-(\lam,n)^2} > 0, \eea where we have abbreviated $\rho_{u_\pm}=\rho_\pm$, $\theta_{u_\pm}=\theta_\pm$. If $H$ is bounded from below we can normalize \begin{equation} 0 < \theta_\mp(\lam,n) < \pi/2, \quad n \in \Z, \quad\lam < \inf\sig(H) \end{equation} and we get as before \begin{equation} \label{limthetapm} \theta_-(\lam,n) = \frac{a(n)}{\lam} + o(\frac{1}{\lam}), \quad \theta_+(\lam,n) = \frac{\pi}{2} - \frac{a(n)}{\lam} + o(\frac{1}{\lam}), \quad n \in \Z \end{equation} as $\lam \to -\infty$. \section{Standard Oscillation Theory} \label{secosc} First of all we recall (\cite{gst}, Lemma 5.2). \bl \label{lemsrc} Let $H,H_n$ be self-adjoint operators and $H_n \to H$ in strong resolvent sense as $n \to \infty$. Then \begin{equation} \dim\Ran\,P_{(\lam_1,\lam_2)}(H)\le \liminf_{n \to \infty} \dim\Ran\, P_{(\lam_1,\lam_2)}(H_n). \end{equation} \el Our first theorem considers half-line operators $H_\pm$ associated with a Dirichlet boundary condition at $n=0$, that is, the following restrictions of $H$ to the subspaces $\ell^2(\pm\N)$, \begin{equation} \ba{llcl} H_\pm :& \db(H_\pm) & \to & \ell^2(\pm\N) \\ & f(n) &\mapsto& \left\{ \ba{l} a({+1 \atop -2})f(\pm 2) - b(\pm1) f(\pm 1), \: n=\pm1 \\ (\tau f)(n), \quad n \gele \pm1 \ea \right. \ea , \end{equation} with \begin{equation} \db(H_\pm) = \{f \in \ell^2(\pm\N) | \tau f \in \ell^2(\pm\N), \: \lim_{n \to \pm\infty} W_n(u_\pm(z_0),f) =0 \}. \end{equation} Similarly one defines finite restriction $H_{n_1,n_2}$ to the subspaces $\ell^2(n_1,n_2)$ with Dirichlet boundary conditions at $n=n_1$ and $n=n_2$. \br \label{rembc} We only consider the case of a Dirichlet boundary condition at $n=0$ since the operators $H_{\pm,n_0}^\beta$ on $\ell^2(n_0,\pm\infty)$ associated with the general boundary condition \begin{equation} \label{boundcon} f(n_0+1) + \beta f(n_0)=0, \qquad \beta \in \R \cup \{ \infty \} \end{equation} at $n=n_0$ can be reduced to this case by a simple shift and altering the sequence $b$ at one point. More precisely, we have \begin{equation} \label{hbetann} H^0_{+,n_0} = H_{+,n_0+1}, \quad H^\beta_{+,n_0} = H_{+,n_0} - a(n_0) \beta^{-1} \spr{\delta_{n_0+1}}{.}\delta_{n_0+1}, \quad \beta \ne 0, \end{equation} and \begin{equation} H^\infty_{-,n_0} = H_{-,n_0}, \quad H^\beta_{-,n_0} = H_{-,n_0+1} - a(n_0) \beta \spr{\delta_{n_0}}{.}\delta_{n_0}, \quad \beta \ne \infty, \end{equation} where $\delta_{n_0}(n) = 1$ if $n=n_0$ and $\delta_{n_0}(n) = 0$ otherwise. Hence all one has to do is alter the definition of $b(n_0)$ or $b(n_0+1)$. Analogously one defines the corresponding finite operators $H_{n_1,n_2}^{\beta_1,\beta_2}$ which will be used in the next section. \er \bth \label{nbhpm} Let $\lam \in \R$. Suppose $\tau$ is $l.p.$ at $+\infty$ or $\lam \in \sig_p(H_+)$. Then \begin{equation} \dim\Ran\, P_{(-\infty,\lam)}(H_+) = \#_{(0,+\infty)}(s(\lam)). \end{equation} The same theorem holds if $+$ is replaced by $-$. \eth \bpf We only carry out the proof for the plus sign (the other part following from reflection). By virtue of (\ref{thetaspos}), (\ref{limthetas}), and Lemma~\ref{nbofu} we infer \begin{equation} \label{dimhhzn} \dim\Ran\, P_{(-\infty,\lam)}(H_{0,n}) = \intp{\theta_s(\lam,n)/\pi} =\#_{(0,n)}(s(\lam)), \quad n>1, \end{equation} since $\lam \in \sig(H_{0,n})$ if and only if $\theta_s(\lam,n) = 0 \mod\pi$. Let $k = \#(s(\lam))$ if $\#(s(\lam))<\infty$, otherwise the following argument works for arbitrary $k \in \N$. If we pick $n$ so large that $k$ nodes of $s(\lam)$ are to the left of $n$ we have $k$ eigenvalues $\hat{\lam}_1 < \cdots < \hat{\lam}_k <\lam$ of $H_{0,n}$. Taking an arbitrary linear combination $\eta(m) = \sum_{j=1}^k c_j s(\hat{\lam}_j,m)$, $c_j \in \C$ for $m\theta_s(\lam_n,m)$ we infer that $\theta_s(\lam_{n+1},m)$ has either more nodes between $0$ and $m$ or there is at least one additional node of $\theta_s(\lam_{n+1},m)$ larger than $m$ by Lemma~\ref{sturm}. In the case where $H$ is not bounded from below we can label the eigenvalues $\lam_n$, $n\in\Z$. The same argument as before shows that the eigenfunction corresponding to $\lam_m$ has $|m-n|$ nodes more than the one corresponding to $\lam_n$. Letting $m \to -\infty$ shows that the eigenfunction corresponding to $\lam_n$ has infinitely many nodes. This completes the proof. \epf \br \label{rembcch} (i) The $l.p.$ / $\lam\in\sig_p(H_+)$ assumption is crucial since we need some information about the boundary condition at $+\infty$.\\ (ii) The previous remark implies the following. Let $\lam \in \R$. Suppose $\tau$ is $l.p.$ at $+\infty$ or $\lam \in \sig_p(H^\beta_{+,n_0})$ and $\beta \ne 0$. Then \begin{equation} \dim\Ran\, P_{(-\infty,\lam)}(H^\beta_{+,n_0}) = \#_{(0,+\infty)}(s_\beta(\lam,.,n_0)), \end{equation} where $s_\beta(\lam,.,n_0)$ is a sequence satisfying $\tau s = \lam s$ and the boundary condition (\ref{boundcon}). Similar modifications apply to Theorems~\ref{thmhartman}, \ref{thmwrfin}, and \ref{thmzerwronshl}. \er As a consequence of Theorem~\ref{nbhpm} we infer \bk \label{corosc} We have \begin{equation} \dim\Ran\, P_{(-\infty,\lam)}(H_\pm) < \infty \end{equation} if and only if $\tau-\lam$ is non-oscillatory near $\pm\infty$, respectively, and hence \begin{equation} \inf\sig_{ess}(H_\pm) = \inf \{ \lam \in \R \,|\, (\tau - \lam) \mbox{ is oscillatory at }\pm\infty \}. \end{equation} Moreover, let $H_\pm$ be bounded from below and $\lam_1 < \cdots < \lam_k < \cdots$ be the eigenvalues of $H_\pm$ below the essential spectrum of $H_\pm$. Then the eigenfunction corresponding to $\lam_k$ has precisely $k-1$ nodes inside $(0,\pm\infty)$. \ek We remark that the first part of Corollary~\ref{corosc} can be found in \cite{glz}, Theorem 32 (see also \cite{hl}). \br Consider the following example \begin{equation} a(n) = -\frac{1}{2}, \: n \in\N, \quad b(1) =1, b(2) =b_2, b(3) =\frac{1}{2}, b(n)=0, \: n \ge 4. \end{equation} The essential spectrum of $H_+$ is given by $\sig_{ess}(H_+) = [-1,1]$ and one might expect that $H_+$ has no eigenvalues below the essential spectrum if $b_2 \to -\infty$. However, since we have \begin{equation} s(-1,0)=0, s(-1,1)=1, s(-1,2)=0, s(-1,n)=-1, \: n \ge 3, \end{equation} Theorem~\ref{nbhpm} shows that, independent of $b_2 \in \R$, there is always precisely one eigenvalue below the essential spectrum. \er In a similar way we obtain \bth \label{thmbelesssph} Let $\lam <\inf\sig_{ess}(H)$. Suppose $\tau$ is $l.p.$ at $-\infty$ or $\lam \in \sig_p(H)$. Then \begin{equation} \dim\Ran\, P_{(-\infty,\lam)}(H) = \#(u_+(\lam)). \end{equation} The same theorem holds if $l.p.$ at $-\infty$ and $u_+(\lam)$ is replaced by $l.p.$ at $+\infty$ and $u_-(\lam)$. \eth \bpf Again it suffices to prove the minus case. If $H$ is not bounded from below the same is true for $H_- \oplus H_+$ (which can be embedded into $\lz$ and considered as a finite rank perturbation of $H$). Hence $H_-$ or $H_+$ (or both) is not bounded from below implying $\tau-\lam$ oscillatory near $-\infty$ or $+\infty$ by Corollary~\ref{corosc} and we can suppose $H$ bounded from below. By virtue of (\ref{thetaminc}) and (\ref{limthetapm}) we infer \begin{equation} \dim\Ran\, P_{(-\infty,\lam)}(H_{-,n}) = \intp{\theta_-(\lam,n)/\pi}, \quad n\in\Z. \end{equation} We first want to show $\intp{\theta_-(\lam,n)/\pi} = \#_{(-\infty,n)}(u_-(\lam))$ or equivalently \begin{equation} \lim_{n \to \infty} \intp{\theta_-(\lam,n)/\pi} =0. \end{equation} Suppose $\lim_{n \to \infty} \intp{\theta_-(\lam_1,n)/\pi} =k\ge 1$ for some $\lam_1 \in \R$ (saying that $u_-(.,n)$ loses at least one node at $-\infty$). In this case we can find $n$ such that $\theta_-(\lam_1,n) > k\pi$ for $m \ge n$. Now pick $\lam_0$ such that $\theta_-(\lam_0,n) = k\pi$. Then $u_-(\lam_0,.)$ has a node at $n$ but no node between $-\infty$ and $n$ (by Lemma \ref{nbofu}). Now apply Lemma \ref{sturm} to $u_-(\lam_0,.)$, $u_-(\lam_1,.)$ to obtain a contradiction. The rest follows as in the proof of Theorem~\ref{nbhpm}. \epf As before we obtain \bk \label{coresssp} We have \begin{equation} \dim\Ran\, P_{(-\infty,\lam)}(H) < \infty \end{equation} if and only if $\tau-\lam$ is non-oscillatory and hence \begin{equation} \inf\sig_{ess}(H) = \inf\{ \lam \in \R \,|\, (\tau - \lam) \mbox{ is oscillatory} \}. \end{equation} Furthermore, let $H$ be bounded from below and $\lam_1 < \cdots < \lam_k < \dots$ be the eigenvalues of $H$ below the essential spectrum of $H$. Then the eigenfunction corresponding to $\lam_k$ has precisely $k-1$ nodes. \ek \br \label{remabovesp} Corresponding results for the projection $P_{(\lam,\infty)}(H)$ can be obtained from $P_{(\lam,\infty)}(H) = P_{(-\infty,-\lam)}(-H)$. In fact, it suffices to change the definition of a node according to $u(n)=0$ or $a(n)u(n)u(n+1)<0$ and $P_{(-\infty,\lam)}(H)$ to $P_{(\lam,\infty)}(H)$ in all results of this section. \er Now we turn to the analog of \cite{har2}, Theorem I. \bth \label{thmhartman} Let $\lam_1<\lam_2$. Suppose $\tau-\lam_2$ is oscillatory near $+\infty$ and $\tau$ is $l.p.$ at $+\infty$. Then \begin{equation} \dim\Ran\, P_{(\lam_1,\lam_2)}(H_+) = \liminf_{n \to +\infty}\Big( \#_{(0,n)}(s(\lam_2)) - \#_{(0,n)}(s(\lam_1)) \Big). \end{equation} The same theorem holds if $+$ is replaced by $-$. \eth \bpf As before we only carry out the proof for the plus sign. Abbreviate $\Delta(n) = \intp{\theta_s(\lam_2,n)/\pi} -\intp{\theta_s(\lam_1,n)/\pi} = \#_{(0,n)}(s(\lam_2)) - \#_{(0,n)}(s(\lam_1))$. By (\ref{dimhhzn}) we infer \begin{equation} \dim\Ran\, P_{[\lam_1,\lam_2)}(H_{0,n}) = \Delta(n), \quad n>2. \end{equation} Let $k=\liminf \Delta(n)$ if $\limsup \Delta(n)<\infty$ and $k \in \N$ otherwise. We contend that there exists a $n \in \N$ such that \begin{equation} \dim\Ran\, P_{(\lam_1,\lam_2)}(H_{0,n}) \ge k. \end{equation} In fact, if $k=\limsup \Delta(n)<\infty$ it follows that $\Delta(n)$ is eventually equal to $k$ and since $\lam_1 \not\in \sig(H_{0,m}) \cap \sig(H_{0,m+1})$, $m \in\N$ we are done in this case. Otherwise we can pick $n$ such that $\dim\Ran\, P_{[\lam_1,\lam_2)}(H_{0,n}) \ge k+1$. Hence $H_{0,n}$ has at least $k$ eigenvalues $\hat{\lam}_j$ with $\lam_1 < \hat{\lam}_1 < \dots < \hat{\lam}_k < \lam_2$. Again let $\eta(m) = \sum_{j=1}^k c_j s(\hat{\lam}_j,n)$, $c_j \in \C$ for $m0$ since $\delta=0$ implies $\delta_1=\delta_2$. Moreover, $k = (k_2 - k_1) \mod 2$ and $k\pi + \Delta = (k_2-k_1)\pi + \Delta_2 - \Delta_1$ implies $\Delta = (\Delta_2 - \Delta_1) \mod 2\pi$. Hence we have $\sin\Delta \ge 0$ and $\Delta \in (\pi,2\pi]$ implies $\Delta = 2\pi$. But this says $\delta_1=\delta_2=\pi/2$ and $\Delta_1=\Delta_2 = \pi$. Since we have at least $\delta(\lam_2-\eps)>0$ and hence $\delta_2(\lam_2-\eps)>\pi/2$, $\Delta_2(\lam_2-\eps)>\pi$ for $\eps>0$ sufficiently small. Thus from $\Delta(\lam_2-\eps) \in (\pi,2\pi)$ we get \begin{equation} 0> \sin\Delta(\lam_2-\eps) = \sin(\Delta_2(\lam_2-\eps) -\pi) >0, \end{equation} contradicting (i). Suppose (ii). Again by (\ref{wueulam}) we have $\sin(\Delta_2-\Delta_1)\ge 0$ since $\delta_1 =\delta_2$. But now $(k + 1) = (k_1-k_2) \mod 2$. Furthermore, $\sin(\Delta_2-\Delta_1) = - \sin(\Delta) \ge 0$ says $\Delta = \pi$ since $\Delta \in (0,\pi]$. Again this implies $\delta_1=\delta_2=\pi/2$ and $\Delta_1=\Delta_2 = \pi$. But since $\delta(\lam)$ increases/decreases precisely if $\Delta(\lam)$ increases/decreases for $\lam$ near $\lam_0$ (\ref{normwr}) stays valid. Suppose (iii) or (iv). Then \begin{equation} W_n(u_1,u(\lam_0)) = -(\lam_0 - \lam_1) u_1(n+1) u(\lam_0,n+1). \end{equation} Inserting Pr\"ufer variables gives \begin{equation} \label{sindelzmdele} \sin(\delta_2 - \delta_1) = -\rho \sin(\Delta_1) \sin(\Delta_2) \end{equation} for some $\rho>0$. We first assume $\delta_2>\delta_1$. In this case we infer $k=(k_2-k_1) \mod 2$ implying $\Delta_2-\Delta_1 = 0 \mod 2\pi$ contradicting (\ref{sindelzmdele}). Next assume $\delta_2 \le \delta_1$. Then we obtain $(k+1)=(k_2-k_1) \mod 2$ implying $\Delta_2-\Delta_1 = \pi \mod 2\pi$ and hence $\sin(\delta_2 - \delta_1) \ge 0$ from (\ref{sindelzmdele}). Thus we get $\delta_1=\delta_2=\pi/2$ $\Delta_1=\Delta_2 = \pi$, and hence $\Delta_2-\Delta_1 = 0 \mod 2\pi$ contradicting (iii), (iv). This settles (\ref{deluoul}). Furthermore, if $\Delta(\lam) \in (0,\pi]$ we have no node at $n$ since $\delta(\lam)=\pi$ implies $\Delta(\lam) =\pi$ by (ii). Conversely, if $\Delta(\lam) \in (\pi,2\pi]$ we have a node at $n$ since $\Delta(\lam)=2\pi$ is impossible by (iii). The rest being straightforward. \epf Equations (\ref{nbunm}), (\ref{deluoul}), and (\ref{eqpropwueuz}) imply \bk \label{cornodw} Let $\lam_1 \le \lam_2$ and suppose $u_{1,2}$ satisfy $\tau u_{1,2} = \lam_{1,2} u_{1,2}$, respectively. Then we have \begin{equation} | \#_{(n,m)}W(u_1,u_2) - (\#_{(n,m)}(u_2) - \#_{(n,m)}(u_1))| \le 2 \end{equation} \ek Now we come to a renormalized version of Theorem \ref{thmhartman}. We first need the result for a finite interval. \bth \label{thmwrfin} Fix $n_10$ for $\lam>\lam_1$. Hence if we chose $n$ so large that all $k$ nodes are to the left of $n$ we have \begin{equation} \Delta(\lam,n) > k \pi. \end{equation} Thus we can find $\lam_1 < \hat{\lam}_1 < \cdots < \hat{\lam}_k < \lam_2$ with $\Delta(\hat{\lam}_j,n) = j \pi$. Now define \begin{equation} \eta_j(m) = \left\{ \ba{ll} s(\hat{\lam}_j,m) - \rho_j s(\lam_1,m) & m \le n \\ 0 & m \ge n \ea\right. , \end{equation} where $\rho_j \ne 0$ is chosen such that $s(\hat{\lam}_j,m) = \rho_j s(\lam_1,m)$ for $m=n,n+1$. Furthermore observe that \begin{equation} \tau \eta_j(m) = \left\{ \ba{ll} \hat{\lam}_j s(\hat{\lam}_j,m) - \lam_1 \rho_1 s(\lam_1,m)& m \le n \\ 0 & m \ge n \ea\right. \end{equation} and that $s(\lam_1,m)$, $s(\hat{\lam}_j,.)$, $1\le j \le k$ are orthogonal on $1,\dots,n$. Next, let $\eta = \sum_{j=1}^k c_j \eta_j$, $c_j$ be an arbitrary linear combination, then a short calculation verifies \begin{equation} \| (H_+ - \frac{\lam_2+\lam_1}{2}) \eta \| < \frac{\lam_2-\lam_1}{2} \| \eta \|. \end{equation} And invoking the spectral theorem gives \begin{equation} \label{ineqwsso} \dim\Ran\, P_{(\lam_1,\lam_2)}(H_+) \ge k. \end{equation} To prove the reversed inequality is only necessary if $\#_{(0,\infty)} W(s(\lam_1),${}$s(\lam_2))<\infty$. In this case we look at $H_{0,n}^{\infty,\beta}$ with $\beta = s(\lam_2,n+1)/s(\lam_2,n)$. By Theorem~\ref{thmwrfin} and Remark~\ref{rembcch} (ii) we have \begin{equation} \dim\Ran\, P_{(\lam_1,\lam_2)}(\ti{H}_{0,n}^{\infty,\beta}) = \#_{(0,n)} W(s(\lam_1),s(\lam_2)). \end{equation} Now use strong resolvent convergence of $\ti{H}_{0,n}^{\infty,\beta} = H_{0,n}^{\infty,\beta} \oplus \lam_1\id$ to $H_+$ (due to our $l.p.$ / $\lam_2 \in \sig_p(H_+)$ assumption) as $n\to\infty$ to obtain \begin{equation} \label{ineqwsst} \dim\Ran\, P_{(\lam_1,\lam_2)}(H_+) \le \liminf_{n\to\infty} \dim\Ran\, P_{(\lam_1,\lam_2)}(\ti{H}_{0,n}^{\infty,\beta}) = k \end{equation} completing the proof. \epf As a consequence we infer. \bk \label{coresslamot} Let $u_{1,2}$ satisfy $\tau u_{1,2} = \lam_{1,2} u_{1,2}$. Then \begin{equation} \#_{(0,\pm\infty)}W(u_1,u_2) < \infty \quad\Leftrightarrow\quad \dim\Ran\, P_{(\lam_1,\lam_2)}(H_\pm) < \infty. \end{equation} \ek \bpf By Corollary~\ref{cornodw} the result does not depend on the choice of $u_{1,2}$. Since the proof of (\ref{ineqwsso}) does not use the $l.p.$ / $\lam_2 \in\sig_p(H_+)$ assumption the first direction follows. Conversely, we can replace the sequence $\beta$ in (\ref{ineqwsst}) by a sequence $\hat{\beta}$ such that $\ti{H}_{0,n}^{\infty,\hat{\beta}}$ converges to $H_+$. Since we have $|\dim\Ran\,\ti{H}_{0,n}^{\infty,\hat{\beta}} - \dim\Ran\,\ti{H}_{0,n}^{\infty,\beta}| \le 1$ the corollary is proven. \epf Finally we turn to our main result for Jacobi operators $H$ on $\Z$. We emphasize that to date, Theorem~\ref{thmzerwon} appears to be the only oscillation theoretic result concerning the number of eigenvalues in essential spectral gaps of Jacobi operators on $\Z$. \bth \label{thmzerwon} Fix $\lam_1<\lam_2$ and suppose $[\lam_1,\lam_2] \cap \sig_{ess}(H) = \emptyset$. Then \begin{equation} \dim\Ran\, P_{(\lam_1,\lam_2)}(H) = \#W(u_\mp(\lam_1),u_\pm(\lam_2)). \end{equation} In addition, if $\tau$ is $l.p.$ at $+\infty$ we even have \begin{equation} \dim\Ran\, P_{(\lam_1,\lam_2)}(H) = \#W(u_+(\lam_1),u_+(\lam_2)). \end{equation} The same result holds if $+$ is replaced by $-$. \eth \bpf Since the proof is similar to the proof of Theorem \ref{thmzerwronshl} we shall only outline the first part. Let $k = \#W(u_+(\lam_1),u_-(\lam_2))$ if this number is finite and $k \in \N$ else. Pick $n>0$ so large that all zeros of the Wronskian are between $-n$ and $n$. We abbreviate \begin{equation} \Delta(\lam,n) = \Delta_{u_+(\lam_1),u_-(\lam)}(n) \end{equation} and normalize $\Delta(\lam_1,n) \in [0,\pi)$ implying $\Delta(\lam,n) >0$ for $\lam>\lam_1$. Hence if we chose $n \in \N$ so large that all $k$ nodes are between $-n$ and $n$ we can assume \begin{equation} \Delta(\lam,n) > k \pi. \end{equation} Thus we can find $\lam_1 < \hat{\lam}_1 < \cdots < \hat{\lam}_k < \lam_2$ with $\Delta(\hat{\lam}_j,n) = 0 \mod\pi$. Now define \begin{equation} \eta_j(m) = \left\{ \ba{ll} u_-(\hat{\lam}_j,m) & m \le n \\ \rho_j u_+(\lam_1,m) & m \ge n \ea\right. , \end{equation} where $\rho_j \ne 0$ is chosen such that $u_-(\hat{\lam}_j,m) = \rho_j u_+(\lam_1,m)$ for $m=n,n+1$. Now proceed as in the previous theorems. \epf Again, we infer as a consequence. \bk \label{coresslamote} Let $u_{1,2}$ satisfy $\tau u_{1,2} = \lam_{1,2} u_{1,2}$. Then \begin{equation} \#W(u_1,u_2) < \infty \quad\Leftrightarrow\quad \dim\Ran\, P_{(\lam_1,\lam_2)}(H) < \infty. \end{equation} \ek \bpf Follows from Corollaries~\ref{cornodw}, \ref{coresslamot}, and $\dim\Ran\, P_{(\lam_1,\lam_2)}(H)<\infty$ if and only if $(\dim\Ran$ $P_{(\lam_1,\lam_2)}(H_-) + \dim\Ran\, P_{(\lam_1,\lam_2)}(H_+))<\infty$. \epf \br The most general three-term recurrence relation \begin{equation} \ti{\tau} f(n) = \ti{a}(n) f(n+1) - \ti{b}(n) f(n) + \ti{c}(n) f(n-1), \end{equation} with $\ti{a}(n)\ti{c}(n)>0$, can be transformed to a Jacobi recurrence relation as follows. First we symmetrise $\ti{\tau}$ via \begin{equation} \ti{\tau} f(n) = \frac{1}{w(n)} \Big( c(n) f(n+1) + c(n-1) f(n-1) - d(n) f(n) \Big), \end{equation} where \bea &w(n) = \left\{ \ba{c@{\quad\mb{ for }}l} \prod\limits_{j=n_0}^{n-1} \frac{\ti{a}(j)}{\ti{c}(j+1)} & n > n_0 \\ 1 & n=n_0\\ \prod\limits_{j=n}^{n_0-1} \frac{\ti{c}(j+1)}{\ti{a}(j)} & n < n_0 \ea \right. >0,&\\ &c(n) = w(n) \ti{a}(n) = w(n+1) \ti{c}(n+1), \quad d(n) = w(n) \ti{b}(n).& \eea The natural Hilbert space for $\ti{\tau}$ is the weighted space $\ell^2(\Z,w)$ with scalar product \begin{equation} \spr{f}{g} = \sum_{n\in \Z} w(n) \ol{f(n)} g(n), \qquad f,g \in \ell^2(\Z,w). \end{equation} Let $\ti{H}$ be a self-adjoint operator associated with $\ti{\tau}$ in $\ell^2(\Z,w)$. Then the unitary operator \begin{equation} \label{unopgentojop} \ba{clcl} U: & \ell^2(\Z,w) & \to & \lz \\ & u(n) & \mapsto & \sqrt{w(n)} u(n) \ea \end{equation} transforms $\ti{H}$ into a Jacobi operator $H$ in $\lz$ associated with the sequences \bea a(n) &=& \frac{c(n)}{\sqrt{w(n)w(n+1)}} = \sgn(\ti{a}(n)) \sqrt{\ti{a}(n) \ti{c}(n+1)}, \\ b(n) &=& \frac{d(n)}{w(n)} = \ti{b}(n). \eea In addition we infer \bea \nn \lefteqn{c(n) \Big( f(n) g(n+1) - f(n+1) g(n) \Big) =}\\ && a(n) \Big( (U f)(n) (U g)(n+1) - (U f)(n+1) (U g)(n) \Big). \eea Hence all the results derived for Jacobi operator thus far apply to generalized Jacobi operators of the type $\ti{H}$ as well. \er \section{Applications} \label{secapp} One important class of Jacobi operators are periodic ones (cf., e.g., \cite{bght}, Appendix B, \cite{kr1}, \cite{vm}). Instead of periodic operators themselves we are interested in short-range perturbations of these operators. In fact, we are going to prove the analog of the Theorem by Rofe-Beketov (\cite{rof}, see also \cite{gs}, \cite{glz}, Section~67, \cite{kl}) about the finiteness of the number of eigenvalues in essential spectral gaps of the perturbed Hill operator. Since constant coefficients $a,b$ are a special case of periodic ones our results contain results from scattering theory (cf., e.g., \cite{dinv1}, \cite{gu}). To set the stage, we first recall some basic facts from the theory of periodic operators. Let $H_p$ be a Jacobi operator associated with periodic sequences $a_p<0,b_p$, that is, \begin{equation} a_p(n+N) =a_p(n), \qquad b_p(n+N) = b_p(n), \end{equation} for some fixed $N \in\N$. The spectrum of $H_p$ is purely absolutely continuous and consists of a finite number of gaps, that is, \begin{equation} \sig(H_p) = \bigcup_{j=0}^g [E_{2j},E_{2j+1}], \qquad g \in\N_0, \end{equation} with $E_0 < E_1 < \cdots < E_{2g+1}$ and $g \le N-1$. Moreover, Floquet theory implies the existence of solutions $u_{p,\pm}(z,.)$ of $\tau_p u = z u$, $z\in\C$ ($\tau_p$ the difference expression corresponding to $H_p$) satisfying \begin{equation} u_{p,\pm}(z,n+N) = m^\pm(z) u_{p,\pm}(z,n), \end{equation} where $m^\pm(z) \in \C$ are called Floquet multipliers. $m^\pm(z)$ satisfy $m^+(z) m^-(z)=1$, $m^\pm(z)^2 =1$ for $z \in \{ E_j \}_{j=0}^{2g+1}$, $|m^\pm(z)|=1$ for $z \in \sig(H_p)$, and $|m^+(z)|<1$ for $z \in \C\bs\sig(H_p)$. (This says in particular, that $u_{p,\pm}(z,.)$ are bounded for $z\in\sig(H_p)$ and linearly independent for $z \in \C\bs\{ E_j \}_{j=0}^{2g+1}$.) We are going to study perturbations $H$ of $H_p$ associated with sequences $a,b$ satisfying $a(n) \to a_p(n)$ and $b(n) \to b_p(n)$ as $|n| \to \infty$. Clearly, $H$ and $H_p$ are both bounded and hence defined on the whole of $\lz$. In fact, we have \begin{equation} \sig(H) \subseteq [\ul{c},\ol{c}], \end{equation} where $\ul{c} = \inf_{n\in\Z} (b(n)+a(n-1)+a(n))$ and $\ol{c} = \sup_{n\in\Z} (b(n)-a(n-1)-a(n))$. Using this notation our theorem reads: \bth \label{thmappl} Suppose $a_p, b_p$ are given periodic sequences and $H_p$ is the corresponding Jacobi operator. Let $H$ be a perturbation of $H_p$ such that \begin{equation} \sum_{n\in\Z} |n(a(n) - a_p(n))| < \infty, \quad \sum_{n\in\Z} |n(b(n) - b_p(n))| < \infty. \end{equation} Then we have $\sig_{ess}(H)=\sig(H_p)$, the point spectrum of $H$ is finite and confined to the spectral gaps of $H_p$, that is, $\sig_p(H) \subset \R\bs\sig(H_p)$. Furthermore, the essential spectrum of $H_p$ is purely absolutely continuous. \eth For the proof we will need the following lemma the proof of which is elementary. \bl \label{voltse} The Volterra sum equation \begin{equation} f(n) = g(n) + \sum_{m=n+1}^\infty K(n,m) f(m), \end{equation} with \begin{equation} |K(n,m)| \le \hat{K}(n,m), \quad \hat{K}(n+1,m) \le \hat{K}(n,m), \quad \hat{K}(n,.) \in \ell^1(0,\infty), \end{equation} has for $g \in \ell^\infty(0,\infty)$ a unique solution $f \in \ell^\infty(0,\infty)$, fulfilling the estimate \begin{equation} |f(n)| \le \Big(\sup_{m>n}|g(m)|\Big) \exp \Big( \sum_{m=n+1}^\infty \hat{K}(n,m) \Big). \end{equation} \el \bpf (of Theorem \ref{thmappl}) The fact that $H-H_p$ is compact implies $\sig_{ess}(H)=\sig_{ess}(H_p)$. To prove the remaining claims it suffices to show the existence of solutions $u_\pm(\lam,.)$ of $\tau u = \lam u$ for $\lam \in \sig(H_p)$ satisfying \begin{equation} \label{asupm} \lim_{|n|\to \infty} |u_\pm(\lam,n) - u_{p,\pm}(\lam,n)| =0. \end{equation} In fact, since $u_\pm(\lam,.)$, $\lam \in \sig(H_p)$ are bounded and do not vanish near $\pm\infty$, there are no eigenvalues in the essential spectrum of $H$ and invoking the principal of subordinacy (cf., \cite{simac}, \cite{st}) shows that the essential spectrum of $H$ is purely absolutely continuous. Moreover, (\ref{asupm}) with $\lam=E_0$ implies that $H -E_0$ is non-oscillatory since we can assume (perhaps after flipping signs) $u_{p,\pm}(E_0,n) \ge \eps >0$, $n\in\Z$ and by Corollary~\ref{coresssp} there are only finitely many eigenvalues below $E_0$. Similarly, (using Remark~\ref{remabovesp}) there are only finitely many eigenvalues above $E_{2g+1}$. Applying Corollary~\ref{coresslamote} in each gap $(E_{2j-1},E_{2j})$, $1 \le j \le g$ shows that the number of eigenvalues in each gap is finite as well. It remains to show (\ref{asupm}). Suppose $u_+(\lam,.)$, $\lam \in\sig(H_p)$ satisfies (disregarding summability for a moment) \bea \nn \lefteqn{u_+(\lam,n) = \frac{a_p(n)}{a(n)} u_{p,+}(\lam,n)}\\ \label{voltseq} && {}+\sum_{m=n+1}^\infty \frac{a_p(n)}{a(n)} K(\lam,n,m) u_+(\lam,m), \eea with \bea \nn \lefteqn{K(\lam,n,m)= \frac{s_p(\lam,n,m-1)}{a_p(m-1)}(a(m-1) - a_p(m-1))}\\ && {}+\frac{s_p(\lam,n,m+1)}{a_p(m+1)}(a(m) - a_p(m)) - \frac{s_p(\lam,n,m)}{a_p(m)}(b(m) - b_p(m)), \eea where $s_p(\lam,.,m)$ is the solution of of $\tau_p u = z u$ satisfying the initial conditions $s_p(z,m,m)=0$ and $s_p(z,m+1,m)=1$. Then $u_+(\lam,.)$ fulfills $\tau u = \lam u$ and (\ref{asupm}). Hence if we can apply Lemma~\ref{voltse} we are done. To do this we need an estimate for $K(\lam,n,m)$ which again follows from Floquet theory \begin{equation} |s_p(\lam,n,m)| \le M |n-m|, \qquad \lam \in \sig(H_p), \end{equation} for some suitable constant $M>0$. \epf As anticipated, specializing to the case $a_p(n)=-1/2$, $b_p(n)=0$ we obtain a corresponding result for the scattering case. \bk \label{corscat} (\cite{gu}) Suppose \begin{equation} \label{decay} \sum_{n\in\Z} |n(1+2a(n))| < \infty, \quad \sum_{n\in\Z} |n\,b(n)| < \infty. \end{equation} Then we have \begin{equation} \sig_{ess}(H)=[-1,1], \quad \sig_p(H) \subseteq [\ul{c},-1) \cup (1,\ol{c}]. \end{equation} Moreover, the essential spectrum of $H$ is purely absolutely continuous and the point spectrum of $H$ is finite. \ek Corollary~\ref{corscat} is stated in \cite{gu} (for the case $a_p(n)=1$ -- but Lemma~\ref{lemaeps} plus a scaling transform takes care of that). Similar results can be obtained using the Birman-Schwinger principle. \chapter{Spectral Deformations} \section{Introduction} For a variety of reasons, techniques to insert and remove eigenvalues in spectral gaps of a given one-dimensional second-order differential (i.e., Sturm-Liouville) respectively difference (i.e., Jacobi) operator have recently attracted great interest. In fact, these techniques are vital in diverse fields such as the inverse scattering approach used by Deift and Trubowitz \cite{dt}, supersymmetric quantum mechanics (cf.\ the literature cited, e.g., in \cite{gss}), level comparison theorems (see, e.g., \cite{ba}), in the construction of soliton solutions of the Korteweg-de Vries (KdV) and Toda hierarchies relative to general KdV and Toda background solutions (see, e.g$.$, \cite{bs}, \cite{bght}, \cite{deift}, \cite{dt}, \cite{eak}, Ch. 4, \cite{fi}, \cite{gg}, \cite{gsv}, \cite{gss}, \cite{km}-\cite{kum}, \cite{lv}, Sect. 6.6, \cite{rs}--\cite{sh2}), and in connection with B\"acklund transformations for the KdV and Toda hierarchies (cf$.$, e.g$.$, \cite{bght}, \cite{ek}, \cite{ef}, \cite{fm}, \cite{gsv}, \cite{gw}, \cite{gss}, \cite{mk1}, \cite{mk2}, \cite{wa}). Historically, methods of inserting eigenvalues in the case of differential operators go back to Jacobi \cite{ja}, Darboux \cite{da}, Crum \cite{cr}, Gel'fand and Levitan \cite{gl}, Schmincke \cite{sc}, and especially Deift \cite{deift}. Two particular such methods, the so called single commutation or Crum-Darboux method and the double commutation method, shortly to be described below, turned out to be of particular importance. The operator theoretic approach developed in \cite{deift} applies to the single commutation method and has been used in \cite{deift} to give a complete spectral characterization in the differential operator case. The double commutation method on the other hand required entirely different methods and was only recently solved in the differential operator case. A solution based on ODE techniques was given in \cite{com} and most recently, a more general and at the same time greatly simplifying operator theoretic approach to a spectral characterization of the double commutation method appeared in \cite{gt}. Surprisingly, a complete spectral characterization of both the single and double commutation methods in the difference operator context is lacking in the literature thus far. Although special cases of the single commutation method with constant or algebro-geometric backgrounds have been discussed in \cite{bght}, \cite{dkv}, \cite{vdo}, no treatment of general backgrounds is known to us. Moreover, with the exception of reference \cite{vdo}, where an eigenvalue is inserted into the spectral gap of a two-band periodic Jacobi operator with period 2, no general formulation of the double commutation method for finite difference operators seems to be available in the literature. The present chapter fills these gaps and provides a complete spectral characterization of the single commutation method (based on Deift's operator theoretic approach) in Sections~\ref{secsc} and \ref{secscit} and develops the corresponding results for the double commutation method in Sections~\ref{secsc}-\ref{secdcit}. Section~\ref{secappc} gives three applications of our results. The discrete analog of the FIT formula for the isospectral torus of periodic Schr\"odinger operators, thereby deriving an explicit realization of the isospectral torus of all algebro-geometric quasi-periodic finite-gap Jacobi operators, and the $N$-soliton solutions of the Toda and Kac-van Moerbeke equations on an arbitrary background solution using the single and double commutation methods. In the remainder of this introduction we provide an informal discussion of commutation methods and restrict ourselves to the case of the whole line and bounded Jacobi operators (so we don't have to bother with domain considerations). We first review the single commutation method \cite{TKvM}: Let $a,b \in \ell(\Z)$ be two bounded, real-valued sequences satisfying \begin{equation} a(n)<0, \qquad b(n) \in \R, \end{equation} and introduce the corresponding Jacobi operator $H$ in $\lz$ \begin{equation} (H f)(n) = a(n) f(n+1) + a(n-1) f(n-1) - b(n) f(n), \quad u \in \ell^2(\Z). \end{equation} Next (cf.\ Lemma \ref{lemaeps}), assume the existence of two weak positive solutions $u_\pm(\lam_1,n)$ of \begin{equation} \label{possol} H u_\pm = \lam_1 u_\pm, \qquad u_\pm(\lam_1,n)>0, \quad u_\pm(\lam_1,n) \in \ell^2(\pm\N) \end{equation} (implying $b(n)+\lam_1<0$, i.e., $H-\lam_1 \ge 0$). $u_\pm$ are the principal solutions as used, e.g., in \cite{crit}. Any positive solution can then be written as \begin{equation} u_{\sig_1}(\lam_1,n) = \frac{1+\sig_1}{2} u_+(\lam_1,n) + \frac{1-\sig_1}{2} u_-(\lam_1,n), \quad \sig_1 \in [-1,1]. \end{equation} Now define the operator $A_{\sig_1}$ in $\lz$ by \begin{equation} (A_{\sig_1} f)(n) = \rho_{o,\sig_1}(n) f(n+1) + \rho_{e,\sig_1}(n) f(n), \quad f \in \lz, \end{equation} where \begin{equation} \rho_{o,\sig_1}(n) = - \sqrt{-\frac{a(n) u_{\sig_1}(\lam_1,n) }{u_{\sig_1}(\lam_1,n+1)}},\quad \rho_{e,\sig_1}(n) = \sqrt{-\frac{a(n) u_{\sig_1}(\lam_1,n+1)}{u_{\sig_1}(\lam_1,n)}}. \end{equation} We will always take the positive branch of all square roots involved. We note that $\rho_{o,\sig_1}$ and $\rho_{e,\sig_1}$ are bounded sequences as can be seen from \begin{equation} |\frac{a(n)u_{\sig_1}(\lam_1,n+1)} {u_{\sig_1}^1(\lam_1,n)}| + |\frac{a(n-1)u_{\sig_1}(\lam_1,n-1)} {u_{\sig_1}(\lam_1,n)}| = |b(n)+\lam_1|. \end{equation} The adjoint operator $A_{\sig_1}^*$ of $A_{\sig_1}$ is given by \begin{equation} (A_{\sig_1}^* f)(n) = \rho_{o,\sig_1}(n-1) f(n-1) + \rho_{e,\sig_1}(n) f(n), \quad f \in \lz, \end{equation} and for the (positive self-adjoint) operator $A_{\sig_1}^* A_{\sig_1}$ one infers \begin{equation} A_{\sig_1}^* A_{\sig_1} = H -\lam_1. \end{equation} This shows that $(H-\lam_1) \ge 0$ is a necessary condition for the existence of a positive solution of (\ref{possol}). We remark that this condition is also sufficient (see, e.g., \cite{crit}, Theorem 2.8). Commuting $A_{\sig_1}^*$ and $A_{\sig_1}$ (observing $(A_{\sig_1}^*)^*=A_{\sig_1}$) yields a second positive self-adjoint bounded operator $A_{\sig_1} A_{\sig_1}^*$ and further the commuted operator \bea H_{\sig_1} = A_{\sig_1} A_{\sig_1}^* + \lam_1. \eea A straightforward calculation shows \bea (H_{\sig_1} f)(n) = a_{\sig_1}(n) f(n+1) + a_{\sig_1}(n-1) f(n-1) - b_{\sig_1}(n) f(n), \eea with \bea a_{\sig_1}(n) &=& -\frac{\sqrt{a(n) a(n+1) u_{\sig_1}(\lam_1,n) u_{\sig_1}(\lam_1,n+2)}}{u_{\sig_1}(\lam_1,n+1)}, \\ b_{\sig_1}(n) &=& a(n) \Big( \frac{u_{\sig_1}(\lam_1,n)}{u_{\sig_1}(\lam_1,n+1)} + \frac{u_{\sig_1}(\lam_1,n+1)} {u_{\sig_1}(\lam_1,n)} \Big) - \lam_1. \eea As proven by Deift \cite{deift}, the operators $H-\lam_1$ and $H_{\sig_1}-\lam_1$, restricted to the orthogonal complements of their respective null-spaces, are unitarily equivalent. Specifically, we have \begin{equation} \label{sigsc} \ba{rcl@{\quad}rcl} \sigma(H_{\sig_1}) \!&=&\! \left\{ \ba{cl} \sigma(H) \cup \{ \lam_1\}, & \sig_1 \in (-1,1)\\ \sigma(H), & \sig_1 \in \{-1,1\} \ea \right. ,& \sig_{ac}(H_{\sig_1}) \!&=&\! \sig_{ac}(H), \\ \sigma_{p}(H_{\sig_1}) \!&=&\! \left\{ \ba{cl} \sigma_{p}(H) \cup \{ \lam_1\}, & \sig_1 \in (-1,1)\\ \sigma_{p}(H), & \sig_1 \in \{-1,1\} \ea \right. , & \sig_{sc}(H_{\sig_1}) \!&=&\! \sig_{sc}(H). \ea \end{equation} Here $\sigma_{p}(.),\sigma_{ac}(.)$, and $\sigma_{sc}(.)$ denote the the point spectrum (i.e., the set of eigenvalues), absolutely, and singularly continuous spectrum, respectively. This method is known as the single commutation method \cite{TKvM} and we will give a complete spectral characterization of it in Sections~\ref{secsc} and \ref{secscit}. Our next aim is to remove the condition that $H$ is bounded from below and thereby introduce the double commutation method. Fix $\gam_\pm>0$ and define \bea \rho_{o,\gam_\pm}(n) &=& \rho_{e,\pm1}(n+1) \sqrt{\frac{c_{\gam_\pm}(\lam_1,n)} {c_{\gam_\pm}(\lam_1,n+1)}},\\ \rho_{e,\gam_\pm}(n) &=& \rho_{o,\pm1}(n) \sqrt{\frac{c_{\gam_\pm}(\lam_1,n+1)} {c_{\gam_\pm}(\lam_1,n)}}, \eea where \bea c_{\gam_\pm}(\lam_1,n) = 1 + \gam_\pm \sum_{j=\pm\infty}^{n+1 \atop n} u_{\pm}(\lam_1,j)^2, \eea and introduce corresponding operators $A_{\gam_\pm},A_{\gam_\pm}^*$ in $\lz$ by \bea (A_{\gam_\pm} f)(n) &=& \rho_{o,\gam_\pm}(n) f(n+1) + \rho_{e,\gam_\pm}(n) f(n),\\ (A_{\gam_\pm}^* f)(n) &=& \rho_{o,\gam_\pm}(n-1) f(n-1) + \rho_{e,\gam_\pm}(n) f(n). \eea A simple calculation shows that $A_{\gam_\pm}^* A_{\gam_\pm} = A_{\pm1} A_{\pm1}^*$ and hence \bea H_{\pm1} = A_{\gam_\pm}^* A_{\gam_\pm} + \lam_1. \eea Performing a second commutation yields the doubly commuted operator \bea H_{\gam_\pm} = A_{\gam_\pm} A_{\gam_\pm}^* + \lam_1. \eea Explicitly, one verifies \bea (H_{\gam_\pm} f)(n) = a_{\gam_\pm}(n) f(n+1) + a_{\gam_\pm}(n-1) f(n-1) - b_{\gam_\pm}(n) f(n), \eea with \bea a_{\gam_\pm}(n) &=& a(n+1) \frac{\sqrt{c_{\gam_\pm}(\lam_1,n) c_{\gam_\pm}(\lam_1,n+2)}}{c_{\gam_\pm}(\lam_1,n+1)},\\ \nn b_{\gam_\pm}(n) &=& b(n+1) \pm \gam_\pm \Big( \frac{a(n) u_{\pm}(\lam_1,n) u_{\pm}(\lam_1,n+1)}{c_{\gam_\pm}(\lam_1,n)} \\&& {}- \frac{a(n+1) u_{\pm}(\lam_1,n+1) u_{\pm}(\lam_1,n+2)} {c_{\gam_\pm}(\lam_1,n+1)}\Big). \eea Now observe that $H_{\gam_\pm}$ remains well-defined even if $u_\pm$ is no longer positive. This applies, in particular, in the case where $u_\pm(\lam_1)$ has zeros and hence all intermediate operators $A_{\pm1}, A_{\gam_\pm}, H_{\pm1}$, etc., become ill-defined. Thus to define $H_{\gam_\pm}$ it suffices to assume the existence of a solution $u_\pm(\lam_1)$ which is square summable near $\pm\infty$. This condition is much less restrictive than the existence of a positive solution (e.g., $\sigma(H) \ne \R$, i.e., the existence of a spectral gap for $H$ around $\lam_1$ is sufficient in this context). One expects that formulas analogous to (\ref{sigsc}) will carry over to this more general setup. That this is actually the case will be shown in our principal Theorem \ref{thmdc} of Section~\ref{secdc}. Hence the double commutation method (contrary to the single commutation method) enables one to insert eigenvalues not only below the spectrum of $H$ but into arbitrary spectral gaps of $H$. \section{The Single Commutation Method} \label{secsc} In this section we intend to give a detailed investigation of the single commutation method. We will assume $a,b$ to satisfy (H.\ref{hypoaposb}) throughout Sections~\ref{secsc} and \ref{secscit}. We shall consider (self-adjoint) Jacobi operators $H$ associated with the difference expression \bea \label{diffex} (\tau f)(n) = a(n) f(n+1) + a(n-1) f(n-1) - b(n) f(n), \eea in the Hilbert space $\lz$. As a preparation we prove We start with operators associated with the difference expression (\ref{diffex}) on the half axis $\pm\N$. For simplicity we will do most calculations only for $\ell^2(\N)$. Let $u(\lam_1)$ be a positive solution of $\tau u = \lam_1 u$ and define \bea \rho_{o,+}(n) &=& -\sqrt{-\frac{a(n)u(\lam_1,n+1)}{u(\lam_1,n)}}, \\ \rho_{e,+}(n) &=& \sqrt{-\frac{a(n-1)u(\lam_1,n-1)}{u(\lam_1,n)}}, \quad n>0. \eea Define the operator $\dot{A}_+$ on $\ell_0(\N)$ \bea (\dot{A}_+ f)(n) = \rho_{o,+}(n) f(n+1) + \rho_{e,+}(n) f(n), \quad f \in \ell_0(\N) \eea and denote its operator closure (in $\ell^2(\N)$) by $A_+$. One verifies, \bea \db(A_+) \subseteq \{ f \in \ell^2(\N) | \rho_{o,+}(n) f(n+1) + \rho_{e,+}(n) f(n) \in \ell^2(\N) \}. \eea The adjoint $A_+^*$ of $A_+$ is then given by \bea &(A_+^* f)(n) = \rho_{o,+}(n-1) f(n-1) + \rho_{e,+}(n) f(n), & \\ \nn &\db(A_+^*) = \{ f \in \ell^2(\N) | f(0)=0; \rho_{o,+}(n-1) f(n-1) + \rho_{e,+}(n) f(n) \in \ell^2(\N) \}. & \eea (The boundary condition $f(0)=0$ is only introduced so that we don't have to specify $(A_+^* f)(1)$ separately.) Due to a well known result of von Neumann (see, e.g., \cite{wd}, Theorem 5.39) the operator $A_+ A_+^*$ is a positive self-adjoint operator when defined naturally \bea \db(A_+ A_+^*) = \{ f \in \db(A_+^*) | A_+^* f \in \db(A_+) \}. \eea A simple calculation shows $A_+ A_+^* f = (\tau -\lam_1)f$ and hence we may define \bea H_+ = A_+ A_+^* + \lam_1, \quad \db(H_+) &\subseteq& \{ f \in \ell^2(\N) | f(0)=0, \: \tau f \in \ell^2(\N) \}, \eea where equality in the last relation is equivalent to $\tau$ being limit point ($l.p.$) at $+\infty$. Similarly one defines for $n<0$ \bea \rho_{o,-}(n) = -\sqrt{-\frac{a(n)u(\lam_1,n)}{u(\lam_1,n+1)}}, \quad \rho_{e,-}(n) = \sqrt{-\frac{a(n)u(\lam_1,n+1)}{u(\lam_1,n)}} \eea and operators $A_-$, and $A_-^*$ in $\ell^2(-\N)$ which satisfy $H_- = A_-^* A_- +\lam_1$. Commuting $A_\pm^*$ and $A_\pm$ yields a second positive self-adjoint operator $A_- A_-^*$, respectively $A_+^* A_+$, and further the commuted operators \bea H_{+,1} = A_+^* A_+ + \lam_1, \qquad H_{-,1} = A_- A_-^* + \lam_1. \eea The next theorem characterizes $H_{\pm,1}$ in terms of $H_\pm$, but first we need to introduce \bh \label{hyposc} Suppose $H_\pm$ satisfies one of the following spectral conditions.\\ (i) $\sig_{ess} (H_\pm) \ne \emptyset$.\\ (ii) $\sig(H_\pm)=\sig_d(H_\pm)= \{ \lam_{\pm,j} \}_{j \in J_\pm}$ with $\sum_{j\in J_\pm} (1+\lam^2_{\pm,j})^{-1} =\infty$. \eh Hypothesis (H.\ref{hyposc}) is satisfied if $a,b$ are bounded near $\pm\infty$. Either one of the conditions (i), (ii) implies that $\tau$ is $l.p.$ at $\pm\infty$. This follows since otherwise the resolvent of $H_\pm$ would be a Hilbert-Schmidt operator contradicting (i), (ii). This further implies that the domain of $H_\pm$ is given by \bea \db(H_\pm) = \{ f \in \ell^2(\pm\N) | f(0)=0, \: \tau f \in \ell^2(\pm\N)\}. \eea \begin{thm} \label{thmpm} Assume (H.\ref{hypoaposb}) and (H.\ref{hyposc}). Then the operators $H_{\pm,1}$ constructed above satisfy (H.\ref{hypoaposb}) and (H.\ref{hyposc}) and are given by \bea \nn (H_{\pm,1} f)(n) &=& (\tau_{\pm,1} f)(n)\\ &=& a_{\pm,1}(n) f(n+1) + a_{\pm,1}(n-1) f(n-1) - b_{\pm,1}(n) f(n),\\ \nn \db(H_{\pm,1}) &=& \{ f \in \ell^2(\pm\N) | f(0)=0, \: \tau_{\pm,1} f \in \ell^2(\pm\N) \}, \eea with \bea a_{+,1}(n) &=& -\frac{\sqrt{a(n-1) a(n) u(\lam_1,n-1) u(\lam_1,n+1)}}{u(\lam_1,n)},\quad n > 0, \\ \nn b_{+,1}(n) &=& a(n-1) \Big(\frac{u(\lam_1,n)}{u(\lam_1,n-1)} + \frac{u(\lam_1,n-1)}{u(\lam_1,n)} \Big) - \lam_1,\quad n>1,\\ b_{+,1}(1) &=& a(0) \frac{u(\lam_1,0)}{u(\lam_1,1)} - \lam_1, \eea and \bea a_{-,1}(n) &=& -\frac{\sqrt{a(n) a(n+1) u(\lam_1,n) u(\lam_1,n+2)}}{u(\lam_1,n+1)},\quad n <-1, \\ \nn b_{-,1}(n) &=& a(n) \Big(\frac{u(\lam_1,n)}{u(\lam_1,n+1)} + \frac{u(\lam_1,n+1)}{u(\lam_1,n)} \Big) - \lam_1,\quad n<-1,\\ b_{-,1}(-1) &=& a(-1) \frac{u(\lam_1,0)}{u(\lam_1,-1)}- \lam_1. \eea Moreover, $H_\pm -\lam_1$ and $H_{\pm,1} -\lam_1$ restricted to the orthogonal complements of their null-spaces are unitarily equivalent and hence \bea \ba{rcl@{\quad}rcl} \sigma(H_{\pm,1}) \bs \{ \lam_1\} \!&=&\! \sigma(H_\pm) \bs \{ \lam_1\}, & \sig_{ac}(H_{\pm,1}) \!&=&\! \sig_{ac}(H_\pm), \\ \sigma_{p}(H_{\pm,1}) \bs \{ \lam_1\} \!&=&\! \sig_{p}(H_\pm) \bs \{ \lam_1\}, & \sig_{sc}(H_{\pm,1}) \!&=&\! \sig_{sc}(H_\pm). \ea \eea \end{thm} \bpf The unitary equivalence follows from \cite{deift}, Theorem 1 and clearly settles the spectral claims. Thus both $H_\pm$ and $H_{\pm,1}$ satisfy (H.\ref{hyposc}) and hence $\tau_\pm$ and $\tau_{\pm,1}$ are $l.p.$ at $\pm\infty$. The rest are straightforward calculations. \epf Next we turn to the case of the whole lattice $\lz$. We pick $\sigma_1 \in [-1,1]$ and $\lam_1 < \inf(\sigma(H))$. Further denote by $u_\pm(\lam,n)$ (for $\lam < \inf(\sigma(H))$) the solutions constructed in Lemma~\ref{lemaeps} and set \bea u_{\sig_1}(\lam_1,n) = \frac{1+\sigma_1}{2} u_+(\lam_1,n) + \frac{1-\sigma_1}{2} u_-(\lam_1,n). \eea Now define sequences \bea \rho_{o,\sig_1}(n) = -\sqrt{-\frac{a(n)u_{\sig_1}(\lam_1,n) }{u_{\sig_1}(\lam_1,n+1)}}, \quad \rho_{e,\sig_1}(n) = \sqrt{-\frac{a(n)u_{\sig_1}(\lam_1,n+1)} {u_{\sig_1}(\lam_1,n)}}, \eea and the corresponding operator $A_{\sig_1}$ (first on $\ell_0(\Z)$ and then take the closure in $\lz$ as before) together with its adjoint $A_{\sig_1}^*$, \bea &(A_{\sig_1} f)(n) = \rho_{o,\sig_1}(n) f(n+1) + \rho_{e,\sig_1}(n) f(n),&\\ \nn &\db(A_{\sig_1}) \subseteq \{ f \in \lz | \rho_{o,\sig_1}(n) f(n+1) + \rho_{e,\sig_1}(n) f(n) \in \lz \}, &\\ &(A_{\sig_1}^* f)(n) = \rho_{o,\sig_1}(n-1) f(n-1) + \rho_{e,\sig_1}(n) f(n),&\\ \nn &\db(A_{\sig_1}^*) = \{ f \in \lz | \rho_{o,\sig_1}(n-1) f(n-1) + \rho_{e,\sig_1}(n) f(n) \in \lz \}.& \eea Again by von Neumann's result $A_{\sig_1}^* A_{\sig_1}$ is a positive self-adjoint operator when defined naturally by \bea \db(A_{\sig_1}^* A_{\sig_1}) = \{ f \in \db(A_{\sig_1}) | A_{\sig_1} f \in \db(A_{\sig_1}^*) \}. \eea A simple calculation shows $A_{\sig_1}^* A_{\sig_1} = \tau -\lam_1$ and we hence may define \bea \label{hpm} H = A_{\sig_1}^* A_{\sig_1} + \lam_1, \qquad \db(H) \subseteq \{ f \in \lz | \tau f \in \lz \}. \eea Commuting $A_{\sig_1}^*$ and $A_{\sig_1}$ yields a second positive self-adjoint operator $A_{\sig_1} A_{\sig_1}^*$ and further the commuted operator \bea H_{\sig_1} = A_{\sig_1} A_{\sig_1}^* + \lam_1, \qquad \db(H_{\sig_1}) \subseteq \{ f \in \lz | \tau_{\sig_1} f \in \lz \}, \eea where $\tau_{\sig_1}$ is the difference expression corresponding to $H_{\sig_1}$. The next theorem characterizes $H_{\sig_1}$ under Assumption (H.2.2) for $H_+$ and $H_-$ implying that $\tau$ is $l.p.$ at $\pm\infty$ and hence that \bea \db(H) = \{ f \in \lz | \tau f \in \lz \}. \eea \begin{thm} \label{thmsc} Assume (H.\ref{hypoaposb}) and (H.\ref{hyposc}). Then the operator $H_{\sig_1}$, \bea\nn (H_{\sig_1} f)(n) &=& (\tau_{\sig_1}f)(n)\\ &=& a_{\sig_1}(n) f(n+1) + a_{\sig_1}(n-1) f(n-1) - b_{\sig_1}(n) f(n),\\ \nn \db(H_{\sig_1}) &=& \{ f \in \lz | \tau_{\sig_1} f \in \lz \}, \eea is self-adjoint. Moreover, \bea a_{\sig_1}(n) &=& -\frac{\sqrt{a(n) a(n+1) u_{\sig_1}(\lam_1,n) u_{\sig_1}(\lam_1,n+2)}}{u_{\sig_1}(\lam_1,n+1)}, \\ b_{\sig_1}(n) &=& a(n) \Big( \frac{u_{\sig_1}(\lam_1,n)} {u_{\sig_1}(\lam_1,n+1)} + \frac{u_{\sig_1}(\lam_1,n+1)} {u_{\sig_1}(\lam_1,n)} \Big) - \lam_1 \eea and $a_{\sig_1}$, $b_{\sig_1}$ satisfy (H.\ref{hypoaposb}). The equation $\tau_{\sig_1} v= \lam_1 v$ has the positive solution \bea v_{\sig_1}(\lam_1,n) = \frac{1}{\sqrt{-a(n) u_{\sig_1}(\lam_1,n) u_{\sig_1}(\lam_1,n+1)}} \eea which is an eigenfunction of $H_{\sig_1}$ if and only if $\sigma_1 \in (-1,1)$. $H-\lam_1$ and $H_{\sig_1}-\lam_1$ restricted to the orthogonal complements of their corresponding one-dimensional null-spaces are unitarily equivalent and hence \begin{equation} \ba{rcl@{\quad}rcl} \sigma(H_{\sig_1}) \!&=&\! \left\{ \ba{cl} \sigma(H) \cup \{ \lam_1\}, & \sig_1 \in (-1,1)\\ \sigma(H), & \sig_1 \in \{-1,1\} \ea \right. ,& \sig_{ac}(H_{\sig_1}) \!&=&\! \sig_{ac}(H), \\ \sigma_{p}(H_{\sig_1}) \!&=&\! \left\{ \ba{cl} \sigma_{p}(H) \cup \{ \lam_1\}, & \sig_1 \in (-1,1)\\ \sigma_{p}(H), & \sig_1 \in \{-1,1\} \ea \right. , & \sig_{sc}(H_{\sig_1}) \!&=&\! \sig_{sc}(H). \ea \end{equation} In addition, the sequence \bea (A_{\sig_1} u)(z,n) = \frac{W_n(u_{\sig_1}(\lam_1),u(z))}{\sqrt{-a(n) u_{\sig_1}(\lam_1,n) u_{\sig_1}(\lam_1,n+1)}} \eea solves $\tau_{\sig_1} u = z u$ if $u(z)$ solves $\tau u = z u$ for some $z \in \C$, where \begin{equation} W_n(u,v) = a(n)(u(n) v(n+1) - u(n+1) v(n)) \end{equation} denotes the modified Wronskian. Moreover, one obtains \bea \label{wronsc} W_{\sig_1,n}(A u(z), A v(z)) = (\lam_1-z) W_n(u(z),v(z)) \eea for solutions $u,v$ of $\tau u = z u$, where $W_{\sig_1,n}(u,v) = a_{\sig_1}(n)(u(n) v(n+1) - u(n+1) v(n))$. The resolvents of $H,H_{\sig_1}$ for $z \in \C \bs (\sig(H) \cup \{\lam_1\})$ are related via \bea \label{res} (H_{\sig_1} - z)^{-1} = \frac{1}{z-\lam_1} \Big(1 - A_{\sig_1} (H - z)^{-1} A_{\sig_1}^* \Big) \eea or, in terms of Green's functions for $n \ge m$, $z \in \C \bs (\sig(H) \cup \{\lam_1\})$, \bea \nn &G(z,n,m) = \frac{\D u_+(z,n)u_-(z,m)} {\D W_n(u_+(z),u_-(z))}&\\ \label{gfsc} &\mb{implies}\quad G_{\sig_1}(z,n,m) = \frac{\D (A_{\sig_1} u_+)(z,n) (-A_{\sig_1} u_-)(z,m)} {\D (z-\lam_1) W_n(u_+(z),u_-(z))}. & \eea Furthermore, $u_{\sig_1,\pm}(z,n)$, the principal solutions of $(H_{\sig_1} - z) u=0$ for $z<\lam_1$, are given by \bea u_{\sig_1,\pm}(z,n) = \pm A_{\sig_1} u_\pm(z,n) = \frac{ \mp W_n(u_{\sig_1}(\lam_1),u_\pm(z))} {\sqrt{-a(n) u_{\sig_1}(\lam_1,n) u_{\sig_1}(\lam_1,n+1)}}. \eea In addition, we have \bea \label{normv} \sum_{n \in \Z} v_{\sig_1}(\lam_1,n)^2 = \frac{4}{1-\sig_1^2} W(u_-(\lam_1),u_+(\lam_1))^{-1}, \quad \sig_1 \in (-1,1) \eea and, if $\tau u(\lam) = \lam u(\lam)$, $u(\lam,.) \in \lz$, \bea \label{normuz} \sum_{n \in \Z} (A_{\sig_1} u)(\lam,n)^2 = (\lam-\lam_1) \sum_{n \in \Z} u(\lam,n)^2. \eea \end{thm} \bpf The unitary equivalence together with equation (\ref{res}) follow from \cite{deift}, Theorem 1. That $H_{\sig_1}$ is $l.p.$ at $\pm\infty$ follows upon looking at the restrictions $H_\pm$, $H_{\pm,1}$ and using Theorem \ref{thmpm}. Equation (\ref{res}) together with (\ref{wronsc}) imply (\ref{gfsc}). The facts concerning the point spectrum follow since $G_{\sig_1}(z,n,n)$ has a pole at $z=\lam_1$ if and only if $\sig_1 \in (-1,1)$. (\ref{normv}) can be obtained by investigating the residue of $G_{\sig_1}(z,n,n)$ at $z=\lam_1$. The rest are straightforward calculations. \epf \begin{rem} \label{remagz} (i). Hypothesis (H.\ref{hyposc}) is only needed in Theorem \ref{thmsc} to characterize the domains of $H$ and $H_{\sig_1}$ explicitly.\\ (ii). Multiplying $u_{\sig_1}$ with a positive constant leaves all formulas and, in particular, $H_{\sig_1}$ invariant.\\ (iii). If $H$ is bounded from above we can insert eigenvalues into the highest spectral gap, i.e., above the spectrum of $H$, upon considering $-H$. Then $\lam>\sup(\sigma(H))$ implies that we don't have positive but rather alternating solutions and all our previous calculations carry over with minor changes.\\ (iv). We can weaken (H.\ref{hypoaposb}) by requiring $a(n) \ne 0$ instead of $a(n)<0$. Everything stays the same with the only difference that $u_\pm$ are not positive but change sign in such a way that (\ref{posauou}) stays positive. Moreover, the signs of $a_{\sig_1}(n)$ can also be prescribed arbitrarily by altering the signs of $\rho_{o,\sig_1}$ and $\rho_{e,\sig_1}$.\\ (v). The fact that $v_{\sig_1} \in \ell^2(\Z)$ if and only if $\sig_1 \in (-1,1)$ gives an alternate proof of \bea \sum_{n=0}^{\pm\infty} \frac{1}{-a(n)u_{\sig_1}(\lam_1,n)u_{\sig_1}(\lam_1,n+1)} < \infty \mb{ if and only if } \sig_1 \in \ba{c} [-1,1) \\ (-1,1] \ea \eea (cf.\ \cite{pat2} and \cite{crit}, Lemma 2.10, Remark 2.11). \end{rem} At the end of this section we will show some connections between the single commutation method and some other theories. We start with the Weyl-Titchmarsh theory and freely use the definitions of Appendices B and C. \begin{lem} Assume (H.\ref{hypoaposb}). The Weyl $\ti{m}$-functions $\ti{m}_{\pm,\sig_1}(z)$ of $H_{\sig_1}$, $\sig_1 \in [-1,1]$ in terms of $\ti{m}_{\pm}(z)$, the ones of $H$, read \bea \ti{m}_{\pm,\sig_1}(z) = \frac{-u_{\sig_1}(\lam_1,1)} {a(1) u_{\sig_1}(\lam_1,2)} \Big( 1 + \frac{(z-\lam_1) \ti{m}_\pm(z)}{1 + \frac{a(0) u_{\sig_1}(\lam_1,0) }{u_{\sig_1}(\lam_1,1)} \ti{m}_\pm(z)} \Big). \eea \end{lem} \bpf The above formulas are straightforward calculations using (\ref{gfsc}) and (\ref{mpl}), (\ref{mmi}). \epf Finally we turn to scattering theory. In order to facilitate comparison with the standard literature on (inverse) scattering theory for second-order difference operators (cf.\ \cite{dinv1}, \cite{dinv2}, \cite{fad}, \cite{conl}, \cite{gu}, \cite{ta}) we now assume \bea a(n)>0,\: b(n)\in\R, \qquad n|1 - 2a(n)|, n b(n) \in \ell^1(\Z) \eea (cf. Remark \ref{remagz}). This implies \bea \sig_{ac}(H) =[-1,1], \quad \sig_{sc}(H) = \emptyset, \quad \sig_{p}(H) = \{ \lam_j \}_{j \in J} \subseteq \R \backslash [-1,1], \eea where $J \subseteq \N$ is a suitable (finite) index set, and the existence of the so called Jost solutions $f_\pm(k,n)$, \bea \Big( \tau - \frac{k+k^{-1}}{2} \Big) f_\pm(k,n) = 0, \quad \lim_{n\to\pm\infty} k^{\mp n} f_\pm(k,n) = 1, \quad |k| \le 1. \eea Transmission $T(k)$ and reflection $R_\pm(k)$ coefficients are then defined via \bea T(k) f_\mp(k,n) = f_\pm(k^{-1},n) + R_\pm(k) f_\pm(k,n), \quad |k|=1, \eea and the norming constants $\gam_{\pm,j}$ corresponding to $\lam_j \in \sig_{p}(H)$ are given by \bea \label{norming} \gam_{\pm,j}^{-1} = \sum_{n \in \Z} |f_\pm(k_j,n)|^2, \quad k_j = \lam_j + \sqrt{\lam_j^2 -1} \in (-1,0),\: j \in J. \eea \begin{lem} \label{scatsc} Suppose $H$ satisfies {\em (\ref{decay})} and let $H_{\sig_1}$ be constructed as in Theorem \ref{thmsc} with \bea u_{\sig_1}(\lam_1,n) = \frac{1+\sig_1}{2} f_+(k_1,n) + \frac{1-\sig_1}{2} f_-(k_1,n). \eea Then the transmission $T_{\sig_1}(k)$ and reflection coefficients $R_{\pm,\sig_1}(k)$ of $H_{\sig_1}$ in terms of the corresponding scattering data $T(k),R_\pm(k)$ of $H$ are given by \bea T_{\sig_1}(k) = \frac{1 - k \, k_1}{k - k_1} T(k), \quad R_{\pm,\sig_1}(k) = k^{\pm 1} \frac{k - k_1}{1 - k \, k_1} R_\pm(k), \:\: \sig_1 \in (-1,1), \eea \bea T_{\sig_1}(k) = T(k), \quad R_{\pm,\sig_1}(k) = \frac{k_1^{\sig_1} - k^{\mp 1}}{k_1^{\sig_1} - k^{\pm 1}} R_\pm(k), \quad \sig_1 \in \{-1,1\}, \eea where $k_1 = \lam_1 + \sqrt{\lam_1^2 -1} \in (-1,0)$. Moreover, the norming constants $\gam_{\sig_1,\pm,j}$ associated with $\lam_j \in \sig_p(H_{\sig_1})$ in terms of $\gam_{\pm,j}$ corresponding to $H$ read \bea \nn \gam_{\sig_1,\pm,j} &=& |k_j|^{\pm1} \frac{1 - k_j k_1} {(k_j - k_1)} \gam_{\pm,j}, \quad j \in J, \: \sig_1 \in (-1,1),\\ \gam_{\sig_1,\pm,1} &=& \left( \frac{1-\sig_1} {1+\sig_1} \right)^{\pm1} |1 - k_1^{\mp2}| \, T(k_1), \quad \sig_1 \in (-1,1), \eea \bea \gam_{\sig_1,\pm,j} = |k_1^{\sig_1} - k_j^{\mp1}| \gam_{\pm,j}, \quad j \in J, \: \sig_1 \in \{-1,1\}. \eea \end{lem} \bpf The claims follow easily after observing that up to normalization the Jost solutions of $H_{\sig_1}$ are given by $A_{\sig_1} f_\pm(k,n)$ (compare (\ref{gfsc})). \epf \section{Iteration of the Single Commutation Method} \label{secscit} By choosing $\lam_2<\lam_1$ and $\sigma_2 \in [-1,1]$ we can define \bea u_{\sig_1,\sig_2}(\lam_2,n) = \frac{1+\sig_2}{2} u_{\sig_1,+}(\lam_2,n) + \frac{1-\sig_2}{2} u_{\sig_1,-}(\lam_2,n) \eea and repeat the process of the previous section by defining $\rho_{o,\sig_1,\sig_2}$, $\rho_{e,\sig_1,\sig_2}$ and corresponding operators $A_{\sig_1,\sig_2}$, $A_{\sig_1,\sig_2}^*$ which satisfy \bea H_{\sig_1} = A_{\sig_1,\sig_2}^* A_{\sig_1,\sig_2} - \lam_2. \eea A further commutation then yields the operator \bea H_{\sig_1,\sig_2} = A_{\sig_1,\sig_2}A_{\sig_1,\sig_2}^* - \lam_2 \eea associated with sequences $a_{\sig_1,\sig_2}$, $b_{\sig_1,\sig_2}$. The result after $N$ steps is summarized in \begin{thm} \label{thmscom} Assume (H.\ref{hypoaposb}) and (H.\ref{hyposc}). Let $H$ be as in Section~\ref{secsc} and choose \bea \lam_N < \dots < \lam_2 < \lam_1 < \inf(\sigma(H)), \quad \sigma_\ell \in [-1,1], \:\: 1\le \ell \le N, \: N \in \N. \eea Then we have \bea a_{\sig_1,\dots,\sig_N}(n) \!\!&=&\!\! - \sqrt{a(n) a(n+N)} \frac{ \sqrt{ C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N) C_{n+2}(u_{\sig_1}^1,\dots,u_{\sig_N}^N) }}{C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)},\\ \nn b_{\sig_1,\dots,\sig_N}(n) \!\!&=&\!\! - \lam_N + a(n) \frac{C_{n+2}(u_{\sig_1}^1,\dots,u_{\sig_{N-1}}^{N-1}) C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)} {C_{n+1}(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)} \\ && {} + a(n+N-1) \frac{C_n(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)} {C_{n+1}(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}, \eea where \bea u_{\sig_\ell}^\ell(n) = \frac{1+\sigma_\ell}{2} u_+(\lam_\ell,n) + (-1)^{\ell+1} \frac{1-\sigma_\ell}{2} u_-(\lam_\ell,n), \eea and $C_n$ denotes the $n$-dimensional Casoratian \bea \label{Nwon} C_n(u_1,\dots,u_N) = \det\{ u_i(n+j-1)\}_{1\le i,j\le N}. \eea Moreover, for $1 \le \ell \le N$, $\lam < \lam_\ell$ \bea \label{defuj} u_{\sig_1,\dots,\sig_\ell,\pm}(\lam,n) = \frac{ \pm \sqrt{ \prod\limits_{j=0}^{\ell-1} (-a(n+j))} C_n(u_{\sig_1}^1,\dots,u_{\sig_\ell}^\ell,u_\pm(\lam))} {\sqrt{ C_n(u_{\sig_1}^1,\dots, u_{\sig_\ell}^\ell) C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_\ell}^\ell) }}, \eea are the principal solutions of $\tau_{\sig_1,\dots,\sig_\ell} u = \lam u$ and \begin{equation} u_{\sig_1,\dots,\sig_\ell}(\lam_\ell,n) = \frac{1+\sig_\ell}{2} u_{\sig_1,\dots,\sig_{\ell-1},+}(\lam_\ell,n) + \frac{1-\sig_\ell}{2} u_{\sig_1,\dots,\sig_{\ell-1},-}(\lam_\ell,n) \end{equation} is used to define $H_{\sig_1,\dots,\sig_\ell}$. We also have \bea \label{rhoo} &\rho_{o,\sig_1,\dots,\sig_N}(n) = -\sqrt{-a(n) \frac{C_{n+2}(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)} {C_{n+1}(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}},&\\ \label{rhoe} &\rho_{e,\sig_1,\dots,\sig_N}(n) = \sqrt{-a(n+N-1) \frac{C_n(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)} {C_{n+1}(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)}}.& \eea The spectrum of $H_{\sig_1,\dots,\sig_N}$ is given by \bea \sigma(H_{\sig_1,\dots,\sig_N}) = \sigma(H) \cup \{ \lam_\ell \; | \; \sigma_\ell \in (-1,1), \; 1 \le \ell \le N \}. \eea \end{thm} \bpf It is enough to prove the formulas for $a_{\sig_1,\dots,\sig_N}(n)$ and $u_{\sig_1,\dots,\sig_N}(n)$, the remaining assertions then follow easily. We will use a proof by induction on $N$. They are valid for $N=1$ and we need to show \begin{equation} u_{\sig_1,\dots,\sig_{N+1},\pm}(\lam,n) = \frac{\sqrt{-a_{\sig_1,\dots,\sig_N}(n)} C_n(u_{\sig_1,\dots,\sig_N}(\lam_N), u_{\sig_1,\dots,\sig_N,\pm1}(\lam))}{ \pm \sqrt{u_{\sig_1,\dots,\sig_N}(\lam_N,n) u_{\sig_1,\dots,\sig_N}(\lam_N,n+1)}}, \end{equation} \bea \nn a_{\sig_1,\dots,\sig_{N+1}}(n) &=& \sqrt{a_{\sig_1,\dots,\sig_N}(n) a_{\sig_1,\dots,\sig_N}(n+1)} \times \\ && \frac{\sqrt{ u_{\sig_1,\dots,\sig_N}(\lam_N,n) u_{\sig_1,\dots,\sig_N}(\lam_N,n+1)} }{u_{\sig_1,\dots,\sig_N}(\lam_N,n+1)}. \eea The first relation follows after a straightforward calculation using Sylvester's determinant identity (cf.\ \cite{gant}, Sect.\ II.3) \bea \nn &C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N,u_\pm(\lam)) C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_{N+1}}^{N+1})& \\ \nn & {}- C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N,u_\pm(\lam)) C_n(u_{\sig_1}^1,\dots,u_{\sig_{N+1}}^{N+1})&\\ & = C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N) C_n(u_{\sig_1}^1,\dots, u_{\sig_{N+1}}^{N+1},u_\pm(\lam)),& \eea and the second is a simple calculation. \epf \begin{rem} If $u(z,n)$ is any solution of $\tau u = z u$, $z \in \C$ define $u_{\sig_1,\dots,\sig_N}(z,n)$ as in (\ref{defuj}) but with $\ell=N$ and $u_\pm(\lam,n)$ replaced by $u(z,n)$. Then $u_{\sig_1,\dots,\sig_N}(z,n)$ solves $\tau_{\sig_1,\dots,\sig_N} u = z u$. \end{rem} Finally we extend Lemma \ref{scatsc} and assume for brevity $\sig_\ell \in (-1,1)$. \begin{lem} \label{scatscN} Suppose $H$ satisfies {\em (\ref{decay})} and let $H_{\sig_1,\dots,\sig_N}$, $\sig_\ell \in (-1,1,)$, $1 \le \ell \le N$ be constructed as in Theorem \ref{thmscom} with \bea u_{\sig_\ell}^\ell(n) = \frac{1+\sig_\ell}{2} f_+(k_\ell,n) + (-1)^{\ell+1} \frac{1-\sig_\ell}{2} f_-(k_\ell,n). \eea Then the transmission $T_{\sig_1,\dots,\sig_N}(k)$ and reflection coefficients $R_{\pm,\sig_1,\dots,\sig_N}(k)$ of the operator $H_{\sig_1,\dots,\sig_N}$ in terms of the corresponding scattering data $T(k),R_\pm(k)$ of $H$ are given by \bea &T_{\sig_1,\dots,\sig_N}(k) = \left( \prod\limits_{\ell=1}^N \D \frac{1 - k \, k_\ell }{k - k_\ell} \right) T(k), &\\ &R_{\pm,\sig_1,\dots,\sig_N}(k) = k^{\pm N} \left( \prod\limits_{\ell=1}^N \D \frac{k - k_\ell}{1 - k \, k_\ell} \right) R_\pm(k),& \eea where $k_\ell = \lam_\ell + \sqrt{\lam_\ell^2 -1} \in (-1,0)$, $1 \le \ell \le N$. Moreover, the norming constants $\gam_{\sig_1,\dots,\sig_N,\pm,j}$ associated with $\lam_j \in \sig_p(H_{\sig_1,\dots,\sig_N})$ in terms of $\gam_{\pm,j}$ corresponding to $H$ read \bea \nn &\gam_{\sig_1,\dots,\sig_N,\pm,j} = \left( \frac{1-\sig_j}{1+\sig_j} \right)^{\pm1} |k_j|^{-2 \mp(N-1)} \D \frac{\prod_{\ell=1}^N |1 - k_j k_\ell|}{\prod_{\ell= 1 \atop \ell \ne j}^N |k_j-k_l|} T(k_j), \quad 1 \le j \le N,&\\ &\gam_{\sig_1,\dots,\sig_N,\pm,j} = |k_j|^{\pm N} \prod\limits_{\ell=1}^N \D \frac{1 - k_j k_\ell}{|k_j - k_\ell|} \gam_{\pm,j}, \quad j \in J.& \eea \end{lem} \bpf Observe that \bea \nn u_{\sig_1,\sig_2}(\lam_2,n) &=& \frac{1+\sig_2}{2} A_{\sig_1} f_+(k_2,n) + \frac{1-\sig_2}{2} A_{\sig_1} f_-(k_2,n)\\ &=& c \Big( \frac{1+\hat{\sig}_2}{2} f_{\sig_1,+}(k_2,n) + \frac{1-\hat{\sig}_2}{2} f_{\sig_1,-}(k_2,n) \Big), \eea where $c>0$ and $\sig_2,\hat{\sig}_2$ are related via \bea \frac{1+\hat{\sig}_2}{1-\hat{\sig}_2} = \frac{1}{k_2} \frac{1+\sig_1}{1-\sig_1}. \eea The claims now follow from Lemma \ref{scatsc} after extending this result by induction. \epf \section{The Double Commutation Method} \label{secdc} In this section we provide a complete characterization of the double commutation method for Jacobi operators. We start with a linear transformation which turns out to be unitary when restricted to proper subspaces of our Hilbert space. We use this transformation to construct an operator $H_{\gam_1}$ from a given background operator $H$. This operator $H_{\gam_1}$ will be the doubly commuted operator of $H$ as discussed in the Introduction. The results of Sections 4-6 appear to be without precedent. Let $\hr=\ell^2(M_--1,M_++1)$ be the underlying Hilbert space ($-\infty \le M_- < M_+ \le \infty$) and let $\psi(n)$ be a given real-valued sequence which is square summable near $M_-$. Choose a positive constant $\gam>0$ and define \bea c_\gam(n) = 1 + \gam \sum_{j=M_-}^n \psi(j)^2, \qquad n \ge M_-. \eea (We set in addition $c_\gam(M_- -1) =1$ if $M_-$ is finite.) Denote the set of sequences in $\ell(M_--1,M_++1)$ which are square summable near $M_-$ by $\hr_-$ and consider the following (linear) transformation \bea \label{unitary1} \ba{llcl} U_\gam: &\hr_- & \to & \hr_- \\ & f(n) &\mapsto& f_\gam(n) = \sqrt{\frac{c_\gam(n)} {c_\gam(n-1)}} f(n) - \gam \psi_\gam(n) \sum\limits_{j=M_-}^n \psi(j) f(j). \ea \eea By inspection, the sequence $f_\gam$ is also square summable near $M_-$ and the inverse transformation is given by \bea \label{transfUf} \ba{llcl} U_\gam^{-1}: & \hr_- &\to& \hr_- \\ & g(n) &\mapsto& \sqrt{\frac{d_\gam(n)}{d_\gam(n-1)}} g(n) + \gam \psi(n) \sum\limits_{j=M_-}^n \psi_\gam(j)g(j) \ea, \eea where \bea \label{psigam} d_\gam(n) = c_\gam(n)^{-1} = 1 - \gam \sum_{j=M_-}^n \psi_\gam(j)^2, \qquad \psi_\gam(n) = \frac{\psi(n)}{\sqrt{c_\gam(n-1)c_\gam(n)}}. \eea \begin{lem} \label{lemuni} Define $\psi_\gam$ as in (\ref{psigam}). Then $\psi_\gam \in \hr$ and \bea \| \psi_\gam \|^2 = \frac{1}{\gam} \Big(1 - \lim_{n \to M_+} c_\gam(n)^{-1} \Big). \eea If $P,P_\gam$ denote the orthogonal projections onto the one-dimensional subspaces of $\hr$ spanned by $\psi, \psi_\gam$ (set $P = 0$ if $\psi \not\in \hr$) the operator $U_\gam$ is unitary from $(1-P) \hr$ onto $(1-P_\gam) \hr$. \end{lem} \bpf For the claims concerning $\psi$ we use \bea \sum_{j=M_-}^n |\psi_\gam(j)|^2 = \frac{1}{\gam} \sum_{j=M_-}^n \Big( \frac{1}{c_\gam(j-1)} - \frac{1}{c_\gam(j)} \Big) = \frac{1}{\gam} \Big( 1 - \frac{1}{c_\gam(n)} \Big). \eea Next we note that \bea c_\gam(n) \sum_{j=M_-}^n \psi_\gam(j) f_\gam(j) = \sum\limits_{j=M_-}^n \psi(j) f(j) \eea and a direct calculation shows \bea \label{un} \sum_{j=M_-}^n |f_\gam(j)|^2 = \sum_{j=M_-}^n |f(j)|^2 - \frac{\gam}{c_\gam(n)} |\sum_{j=M_-}^n f(j) \psi(j)|^2. \eea This clearly proves the lemma if $\psi \in \hr$. Otherwise, i.e., if $\psi \not\in \hr$, consider $U_\gam$, $U_\gam^{-1}$ on the dense subspace $\ell_0((M_-,M_+))$ and take closures (cf$.$, e.g$.$, \cite{wd}, Theorem 6.13). \epf Using, e.g., the polarization identity, we further get \bea \label{sprgam} \sum_{j=M_-}^n \ol{g_\gam(j)} f_\gam(j) = \sum_{j=M_-}^n \ol{g(j)} f(j) - \frac{\gam} {c_\gam(n)} \sum_{j=M_-}^n \psi(j) f(j) \sum_{j=M_-}^n \psi(j) \ol{g(j)} . \eea Next we take two sequences $a,b$ satisfying \bh \label{hypoabf} Suppose \bea a,b \in \ell(M_- -1,M_+ +1), \qquad a(n) \in \R \bs \{ 0 \}, \: b(n) \in \R \eea \eh and introduce the difference expression \bea (\tau f) (n) = a(n) f(n+1) +a(n-1) f(n-1) -b(n) f(n). \eea We want to consider a self-adjoint operator $H$ associated with $\tau$ and separated boundary conditions at $M_\pm$ and assume the existence of a sequence $\psi(\lam_1,n)$ of the following kind. \bh \label{hypodc} Suppose $\psi(\lam)$ satisfies the following conditions.\\ (i) $\psi(\lam,n)$ is a real-valued solution of $\tau \psi(\lam) = \lam \psi(\lam)$.\\ (ii) $\psi(\lam,n)$ is square summable near $M_-$ and fulfills the boundary condition (of $H$) at $M_-$ (if any, i.e., if $\tau$ is $l.c.$ at $M_-$).\\ (iii) $\psi(\lam,n)$ also fulfills the boundary condition (of $H$) at $M_+$ if $\tau$ is $l.c.$ at $M_+$ ($\psi(\lam,n)$ is then an eigenfunction of $H$). \eh Sufficient conditions for the above function to exist are\\ (i) $\lam \in \sigma_{p}(H)$, or\\ (ii) $\tau$ is $l.c.$ at $M_-$ but not at $M_+$, or\\ (iii) $\sigma(H) \ne \R$ (and $\lam \in \R \bs \sigma(H)$), or\\ (iv) $\sigma(H_-) \ne \R$ (and $\lam \in \R \bs \sigma(H_-)$), where $H_-$ is a restriction of $H$ to $\ell^2(M_--1,\hat{M}+1)$ with $\hat{M} \in \Z$ and (for instance) a Dirichlet boundary condition at $\hat{M}+1$. It follows that $H$ is explicitly given by \bea \label{defbh} \db(H) = \{ f \in \hr | \ba[t]{l} \tau f \in \hr ; \: W_{M_- -1}(\psi(\lam_1),f)=0 \mb{ if $\tau$ is $l.c.$ at $M_-$} , \\ W_{M_+}(\psi(\lam_1),f)=0 \mb{ if $\tau$ is $l.c.$ at $M_+$} \}. \ea \eea We now use Lemma \ref{lemuni} with $\psi(n) = \psi(\lam_1,n)$, $\gam=\gam_1$, $U_\gam=U_{\gam_1}$ to prove \begin{thm} \label{thmdc} Suppose (H.\ref{hypoabf}) and (H.\ref{hypodc}) and let $\tau_{\gam_1}$ be the difference expression \bea (\tau_{\gam_1} f)(n) = a_{\gam_1}(n) f(n+1) + a_{\gam_1}(n-1) f(n-1) - b_{\gam_1}(n) f(n), \eea where \bea a_{\gam_1}(n) &=& a(n) \frac{\sqrt{c_{\gam_1} (\lam_1,n-1)c_{\gam_1}(\lam_1,n+1)}} {c_{\gam_1}(\lam_1,n)},\\ \nn b_{\gam_1}(n) &=& b(n) + \gam_1 \Big( \frac{a(n-1) \psi(\lam_1,n-1) \psi(\lam_1,n)}{c_{\gam_1}(\lam_1,n-1)} \\ &&{}- \frac{a(n) \psi(\lam_1,n) \psi(\lam_1,n+1)}{c_{\gam_1}(\lam_1,n)}\Big). \eea Then the operator $H_{\gam_1}$ defined by \bea & H_{\gam_1} f = \tau_{\gam_1} f,& \\ \nn &\db(H_{\gam_1}) = \{ f \in \hr | \tau_{\gam_1} f \in \hr ; W_{\gam_1,M_- -1} (\psi_{\gam_1}(\lam_1),f) = W_{\gam_1,M_+}(\psi_{\gam_1}(\lam_1),f)=0 \},& \eea where $W_{\gam_1,n}(u,v) = a_{\gam_1}(n)(u(n) v(n+1) - u(n+1) v(n))$, is self-adjoint and has the eigenfunction \bea \psi_{\gam_1}(\lam_1,n) = \frac{\psi(\lam_1,n)}{\sqrt{c_{\gam_1}(\lam_1,n-1) c_{\gam_1}(\lam_1,n)}} \eea associated with the eigenvalue $\lam_1$. If $\psi(\lam_1) \not\in \hr$ (and hence $\tau$ is $l.p.$ at $M_+$) we have \bea (1-P_{\gam_1}(\lam_1)) H_{\gam_1} = U_{\gam_1} H U_{\gam_1}^{-1} (1-P_{\gam_1}(\lam_1)), \eea where $U_{\gam_1}$ is the unitary transformation of Lemma \ref{lemuni} and thus \bea \ba{rcl@{\qquad}rcl} \sigma(H_{\gam_1}) &=& \sigma(H) \cup \{ \lam_1\}, & \sigma_{ac}(H_{\gam_1}) &=& \sigma_{ac}(H), \\ \sigma_{p}(H_{\gam_1}) &=& \sigma_{p}(H) \cup \{ \lam_1\}, & \sigma_{sc}(H_{\gam_1}) &=& \sigma_{sc}(H). \ea \eea If $\psi(\lam_1) \in \hr$ there is a unitary operator $\ti{U}_{\gam_1} = U_{\gam_1} \oplus \sqrt{1+\gam_1 \| \psi(\lam_1) \|^2} \mb{\bf 1}$ on $(1-P_{\gam_1}(\lam_1)) \hr \oplus P_{\gam_1}(\lam_1) \hr $ such that \bea H_{\gam_1} = \ti{U}_{\gam_1} H \ti{U}_{\gam_1}^{-1} \eea and thus \bea \ba{rcl@{\qquad}rcl} \sigma(H_{\gam_1}) &=& \sigma(H), & \sigma_{ac}(H_{\gam_1}) &=& \sigma_{ac}(H), \\ \sigma_{p}(H_{\gam_1}) &=& \sigma_{p}(H), & \sigma_{sc}(H_{\gam_1}) &=& \sigma_{sc}(H). \ea \eea \end{thm} \bpf It suffices to prove \bea (1-P_{\gam_1}(\lam_1)) H_{\gam_1} = U_{\gam_1} H U_{\gam_1}^{-1} (1-P_{\gam_1}(\lam_1)). \eea Let $f$ be a sequence which is square summable near $M_-$ such that $\tau f$ is also square summable near $M_-$ and assume that $f$ fulfills the boundary condition at $M_-$, if any. Then a straightforward calculation shows \bea \tau_{\gam_1} (U_{\gam_1} f) = U_{\gam_1} (\tau f) \eea and we only have to check the boundary conditions at $M_\pm$. Equation (\ref{un}) shows that $\tau_{\gam_1}$ is $l.c.$ at $M_-$ if and only if $\tau$ is and that $\tau_{\gam_1}$ is $l.c.$ at $M_+$ if $\tau$ is. The formula \bea W_{\gam_1,n}(\psi_{\gam_1}(\lam_1),U_{\gam_1} f) = \frac{W_n(\psi(\lam_1),f)}{c_{\gam_1}(\lam_1,n)} \eea shows that \bea \label{WMm} W_{\gam_1,M_- -1}(\psi_{\gam_1}(\lam_1),U_{\gam_1} f) =0, \qquad f \in \db(H). \eea We further claim that \bea \label{WMp} W_{\gam_1,M_+}(\psi_{\gam_1}(\lam_1),U_{\gam_1} f) = 0, \qquad f \in \db(H). \eea This is clear if $\psi(\lam_1) \in \hr$. Otherwise, i.e., if $\psi(\lam_1) \not\in \hr$, we use \bea \frac{W_n(\psi(\lam_1),f)}{c_{\gam_1}(\lam_1,n)} = \frac{\sum_{j=M_-}^n \psi(\lam_1,j) (\lam_1 - \tau) f(j)}{c_{\gam_1}(\lam_1,n)}. \eea The right hand side tends to zero for $ f \in \db(H)$ as can be seen from (\ref{un}) and the fact that $U_{\gam_1}$ is unitary. Combining (\ref{WMm}) and (\ref{WMp}) yields \bea (1-P_{\gam_1}(\lam_1)) U_{\gam_1} \db(H) \subseteq (1-P_{\gam_1}(\lam_1)) \db(H_{\gam_1}). \eea But $(1-P_{\gam_1}(\lam_1)) U_{\gam_1} \db(H)$ cannot be properly contained in $(1-P_{\gam_1}(\lam_1)) \db(H_{\gam_1})$ by the property of self-adjoint operators being maximally defined. \epf \begin{rem} (i). By choosing $\lam_1 \in \sigma_{ac}(H) \cup \sigma_{sc}(H)$ (provided the continuous spectrum is not empty and a solution satisfying (H.\ref{hypodc}) exists) we can use the double commutation method to construct operators with eigenvalues embedded in the continuous spectrum.\\ (ii). If $M_+=\infty$ and $H$ has an eigenfunction $\psi(\lam_1)$ one can remove this eigenfunction from the spectrum upon choosing $\gam_1 = - \| \psi(\lam_1) \|^{-2}$. The corresponding function $\psi_{\gam_1}(\lam_1)$ is then no longer in $\hr$, implying that $\tau_{\gam_1}$ is $l.p.$ at $M_+$.\\ (iii). Especially, removing an eigenvalue from an operator which is $l.c.$ at $\infty$ yields an operator which is $l.p.$. Thus $\tau_{\gam_1}$ is not necessary $l.p.$ if $\tau$ is. Moreover, this shows that one cannot insert additional eigenvalues into an operator which is $l.c.$ at $M_+$ (remove this eigenvalue again to obtain a contradiction).\\ (iv). The limiting case $\gam_1 =\infty$ can be handled analogously producing a unitarily equivalent operator if $\psi(\lam_1) \not\in \hr$ and removes the eigenvalue $\lam_1$ otherwise. \end{rem} The previous theorem tells us only how to transfer solutions of $\tau u = z u$ into solutions of $\tau_{\gam_1} v = z v$ if $u$ is square summable near $M_-$. The following lemma treats the general case. \begin{lem} \label{addprophg} The sequence \bea u_{\gam_1}(z,n) = \frac{c_{\gam_1}(\lam_1,n) u(z,n) - \frac{\gam_1}{z-\lam_1} \psi(\lam_1,n) W_n(\psi(\lam_1),u(z))}{\sqrt{c_{\gam_1}(\lam_1,n-1) c_{\gam_1}(\lam_1,n)}}, \:\: z \in \C \bs \{ \lam_1 \} \eea solves $\tau_{\gam_1} u = z u$ if $u(z)$ solves $\tau u = z u$. If $u(z)$ is square summable near $M_-$ and fulfills the boundary condition at $M_-$ (if any) we have $u_{\gam_1}(z,n)=(U_{\gam_1} u)(z,n)$ justifying our notation. Furthermore, we note \bea \nn |u_{\gam_1}(z,n)|^2 &=& |u(z,n)|^2 - \frac{\gam_1}{|z-\lam_1|^2} \times\\ &&{} \Big( \frac{|W_n(\psi(\lam_1),u(z))|^2} {c_{\gam_1}(\lam_1,n)} - \frac{|W_{n-1}(\psi(\lam_1),u(z))|^2} {c_{\gam_1}(\lam_1,n-1)}\Big), \eea and \bea \label{wrongamp} W_{\gam_1,n}(\psi_{\gam_1}(\lam_1),u_{\gam_1}(z)) = \frac{W_n(\psi(\lam_1),u(z))}{c_{\gam_1}(\lam_1,n)}. \eea Hence $u_{\gam_1}$ is square summable near $M_+$ if $u$ is. If $\hat{u}_\gam(\hat{z})$ is constructed analogously then \bea \nn W_{\gam_1,n}(u_{\gam_1}(z),\hat{u}_{\gam_1}(\hat{z})) &=& W_n(u(z),\hat{u}(\hat{z})) + \frac{\gam_1}{c_{\gam_1}(\lam_1,n)} \frac{z-\hat{z}}{(z-\lam_1)(\hat{z}-\lam_1)} \times \\ \label{wrongam} && W_n(\psi(\lam_1),u(z))W_n(\psi(\lam_1),\hat{u}(\hat{z})). \eea \end{lem} \bpf All facts are tedious but straightforward calculations. \epf Next we want to give some conditions implying the $l.p.$ case of $\tau_{\gam_1}$ at $M_+$, assuming $M_+=\infty$. Let $M_-<\hat{M}<\infty$ and let $H_+$ denote a self-adjoint operator associated with $\tau$ on $(\hat{M}-1,\infty)$ and the boundary condition induced by $\psi(\lam_1)$ at $\hat{M}$ (cf. equation (\ref{defbh})). \bh \label{hypodcsp} Suppose $H_+$ satisfies one of the following spectral conditions:\\ (i). $\sig_{ess} (H_+) \ne \emptyset$.\\ (ii). $\sig(H_+)=\sig_d(H_+)= \{ \lam_{+,j} \}_{j\in J_+}$ with $\sum_{j\in J_+} (1+\lam^2_{+,j})^{-1} =\infty$. \eh Clearly Hypothesis (H.\ref{hypodcsp}) is satisfied if $a,b$ are bounded near $\infty$ (which is equivalent to $H_+$ being bounded) since then $\tau$ is $l.p.$ at $\infty$. \begin{thm} \label{thmlp} Assume (H.\ref{hypoabf}), (H.\ref{hypodc}), and (H.\ref{hypodcsp}). Then $\tau_{\gam_1}$ is $l.p.$ at $M_+=\infty$. \end{thm} \bpf Let $\gam_{1,+} = c_{\gam_1}(\lam_1,\hat{M})^{-1}\gam_1$ and consider the doubly commuted operator $H_{+,\gam_{1,+}}$ of $H_+$. Then $\tau_{\gam_1}|_{(\hat{M},\infty)} = \tau_{\gam_{1,+}}$ and $H_{+,\gam_{1,+}}$ also satisfies (H.\ref{hypodcsp}). Hence $\tau_{\gam_1}$ is $l.p.$ at $\infty$ as claimed. \epf \begin{rem} We can interchange the role of $M_-$ and $M_+$ in this section by substituting $M_- \leftrightarrow M_+$, $\sum_{j=M_-}^n \to \sum_{j=n+1}^{M_+}$ and $\gam_1 \to -\gam_1$. \end{rem} Let $M_\pm=\pm\infty$ and $H$ be a given Jacobi operator satisfying (\ref{decay}). Our next aim is to show how the scattering data of the operators $H, H_{\gam_1}$ are related, where $H_{\gam_1}$ is defined as in Theorem \ref{thmdc}. \begin{lem} \label{sctdc} Let $H$ be a given Jacobi operator satisfying (\ref{decay}). Then the doubly commuted operator $H_{\gam_1}$, defined via $\psi(\lam_1,n) = f_-(k_1,n), \quad \lam_1=(k_1+k_1^{-1})/2$ as in Theorem \ref{thmdc}, has the transmission and reflection coefficients \bea T_{\gam_1}(k) = \sgn(k_1) \frac{k \, k_1 -1}{k - k_1} T(k), \eea \bea R_{-,\gam_1}(k) = R_-(k), \qquad R_{+,\gam_1}(k) = \left( \frac{k - k_1}{k \, k_1 -1} \right)^2 R_+(k), \eea where $k$ and $z$ are related via $z = (k + k^{-1})/2$. Furthermore, the norming constants $\gam_{-,j}$ corresponding to $\lam_j \in \sig_{p}(H)$, $j \in J$ (cf.~$(\ref{norming})$) remain unchanged except for an additional eigenvalue $\lam_1$ with norming constant $\gam_{-,1}=\gam_1$ if $\psi(\lam_1) \not\in \hr$ respectively with norming constant $\ti{\gam}_{-,1}= \gam_{-,1} + \gam_1$ if $\psi(\lam_1) \in \hr$ and $\gam_{-,1}$ denotes the original norming constant of $\lam_1 \in \sig_{p}(H)$. \end{lem} \bpf By Lemma \ref{addprophg} the Jost solutions $f_{\gam_1,\pm}(k,n)$ are up to a constant given by \bea \frac{c_{\gam_1}(\lam_1,n-1) f_\pm(k,n) - \frac{\gam_1}{z-\lam_1} \psi(\lam_1,n) W_{n-1}(\psi(\lam_1),f_\pm(k))} {\sqrt{c_{\gam_1}(\lam_1,n-1)c_{\gam_1}(\lam_1,n)}}. \eea This constant is easily seen to be 1 for $f_{\gam_1,-}(k,n)$. Thus we can compute $R_-(\lam)$ using (\ref{wrongam}) (the second unknown constant cancels). The rest follows by a straightforward calculation. \epf \section{Double Commutation and Weyl--Titchmarsh Theory} In this section we want to reveal the connections between Weyl--Titchmarsh theory and the double commutation method. Without loss of generality we consider only the cases $\ell^2(\N)$ and $\ell^2(\Z)$. We start with the half-line $\N$ and freely use the notation employed in Appendices A--D. Let $H_+$ be a self-adjoint operator associated with $\tau$ on $\N$ and a Dirichlet boundary condition at 0. Without loss of generality we assume $\psi(\lam_1,1)=1$. \begin{thm} Assume (H.\ref{hypoabf}), $\psi(\lam_1,1)=1$ and let $m_+(z,0)$, $m_{+,\gam_1}(z,0)$ denote the Weyl $m$-functions of $H_+$, $H_{+,\gam_1}$. Then we have \bea m_{+,\gam_1}(z,0) = \frac{1}{1+\gam_1} \Big( m_+(z,0) - \frac{\gam_1}{z - \lam_1} \Big). \eea If $\mu_+$ and $\mu_{+,\gam_1}$ denote the corresponding spectral functions of $H_+$ and $H_{+,\gam_1}$ it follows that \bea \mu_{+,\gam_1}(\lam) = \frac{1}{1+\gam_1} \Big( \mu_+(\lam) + \gam_1 \Theta(\lam-\lam_1) \Big), \eea where $\Theta(.)$ denotes the (right continuous) step function \bea \Theta(x) = \left\{ \ba{c@{\quad}l} 1, & x \ge 0 \\ 0, & x < 0 \ea \right. . \eea \end{thm} \bpf As in Section~\ref{secwm} we use the finite approximations $m_N(z,0)$ and $m_{N,\gam_1}(z,0)$. If $\gam_j(N)$, $\gam_{j,\gam_1}(N)$ are the corresponding norming constants we have \bea \gam_{j,\gam_1}(N) = \frac{1}{1+\gam_1} \left\{ \ba{c@{\quad}l} \gam_j(N) + \gam_1, & \lam_j=\lam_1 \\ \gam_j(N), & \lam_j \ne \lam_1 \ea \right. . \eea This follows since $\psi(z,0)=0$, $\psi(z,1)=1$ implies $\psi_{\gam_1}(z,0)=0$, $\psi_{\gam_1}(z,1)= (1+\gam_1)^{-1/2}$. Hence we infer \bea m_{N,\gam_1}(z,0) = \frac{1}{1+\gam_1} \Big( m_N(z,0) - \frac{\gam_1}{z - \lam_1} \Big) \eea and the theorem follows upon taking the limit $N \to \infty$. \epf \begin{rem} If we transform the operator $H_+$ into it's diagonal form as in Section~\ref{secwtn} the double commutation method gets particularly transparent: it corresponds to adding a step function to the spectral function. This approach can also be used to derive the unitary transformation stated in Section 2 in the following way. Take the spectral function $\mu_+$ of a given Jacobi operator, switch to $\mu_{+,\gam_1}$, and compute the orthogonal polynomials with respect to this new measure (compare Section~\ref{secwtn} and \cite{ak}, Ch. 1). Now take a sequence $f(n)$ and its transform $F(z)$ and use (\ref{unitary2}) to obtain (\ref{unitary1}). \end{rem} Next we turn to operators in $\ell^2(\Z)$. Without loss of generality we assume \bea \label{initv} \psi(\lam_1,0) = - \sin(\alpha), \quad \psi(\lam_1,0) = \cos(\alpha), \quad \alpha \in [0,\pi). \eea \begin{thm} Assume (H.\ref{hypoabf}) and let $\ti{m}_\pm(z,\alpha)$, $\ti{m}_{\pm,\gam_1}(z,\alpha)$ denote the Weyl $\ti{m}$-functions of $H$, $H_{\gam_1}$ as introduced in Section~\ref{secwm}. Then we have \bea \ti{m}_{\pm,\gam_1}(z,\ti{\alpha}) = \frac{(1+\ti{\gam}_1 (\cos(\alpha)^4 - \sin(\alpha)^4))^{-1/2}}{((1+\ti{\gam}_1\cos(\alpha)^2) (1-\ti{\gam}_1\sin(\alpha)^2))^{1/2}} \Big( \ti{m}_\pm(z,\alpha) - \frac{\ti{\gam}_1}{z - \lam_1} \Big), \eea where \bea \ti{\gam}_1 = \frac{\gam_1}{c_{\gam_1}(\lam_1,0)}, \qquad \tan(\ti{\alpha}) = \sqrt{\frac{c_{\gam_1}(\lam_1,1)} {c_{\gam_1}(\lam_1,-1)}} \tan(\alpha). \eea \end{thm} \bpf Consider the sequences \bea \phi_{\alpha,\gam_1}(z,n), \quad \theta_{\alpha,\gam_1}(z,n) - \frac{\ti{\gam}_1}{z-\lam_1} \phi_{\alpha,\gam_1}(z,n) \eea constructed from the fundamental system $\theta_\alpha(z,n)$, $\phi_\alpha(z,n)$ for $\tau$ (cf. (\ref{funsy})) as in Lemma \ref{addprophg}. They form a fundamental system for $\tau_{\gam_1}$ corresponding to the initial conditions associated with $\ti{\alpha}$ up to constant multiples. Now use (\ref{wrongamp}) to evaluate (\ref{mfun}). \epf The Weyl $M$-matrix and the corresponding spectral matrix can now be computed in a straightforward manner (cf.\ Section~\ref{secwtz}). \section{Iteration of the double commutation method} \label{secdcit} Finally we demonstrate how to iterate the double commutation method. We choose a given background operator $H$ (with coefficients $a$, $b$ satisfying (H.\ref{hypoabf})) and further $\gamma_1>0, \lam_1\in \R$. Next choose $\psi(\lam_1)$ as in Hypothesis (H.\ref{hypodc}) to define the transformation $U_{\gam_1}$ and the operator $H_{\gam_1}$. In the second step, we choose $\gamma_2>0, \lam_2 \in \R$ and another function $\psi(\lam_2)$ to define $\psi_{\gam_1}(\lam_2) = U_{\gam_1} \psi(\lam_2)$, a corresponding transformation $U_{\gam_1,\gam_2}$, and an operator $H_{\gam_1,\gam_2}$. Applying this procedure $N$-times results in \begin{thm} \label{dcommN} Assuming (H.\ref{hypoabf}) let $H$ be a given background Jacobi operator in $\hr = \ell^2(M_--1,M_++1)$ and let $\gam_j>0$, $\lam_j$, $1 \le j \le N$ be such that there exist corresponding solutions $\psi(\lam_j,n)$ of $\tau \psi = \lam_j \psi$ satisfying Hypothesis (H.\ref{hypodc}). We set $\psi_{\gam_1,\dots,\gam_k}(\lam_j) = U_{\gam_1,\dots,\gam_k} \cdots U_{\gam_1} \psi(\lam_j)$ and define the following matrices $(1 \le \ell \le N)$ \bea C^\ell(n) = \left\{ \delta_r(s) + \sqrt{\gam_r \gam_s} \sum_{m=M_-}^n \psi(\lam_r,m) \psi(\lam_s,m) \right\}_{1 \le r,s \le \ell}, \eea \bea \label{cij} C^\ell_{i,j}(n) = \left\{ \ba{c@{\quad}l} C^{\ell-1}(n)_{r,s} & \scriptstyle r,s \le \ell-1\\ \sqrt{\gam_s} \sum\limits_{m=M_-}^n \psi(\lam_i,m) \psi(\lam_s,m) & \scriptstyle s \le \ell-1, r=\ell\\ \sqrt{\gam_r} \sum\limits_{m=M_-}^n \psi(\lam_r,m) \psi(\lam_j,m) & \scriptstyle r \le \ell-1, s=\ell\\ \sum\limits_{m=M_-}^n \psi(\lam_i,m) \psi(\lam_j,m) & \scriptstyle r=s=\ell \ea \right\}_{1 \le r,s \le \ell}, \eea \bea \Psi^\ell(\lam_j,n) = \left\{ \ba{c@{\quad}l} C^\ell(n)_{r,s} & \scriptstyle r,s \le \ell\\ \sqrt{\gam_s} \sum\limits_{m=M_-}^n \psi(\lam_j,m) \psi(\lam_s,m) & \scriptstyle s \le \ell, r=\ell+1\\ \sqrt{\gam_r}\psi(\lam_r,n) & \scriptstyle r \le \ell, s=\ell+1\\ \psi(\lam_j,n) & \scriptstyle r=s=\ell+1 \ea \right\}_{1 \le r,s \le \ell+1}. \eea Then we have (set $C^0(n)=1$) \bea \label{ckn} c_{\gam_\ell}(\lam_\ell,n) = 1 + \gam_\ell \sum_{m=M_-}^n \psi_{\gam_1,\dots,\gam_\ell} (\lam_\ell,m)^2 = \frac{\det C^\ell(n)}{\det C^{\ell-1}(n)}, \eea and hence \bea \label{prck} \prod_{\ell=1}^N c_{\gam_\ell}(\lam_\ell,n) = \det C^N(n). \eea Moreover, \bea \label{sprk} \sum_{m=M_-}^n \psi_{\gam_1,\dots,\gam_\ell}(\lam_i,m) \psi_{\gam_1,\dots,\gam_\ell}(\lam_j,m) = \frac{\det C^\ell_{i,j}(n)}{\det C^{\ell-1}(n)} \eea and \bea \label{phik} \psi_{\gam_1,\dots,\gam_\ell}(\lam_j,n) = \frac{\det \Psi^\ell(\lam_j,n)} {\sqrt{\det C^\ell(n-1)\det C^\ell(n)}}. \eea In addition, we get \bea a_{\gam_1,\dots,\gam_N}(n) &=& a(n) \frac{\sqrt{\det C^N(n-1) \det C^N(n+1)}}{\det C^N(n)}, \\ \nn b_{\gam_1,\dots,\gam_N}(n) &=& b(n) - \sum_{\ell=1}^N \gam_\ell \left( a(n) \frac{ \det \Psi^\ell(\lam_\ell,n) \det \Psi^\ell(\lam_\ell,n+1)}{\det C^{\ell-1}(n)\det C^\ell(n)} \right. \\ \nn && \left. {} - a(n-1) \frac{ \det \Psi^\ell(\lam_\ell,n-1) \det \Psi^\ell(\lam_\ell,n)} {\det C^{\ell-1}(n-1) \det C^\ell(n-1)} \right)\\ \nn &=& - \lam_N + a(n) \frac{\det C^N(n-1)} {\det C^N(n)}\frac{\det \Psi^N(\lam_N,n+1)} {\det \Psi^N(\lam_N,n)}\\ &&{}+ a(n-1) \frac{\det C^N(n)}{\det C^N(n-1)}\frac{\det \Psi^N(\lam_N,n-1)} {\det \Psi^N(\lam_N,n)}, \eea the last equation only being valid if $\det \Psi^N(\lam_N,n) \ne 0$ (e.g., if $\lam_N\le\inf \sig(H)$). The spectrum of $H_{\gam_1,\dots,\gam_N}$ is given by \bea \ba{rcl@{\quad}rcl} \sig(H_{\gam_1,\dots,\gam_N}) \!\!&=&\!\! \sig(H) \cup \{ \lam_j\}_{j=1}^N, & \sig_{ac}(H_{\gam_1,\dots,\gam_N}) \!\!&=&\!\! \sig_{ ac}(H),\\ \sig_{p}( H_{\gam_1,\dots,\gam_N}) \!\!&=&\!\! \sig_{p}(H) \cup \{ \lam_j \}_{j=1}^N , & \sig_{sc}(H_{\gam_1,\dots,\gam_N}) \!\!&=&\!\! \sig_{sc}(H). \ea \eea Moreover, \bea \nn && H_{\gam_1,\dots,\gam_N} (1-\sum_{j=1}^N P_{\gam_1,\dots,\gam_N}(\lam_j)) \\ && \quad = (U_{\gam_1,\dots,\gam_N} \cdots U_{\gam_1}) H (U_{\gam_1}^{-1} \cdots U_{\gam_1,\dots,\gam_N}^{-1}) (1-\sum_{j=1}^N P_{\gam_1,\dots,\gam_N}(\lam_j)), \eea where $P_{\gam_1,\dots,\gam_N}(\lam_j)$ denotes the projection onto the one-dimensional subspace of $\hr$ spanned by $\psi_{\gam_1,\dots,\gam_N}(\lam_j)$. \end{thm} \bpf We start with (\ref{sprk}). Using Sylvester's determinant identity (cf.\ \cite{gant}, Sect.\ II.3) we obtain \bea \nn && \det C^{\ell-1}(n) \det C^{\ell+ 1}_{i,j}(n)\\ &&= \det C^\ell(n) \det C^\ell_{i,j}(n) - \gam_\ell \det C^\ell_{\ell,j}(n) \det C^\ell_{i,\ell}(n), \eea which proves (\ref{sprk}) together with a look at (\ref{sprgam}) by induction on $N$. Next, (\ref{ckn}) easily follows from (\ref{sprk}). Similarly, \bea \nn && \det C^\ell(n) \det \Psi^{\ell+ 1}(\lam_j,n) \\ && = \det C^{\ell+1}(n) \det \Psi^\ell(\lam_j,n) - \gam_{\ell} \det \Psi^\ell(\lam_\ell,n) \det C^\ell_{j,\ell}(n), \eea and (\ref{transfUf}) prove (\ref{phik}). The rest follows in a straightforward manner. \epf \begin{rem} (i). For a sequence $f$, which is square summable near $M_-$, $f_{\gam_1,\dots,\gam_j} = U_{\gam_1,\dots,\gam_j} \cdots U_{\gam_1} f$ is given by substituting $\psi(\lam_j) \to f$ in (\ref{phik}). Similarly we get the scalar product of $f_{\gam_1,\dots,\gam_i}$ and $g_{\gam_1,\dots,\gam_j}$ from (\ref{sprk}) by substituting $f \to \psi(\lam_i)$ and $g \to \psi(\lam_j)$ in (\ref{cij}).\\ (ii). Equation (\ref{phik}) can be rephrased as \bea \nn && (\gam_1 \psi_{\gam_1,\dots,\gam_\ell}(\lam_1,n), \dots, \gam_\ell \psi_{\gam_1,\dots,\gam_\ell}(\lam_\ell,n)) =\\ && \sqrt{\frac{\det C^\ell(n)} {\det C^\ell(n-1)}}(C^\ell(n))^{-1} (\gam_1 \psi(\lam_1,n), \dots, \gam_\ell \psi(\lam_\ell,n)), \eea where $(C^\ell(n))^{-1}$ is the inverse matrix of $C^\ell(n)$. \end{rem} Clearly Theorem \ref{thmlp} extends (by induction) to this more general situation. \begin{thm} Assume (H.\ref{hypoabf}) and (H.\ref{hypodcsp}). Then $\tau_{\gam_1,\dots,\gam_N}$ is $l.p.$ at $M_+$. \end{thm} Finally we also extend Lemma \ref{sctdc}. For simplicity we assume $\psi(\lam_j,n) \not\in \hr$, $1 \le j \le N$. \begin{lem} Let $H$ be a given Jacobi operator satisfying (\ref{decay}). Then $H_{\gam_1,\dots,\gam_N}$, defined via $\psi(\lam_\ell,n) = f_-(k_\ell,n), \quad \lam_\ell=(k_\ell+k_\ell^{-1})/2 \in \R \bs \sig(H_{\gam_1,\dots, \gam_{\ell-1}})$, $1 \le \ell \le N$ has the transmission and reflection coefficients \bea &T_{\gam_1,\dots,\gam_N}(k) = \prod_{\ell=1}^N \sgn(k_\ell) \frac{k \, k_\ell -1}{k - k_\ell} T(k),&\\ &R_{-,\gam_1,\dots,\gam_N}(k) = R_-(k), \quad R_{+,\gam_1,\dots,\gam_N}(k) = \left( \prod\limits_{\ell=1}^N \Big( \D \frac{k - k_\ell}{k \, k_\ell -1} \Big)^2 \right) R_+(k),& \eea where $z = (k + k^{-1})/2$. Furthermore, the norming constants $\gam_{-,j}$ corresponding to $\lam_j \in \sig_{p}(H)$, $j \in J$ (cf.\ $(\ref{norming})$) remain unchanged and the additional eigenvalues $\lam_\ell$ have norming constants $\gam_{-,\ell}=\gam_\ell$. \end{lem} \begin{rem} Of special importance is the case $a(n) =1/2$, $b(n)=0$. Here we have $f_\pm(k,n) = k^{\pm n}$, $T(k)=1$, and $R_\pm(k)=0$. It is well known from inverse scattering theory that $R_\pm(k)$, $|k|=1$ together with the point spectrum and corresponding norming constants uniquely determine $a(n),b(n)$. Hence we infer from Lemma \ref{scatscN} that $H_{\gam_1,\dots,\gam_N}$ constructed from $\psi(\lam_\ell,n) = f_-(k_\ell,n)$ as in Theorem \ref{dcommN} and $H_{\sig_1,\dots,\sig_N}$ constructed from $u_{\sig_\ell}^\ell = \frac{1+\sig_\ell}{2} f_+(k_\ell,n) + (-1)^{\ell+1} \frac{1-\sig_\ell}{2} f_-(k_\ell,n)$ as in Theorem \ref{thmscom} coincide if \bea \gam_j = \left( \frac{1-\sig_j}{1+\sig_j} \right)^{-1} |k_j|^{-1-N} \frac{\prod_{\ell=1}^N |1 - k_j k_\ell|}{\prod_{ {\ell=1 \atop \ell \ne j}}^N |k_j-k_\ell|} T(k_j), \quad 1 \le j \le N. \eea For a direct proof compare \cite{TKvM}. \end{rem} \section{Applications} \label{secappc} First we state the discrete analogue of the FIT-formula derived in \cite{fit} for the isospectral torus of periodic Schr\"odinger operators. This yields an explicit realization of the isospectral torus of all algebro-geometric quasi-periodic finite-gap Jacobi operators. Let $a(n),b(n)$ be given algebro-geometric quasi-periodic $g$-gap sequences characterized by the band-edges $E_0 < E_1 < \dots < E_{2g+1}$ and Dirichlet data $\{ (\mu_j,\sig_j) \}_{j=1}^g$ at the reference point $n=0$ (cf.\ \cite{bght}), where $\mu_j \in [E_{2j-1},E_{2j}]$ and $\sig_j \in \{\pm\}$, $1 \le j \le g$. Then the spectrum of the associate Jacobi operator $H$ is of the type \bea \nn &\sig(H) = \sig_{ac}(H) = \bigcup_{n=1}^{g+1} [E_{2n-2},E_{2n-1}],&\\ \label{ggapspec} &\sig_{sc}(H) = \sig_p(H) = \emptyset.& \eea and (cf.\ (\ref{hpm})) \bea \sig(H_\pm) = \sig(H) \cup \{ \mu_j | \sig_j = \pm, \: 1 \le j \le g \}. \eea Then considerations as in Theorem \ref{thmscom} readily yield that all other isospectral algebro-geometric $g$-gap sequences can be realized in the following way \bea \nn \lefteqn{a_{(\ti{\mu}_1,\ti{\sig}_1), \dots,(\ti{\mu}_g,\ti{\sig}_g)}(n) = - \sqrt{a(n-g) a(n-g+2)} \times} \\ \nn && \sqrt{\frac{ C_{n-g}(\psi_{\sig_1}(\mu_1), \psi_{-\ti{\sig}_1}(\ti{\mu}_1),\dots, \psi_{\sig_g}(\mu_g), \psi_{-\ti{\sig}_g}(\ti{\mu}_g))}{ C_{n-g+1}(\psi_{\sig_1}(\mu_1), \psi_{-\ti{\sig}_1}(\ti{\mu}_1),\dots, \psi_{\sig_g}(\mu_g), \psi_{-\ti{\sig}_g}(\ti{\mu}_g))}} \times \\ \label{fita} && \sqrt{\frac{ C_{n-g+2} (\psi_{\sig_1}(\mu_1),\psi_{-\ti{\sig}_1} (\ti{\mu}_1), \dots, \psi_{\sig_g}(\mu_g), \psi_{-\ti{\sig}_g}(\ti{\mu}_g)) }{C_{n-g+1}(\psi_{\sig_1}(\mu_1), \psi_{-\ti{\sig}_1}(\ti{\mu}_1),\dots, \psi_{\sig_g}(\mu_g), \psi_{-\ti{\sig}_g}(\ti{\mu}_g))}}, \eea \bea \nn \lefteqn{b_{(\ti{\mu}_1,\ti{\sig}_1), \dots,(\ti{\mu}_g,\ti{\sig}_g)}(n) = a(n-g) \frac{ C_{n-g+2}(\psi_{\sig_1}(\mu_1), \psi_{-\ti{\sig}_1}(\ti{\mu}_1), \dots, \psi_{\sig_g}(\mu_g)) }{ C_{n-g+1} (\psi_{\sig_1}(\mu_1), \psi_{-\ti{\sig}_1}(\ti{\mu}_1), \dots,\psi_{\sig_g}(\mu_g))} \times} \\ && \nn \frac{C_{n-g}(\psi_{\sig_1} (\mu_1),\psi_{-\ti{\sig}_1}(\ti{\mu}_1), \dots, \psi_{\sig_g}(\mu_g), \psi_{-\ti{\sig}_g}(\ti{\mu}_g))}{C_{n-g+1}( \psi_{\sig_1}(\mu_1), \psi_{-\ti{\sig}_1}(\ti{\mu}_1), \dots, \psi_{\sig_g}(\mu_g), \psi_{-\ti{\sig}_g}(\ti{\mu}_g))} \\ \nn && {} + a(n+1) \frac{C_{n-g} (\psi_{\sig_1}(\mu_1), \psi_{-\ti{\sig}_1}(\ti{\mu}_1),\dots , \psi_{\sig_g}(\mu_g))}{ C_{n-g+1}(\psi_{\sig_1}(\mu_1), \psi_{-\ti{\sig}_1}(\ti{\mu}_1), \dots, \psi_{\sig_g}(\mu_g)) } \times \\ && \label{fitb} \frac{C_{n-g+1} (\psi_{\sig_1}(\mu_1), \psi_{-\ti{\sig}_1}(\ti{\mu}_1), \dots,\psi_{\sig_g}(\mu_g), \psi_{-\ti{\sig}_g}(\ti{\mu}_g))}{ C_{n-g}(\psi_{\sig_1}(\mu_1), \psi_{-\ti{\sig}_1}(\ti{\mu}_1), \dots, \psi_{\sig_g}(\mu_g), \psi_{-\ti{\sig}_g}(\ti{\mu}_g))} - \ti{\mu}_g, \eea where $\psi_\pm(z,n)$ are the branches of the Baker-Akhiezer function associated with $a,b$ (i.e., the solutions of $\tau \psi = z \psi$ which are square summable near $\pm\infty$) and the new sequences are associated with the new Dirichlet data $\{ (\ti{\mu}_j,\ti{\sig}_j) \}_{j=1}^g$ at the same reference point $n=0$. Even though $\psi_\pm(z,n)$ is not necessarily positive as required in our Theorem \ref{thmscom}, the above sequences can be shown to be well-defined by using the explicit theta-function representations for $\psi_\pm(z,n)$ (cf., e.g., \cite{bght}) as long as $\ti{\mu}_j \in [E_{2j-1},E_{2j}]$ and $\ti{\sig}_j \in \{\pm\}$, $1 \le j \le g$. In fact, consider the hyperelliptic Riemann surface $K_g$ associated with the function \bea R_{2g+2}(z)^{1/2} = \prod_{j=0}^{2g+1} (z-E_j)^{1/2} \eea and branch points $E_0 < E_1 < \dots < E_{2g+1}$. A point $P\in K_g$ will be denoted by $P = (z,\pm R_{2g+2}(z)^{1/2})$ and we add two points $\infty_\pm \in K_g$ such that $K_g$ is compact. Introduce \bea \underline{z}(P,n) =\underline{\hat{A}}_{P_0}(P) - \sum_{j=1}^g \underline{\hat{A}}_{P_0}(\hat{\mu}_j) + 2 n \underline{\hat{A}}_{P_0}(\infty_+) - \hat{\underline{\Xi}}_{P_0}, \eea where $\underline{\hat{A}}_{P_0}$ is Abel's map with base point $P_0=(E_0,0)$ and $\hat{\underline{\Xi}}_{P_0}$ is the vector of Riemann constants (cf.\ \cite{bght} for more details). Then \bea \label{thetaa} a(n) &=& \ti{a} [\theta (\underline{z}(\infty_+,n-1)) \theta (\underline{z}(\infty_+,n+1))/ \theta (\underline{z} (\infty_+,n))^2 ] ^{1/2},\\ \nn b(n) &=& -E_0 + \ti{a} \frac{\theta (\underline{z} (\infty_+,n-1))\theta (\underline{z} (P_0, n+1))}{\theta (\underline{z} (\infty_+,n)) \theta (\underline{z}(P_0, n))}\\ \label{thetab} && {} + \ti{a} \frac{\theta (\underline{z} (\infty_+,n)) \theta (\underline{z}(P_0,n-1))}{\theta (\underline{z}(\infty_+,n-1))\theta (\underline{z}(P_0,n))}, \eea where $\theta$ is Riemann's theta function associated with $K_g$ and $\ti{a}$ is a constant depending only on $K_g$ (i.e.,on $ \{ E_j\}_{j=0}^{2g+1}$). Performing one single commutation at a point $Q=(z,\sig R_{2g+2}(z)^{1/2})\in K_g$ (i.e., choosing $\psi_\sig(z,n)$ to perform the commutation) it is shown in \cite{bght}, Chapter 9 that the new sequences are again given by (\ref{thetaa}), (\ref{thetab}) if $\underline{z}(P,n)$ is replaced by \bea \underline{\ti{z}}(P,n) = \underline{z}(P,n) + \underline{\hat{A}}_{P_0}(Q) + \underline{\hat{A}}_{P_0}(\infty_+). \eea As a consequence we note that for the standard procedure as in Theorem \ref{thmsc} (i.e., with $Q=(\lam_1,\sig_1 R_{2g+2}(\lam_1)^{1/2})$, $\sig_1 \in \{ \pm1\}$) the corresponding commuted operator $H_{\sig_1}$ is again quasi-periodic and isospectral to $H$. Hence, choosing $Q=\hat{\mu}_j$ we obtain \bea \label{utizbla} \underline{\ti{z}}(P,n) = \underline{z}(P,n) + \underline{\hat{A}}_{P_0}(\hat{\mu}_j) + \underline{\hat{A}}_{P_0}(\infty_+) \eea and the Dirichlet eigenvalue at $\hat{\mu}_j$ is formally replaced by one at $\infty_-$ (since $\underline{\hat{A}}_{P_0}(\infty_-) = - \underline{\hat{A}}_{P_0}(\infty_+)$). The corresponding sequences are neither real-valued nor well-defined. To repair this we perform a second single commutation making the following choice $Q=(\ti{\mu}_j,-\ti{\sig}_j R_{2g+2}(\ti{\mu}_j)^{1/2})$. The resulting sequences $a_{(\ti{\mu}_j,\ti{\sig}_j)}$, $b_{(\ti{\mu}_j,\ti{\sig}_j)}$ are associated with \bea \label{divisors} \underline{z}_{(\ti{\mu}_j,\ti{\sig}_j)}(P,n) = \underline{z}(P,n+1) + \underline{\hat{A}}_{P_0}(\hat{\mu}_j) -\underline{\hat{A}}_{P_0}((\ti{\mu}_j,\ti{\sig}_j R_{2g+2}(\ti{\mu}_j)^{1/2})) \eea and are again real-valued. Moreover, we have replaced the Dirichlet eigenvalue $(\mu_j,\sig_j)$ by $(\ti{\mu}_j,\ti{\sig}_j)$ and we have shifted the reference point for the Dirichlet boundary condition by one (since $\underline{z}(P,n+1)$ and not $\underline{z}(P,n)$ occurs in (\ref{divisors})) whereas everything else remains unchanged. From Section 3 we know that $a_{(\ti{\mu}_j,\ti{\sig}_j)}$, $b_{(\ti{\mu}_j,\ti{\sig}_j)}$ are equivalently given by \bea\nn a_{(\ti{\mu}_j,\ti{\sig}_j)}(n+1) \!\!&=&\!\! - \sqrt{a(n) a(n+2)} \times\\ &&\sqrt{\frac{ C_{n}(\psi_{\sig_j}(\mu_j), \psi_{-\ti{\sig}_j}(\ti{\mu}_j)) C_{n+2} (\psi_{\sig_j}(\mu_j),\psi_{-\ti{\sig}_j} (\ti{\mu}_j))}{ C_{n+1}(\psi_{\sig_j}(\mu_j), \psi_{-\ti{\sig}_j}(\ti{\mu}_j))^2}},\\ \nn b_{(\ti{\mu}_j,\ti{\sig}_j)}(n+1) \!\!&=&\!\! a(n) \frac{\psi_{\sig_j}(\mu_j,n+2) C_{n}(\psi_{\sig_j} (\mu_j),\psi_{-\ti{\sig}_j}(\ti{\mu}_j))}{ \psi_{\sig_j}(\mu_j,n+1) C_{n+1}( \psi_{\sig_j}(\mu_j), \psi_{-\ti{\sig}_j}(\ti{\mu}_j))} + \\ && a(n+1) \frac{\psi_{\sig_j}(\mu_j,n) C_{n+1}(\psi_{\sig_j} (\mu_j),\psi_{-\ti{\sig}_j}(\ti{\mu}_j))}{ \psi_{\sig_j}(\mu_j,n+1) C_{n}( \psi_{\sig_j}(\mu_j), \psi_{-\ti{\sig}_j}(\ti{\mu}_j))} -\ti{\mu}_j, \eea where the $n+1$ on the left-hand-side takes the aforementioned shift of reference point into account. Thus, applying this procedure $g$ times we can replace all Dirichlet eigenvalues proving (\ref{fita}), (\ref{fitb}). The reader might be puzzled by the fact that the Dirichlet eigenvalue $\hat{\mu}_j$ is shifted to $\infty_-$ (as opposed to $\infty_+$) which seemingly distinguishes $\infty_-$ from $\infty_+$. However, this apparent asymmetry between $\infty_+$ and $\infty_-$ is related to our way of factoring $H$. If we would instead split up $H$ as \begin{equation} H = \ti{A}_{\sig_j}^* \ti{A}_{\sig_j} + \mu_j, \end{equation} where \begin{equation} (\ti{A}_{\sig_j}) f(n) = - \sqrt{-\frac{a(n-1) \psi_{\sig_j}(\mu_j,n)}{\psi_{\sig_j}(\mu_j,n-1)}} f(n-1) + \sqrt{-\frac{a(n-1) \psi_{\sig_j}(\mu_j,n-1)}{\psi_{\sig_j}(\mu_j,n)}} f(n), \end{equation} with $\ti{A}_{\sig_j}^*$ being the adjoint of $\ti{A}_{\sig_j}$, the role of $\infty_+$ and $\infty_-$ would be interchanged. We stress again that (\ref{fita}), (\ref{fitb}) represent an explicit realization of the isospectral torus of all algebro-geometric quasi-periodic $g$-gap Jacobi operators with spectrum (\ref{ggapspec}). Next we turn to bounded solutions $(a(n,t),b(n,t))$ of the Toda equations and construct $N$-soliton solutions on these (arbitrary) background solutions using the single commutation method. The corresponding Jacobi operators $H(t)$ satisfy $\inf(\sig(H(t))) = \inf(\sig(H(0))) > -\infty$ for all $t\in \R$. Furthermore, this implies the existence of principal solutions $u_\pm(\lam,n,t)$ which satisfy \bea \label{systo} H(t) u_\pm(\lam,n,t) &=& \lam u_\pm(\lam,n,t),\\ \label{systt} \frac{d}{dt} u_\pm(\lam,n,t) &=& P(t) u_\pm(\lam,n,t), \quad (n,t) \in \Z \times \R, \eea where the difference expression $P(t)$ associated with $(a(t),b(t))$ is defined by \bea \label{poft} (P(t) f)(n) = a(n,t) f(n+1) - a(n-1,t) f(n-1). \eea (\ref{systo}) and (\ref{systt}) then imply the Toda lattice equations, \bea \label{tleq} \ba{rcl} \D \frac{d}{dt} a(n,t) &=& a(n,t) \Big( b(n,t) - b(n+1,t)\Big)\\ \D \rule{0pt}{4ex} \frac{d}{dt} b(n,t) &=& 2 \Big( a(n-1,t)^2 - a(n,t)^2\Big) \ea, \quad (n,t) \in \Z \times \R \eea which are well-known to be equivalent to the Lax equation \bea \frac{d}{dt} H(t) - [P(t),H(t)] =0, \quad t \in \R \eea (where $[.,.]$ denotes the commutator). Next, let $H(t)$ be as above and choose \bea \lam_N < \dots < \lam_1 < \inf(\sigma(H(0))), \quad \sigma_j \in [-1,1], \quad 1\le j \le N \in \N. \eea Then Theorem \ref{thmscom} implies \bea \nn \lefteqn{a_{\sig_1,\dots,\sig_N}(n,t) = - \sqrt{a(n,t) a(n+N,t)} \times}\\ && \frac{ \sqrt{ C_n(u_{\sig_1}^1,\dots, u_{\sig_N}^N) C_{n+2}(u_{\sig_1}^1,\dots,u_{\sig_N}^N) }}{C_{n+1}(u_{\sig_1}^1,\dots, u_{\sig_N}^N)}, \eea \bea \nn \lefteqn{b_{\sig_1,\dots,\sig_N}(n,t) = - \lam_N} \\ && \nn + a(n,t) \frac{C_{n+2}(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)} {C_{n+1}(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_{n+1}(u_{\sig_1}^1,\dots, u_{\sig_N}^N)} \\ && {} + a(n+N-1,t) \frac{C_n(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)} {C_{n+1}(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_n(u_{\sig_1}^1,\dots, u_{\sig_N}^N)}, \eea where \bea u_{\sig_\ell}^\ell(n,t) = \frac{1+\sigma_\ell}{2} u_+(\lam_\ell,n,t) + (-1)^{\ell+1} \frac{1-\sigma_\ell}{2} u_-(\lam_\ell,n,t). \eea Moreover, for $\lam < \lam_N$, \begin{equation} u_{\sig_1,\dots,\sig_N,\pm}(\lam,n,t) = \frac{ \pm \sqrt{ \prod\limits_{j=0}^{N-1} (-a(n+j,t))} C_n(u_{\sig_1}^1,\dots, u_{\sig_N}^\ell,u_\pm(\lam))}{\sqrt{ C_n(u_{\sig_1}^1,\dots, u_{\sig_N}^N) C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N) }} \end{equation} are the principal solutions of $\tau_{\sig_1,\dots,\sig_N}(t) u = \lam u$ satisfying \begin{equation} \frac{d}{dt} u_{\sig_1,\dots,\sig_N,\pm}(\lam,n,t) = P_{\sig_1,\dots,\sig_N}(t) u_{\sig_1,\dots,\sig_N,\pm}(\lam,n,t), \end{equation} where $P_{\sig_1,\dots,\sig_N}(t)$ is defined as in (\ref{poft}) with $a$ replaced by $a_{\sig_1,\dots,\sig_N}$. We also have (cf.\ (\ref{rhoo}), (\ref{rhoe})) \bea &\rho_{o,\sig_1,\dots,\sig_N}(n,t) = -\sqrt{-a(n,t) \frac{C_{n+2} (u_{\sig_1}^1,\dots,u_{\sig_{N-1}}^{N-1}) C_n(u_{\sig_1}^1,\dots,u_{\sig_N}^N)} {C_{n+1}(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_{n+1} (u_{\sig_1}^1,\dots,u_{\sig_N}^N)}},&\\ &\rho_{e,\sig_1,\dots,\sig_N}(n,t) = \sqrt{-a(n+N-1,t) \frac{C_n(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_{n+1}(u_{\sig_1}^1,\dots,u_{\sig_N}^N)} {C_{n+1}(u_{\sig_1}^1,\dots, u_{\sig_{N-1}}^{N-1}) C_n(u_{\sig_1}^1,\dots, u_{\sig_N}^N)}}.& \eea Finally, the sequences $a_{\sig_1,\dots,\sig_N}(n,t)$, $b_{\sig_1,\dots,\sig_N}(n,t)$ fulfill the Toda lattice equations (\ref{tleq}) and the sequence \bea \rho_{\sig_1,\dots,\sig_N}(n,t) = \left\{ \ba{c@{\quad}l} \rho_{e,\sig_1,\dots,\sig_N}(m,t), & n=2m \\ \rho_{o,\sig_1,\dots,\sig_N}(m,t), & n=2m+1\ea \right. , \eea fulfills the Kac--van Moerbeke lattice equation \bea \frac{d}{dt} \rho(n,t) = \rho(n,t) \Big( \rho(n+1,t)^2 - \rho(n-1,t)^2 \Big). \eea At the end we derive the $N$-soliton solutions relative to an arbitrary Toda background solution $(a(t),b(t))$ using the double commutation method. Denote by $\psi(\lam,n,t)$ the solutions of $\tau(t) \psi = \lam \psi$ which are square summable near $-\infty$ and satisfy \bea \frac{d}{dt} \psi(\lam,n,t) = P(t) \psi(\lam,n,t). \eea As in Theorem \ref{dcommN} we define the following matrices \bea C^N(n,t) = \left\{ \delta_r(s) + \sqrt{\gam_r \gam_s} \sum_{m=M_-}^n \psi(\lam_r,m,t) \psi(\lam_s,m,t) \right\}_{1 \le r,s \le N}, \eea \begin{equation} \Psi^N(\lam_j,n,t) = \left\{ \ba{c@{\quad}l} C^N(n,t)_{r,s} & \scriptstyle r,s \le N\\ \sqrt{\gam_s} \sum\limits_{m=M_-}^n \psi(\lam_j,m,t) \psi(\lam_s,m,t) & \scriptstyle s \le \ell, r=N+1\\ \sqrt{\gam_r}\psi(\lam_r,n,t) & \scriptstyle r \le \ell, s=N+1\\ \psi(\lam_j,n,t) & \scriptstyle r=s=N+1 \ea \right\}_{1 \le r,s \le N+1}. \end{equation} Then the sequences \bea a_{\gam_1,\dots,\gam_N}(n,t) &=& a(n,t) \frac{\sqrt{\det C^N(n-1,t) \det C^N(n+1,t)}}{\det C^N(n,t)}, \\ b_{\gam_1,\dots,\gam_N}(n,t) &=& b(n,t) - \frac{1}{2} \frac{d}{dt} \ln \frac{\det C^N(n,t)}{\det C^N(n-1,t)}. \eea satisfy the Toda lattice equations (\ref{tleq}). Moreover, \bea \psi_{\gam_1,\dots,\gam_N}(\lam_j,n,t) = \frac{\det \Psi^N(\lam_j,n,t)} {\sqrt{\det C^N(n-1,t)\det C^N(n,t)}} \eea satisfies \bea \frac{d}{dt} \psi_{\gam_1,\dots,\gam_N}(\lam_j,n,t) = P_{\gam_1,\dots,\gam_N}(t) \psi_{\gam_1,\dots,\gam_N}(\lam_j,n,t), \eea where again $P_{\gam_1,\dots,\gam_N}(t)$ is defined as in (\ref{poft}) with $a$ replaced by $a_{\gam_1,\dots,\gam_N}$. \begin{thebibliography}{XXXX} \addcontentsline{toc}{chapter}{References} \baselineskip12pt \bibitem{ak} N. I. Akhiezer, {\em The Classical Moment Problem}, Oliver and Boyd, London, 1965. \bibitem{ad} N. Aronszajn and W. Donoghue, {\em On the exponential representation of analytic functions in the upper half-plane with positive imaginary part}, J. Analyse Mathematique {\bf 5}, 321-388 (1956-57). \bibitem{at} F. V. Atkinson, {\em Discrete and Continuous Boundary Problems}, Academic Press, New York, 1964. \bibitem{ba} B. Baumgartner, {\em Level comparison theorems}, Ann. Phys. 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Weidmann, {\em Linear Operators in Hilbert Spaces}, Springer, New York, 1980. \end{thebibliography} \newpage \addcontentsline{toc}{chapter}{Vita} \vspace*{1cm} \begin{center} \large VITA \end{center} \vspace*{0.5cm} \begin{tabular}{ll} {\bf Name:} & Gerald Eric Teschl \\ \vspace*{1mm} {\bf Date of Birth:} & May 12, 1970 in Graz, Austria\\ \vspace*{1mm} {\bf Nationality:} & Austria \\ \vspace*{1mm} {\bf Education:} & \\ \vspace*{1mm} 1995 December & \begin{minipage}[t]{9cm} Ph.D., University of Missouri, Columbia \end{minipage}\\ \vspace*{1mm} 1994-1995& \begin{minipage}[t]{9cm} Ph.D student, Department of Mathematics, University of Missouri, Columbia. Doctoral dissertation: ``Spectral Theory for Jacobi Operators''. Thesis advisor: Prof.~F.~Gesztesy \end{minipage}\\ 1993-1994 & \begin{minipage}[t]{9cm} Visiting the Department of Mathematics, University of Missouri, Columbia (December 93 - March 94) \end{minipage}\\ \vspace*{1mm} 1993 July & \begin{minipage}[t]{9cm} Dipl.~Ing. (MS), with distinction, Technical University of Graz, Austria \end{minipage}\\ \vspace*{1mm} 1988-1993 & \begin{minipage}[t]{9cm} Undergraduate studies at the Technical University of Graz. Diploma Thesis: ``Schr\"odinger Operators with Interaction Concentrated on a Spheroid'', May 93. Advisor: Prof. W. Bulla \end{minipage} \end{tabular} \end{document}