Content-Type: multipart/mixed; boundary="-------------0312091241375" This is a multi-part message in MIME format. ---------------0312091241375 Content-Type: text/plain; name="03-530.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-530.keywords" Spectral Theory, Schr\"odinger Operators, Jacobi Matrices, Orthogonal Polynomials, Sum Rules, Szego Asymptotics ---------------0312091241375 Content-Type: application/x-tex; name="Nec.TEX" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Nec.TEX" \documentclass[reqno,12pt]{amsart} \usepackage{amsmath,amsthm,amscd,amssymb} %\usepackage[notref,notcite]{showkeys} %\usepackage{showkeys} \sloppy %%%%%%%%%%%%% fonts/sets %%%%%%%%%%%%%%%%%%%%%%% \newcommand{\bbC}{{\mathbb{C}}} \newcommand{\bbD}{{\mathbb{D}}} \newcommand{\bbR}{{\mathbb{R}}} \newcommand{\bbZ}{{\mathbb{Z}}} %%%%%%%%%%%%%%%%%% abbreviations %%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\dott}{\,\cdot\,} \newcommand{\no}{\nonumber} \newcommand{\lb}{\label} \newcommand{\f}{\frac} \newcommand{\ul}{\underline} \newcommand{\ol}{\overline} \newcommand{\ti}{\tilde } \newcommand{\wti}{\widetilde } \newcommand{\Oh}{O} \newcommand{\oh}{o} \newcommand{\marginlabel}[1]{\mbox{}\marginpar{\raggedleft\hspace{0pt}#1}} \newcommand{\tr}{\text{\rm{Tr}}} \newcommand{\dist}{\text{\rm{dist}}} \newcommand{\loc}{\text{\rm{loc}}} \newcommand{\spec}{\text{\rm{spec}}} \newcommand{\rank}{\text{\rm{rank}}} \newcommand{\ran}{\text{\rm{ran}}} \newcommand{\dom}{\text{\rm{dom}}} \newcommand{\ess}{\text{\rm{ess}}} \newcommand{\ac}{\text{\rm{ac}}} \newcommand{\s}{\text{\rm{s}}} \newcommand{\sing}{\text{\rm{sc}}} \newcommand{\pp}{\text{\rm{pp}}} \newcommand{\supp}{\text{\rm{supp}}} \newcommand{\AC}{\text{\rm{AC}}} \newcommand{\bi}{\bibitem} \newcommand{\hatt}{\widehat} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\ba}{\begin{align}} \newcommand{\ea}{\end{align}} \newcommand{\veps}{\varepsilon} %\newcommand{\Ima}{\operatorname{Im}} %\newcommand{\Real}{\operatorname{Re}} %\newcommand{\diam}{\operatorname{diam}} % use \hat in subscripts % and upperlimits of int. % % Rowan's unspaced list % \newcounter{smalllist} \newenvironment{SL}{\begin{list}{{\rm\roman{smalllist})}}{% \setlength{\topsep}{0mm}\setlength{\parsep}{0mm}\setlength{\itemsep}{0mm}% \setlength{\labelwidth}{2em}\setlength{\leftmargin}{2em}\usecounter{smalllist}% }}{\end{list}} %%%%%%%%%%%%%%%%%%%%%% operators %%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator{\Real}{Re} \DeclareMathOperator{\Ima}{Im} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator*{\slim}{s-lim} \DeclareMathOperator*{\wlim}{w-lim} \DeclareMathOperator*{\simlim}{\sim} \DeclareMathOperator*{\eqlim}{=} \DeclareMathOperator*{\arrow}{\rightarrow} \allowdisplaybreaks \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% end of definitions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} %\newtheorem{hypothesis}[theorem]{Hypothesis} %\theoremstyle{hypothesis} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} % Absolute value notation \newcommand{\abs}[1]{\lvert#1\rvert} %%%%%%%%%%%%% marginal warnings %%%%%%%%%%%%%%%% % ON: \newcommand{\TK}{{\marginpar{x-ref?}}} % OFF: %\newcommand{\TK}{} \begin{document} \title[Necessary and Sufficient Conditions] {Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schr\"odinger Operators} \author[D.~Damanik, R.~Killip, and B. Simon]{David Damanik$^{1}$, Rowan Killip$^{2}$, and Barry Simon$^{3}$} \thanks{$^1$ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA. E-mail: damanik@its.caltech.edu. Supported in part by NSF grant DMS-0227289} \thanks{$^2$ Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095. E-mail: killip@math.ucla.edu} \thanks{$^3$ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592} \date{September 5, 2003} \begin{abstract} We announce three results in the theory of Jacobi matrices and Schr\"odinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schr\"odinger operator $-\f{d^2}{dx^2} +V(x)$ on $L^2 (0,\infty)$ with $V\in L^2 (0,\infty)$ and $u(0)=0$ boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szeg\H{o} asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate. \end{abstract} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \lb{s1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this note, we want to describe some new results in the spectral and inverse spectral theory of half-line Schr\"odinger operators and Jacobi matrices. Given $V\in L_\loc^1 (0,\infty)$ with a mild regularity condition at infinity (ensuring limit-point case there, cf.\ \cite{LS}), one can define a unique selfadjoint operator which is formally \begin{equation} \lb{1.1} H=-\f{d^2}{dx^2} + V(x) \end{equation} with $u(0) =0$ boundary condition (see, e.g., \cite{LS}). For any $z\notin\bbR$, there is a solution $u_+(x;z)$ of $-u'' +Vu =zu$ which is $L^2$ at infinity and unique up to a constant. The Weyl $m$-function is then defined by \begin{equation} \lb{1.2} m(z) = \f{u_+'(0;z)}{u_+(0;z)} \end{equation} It obeys $\Ima m (z) >0$ when $\Ima z >0$, which implies that $\Ima m(E+i\veps)$ has a boundary value as $\veps \downarrow 0$ in distributional sense: \begin{equation} \lb{1.3} d\rho(E) = \wlim_{\veps\downarrow 0}\, \f{1}{\pi}\, \Ima m(E+i\veps)\, dE \end{equation} $d\rho$ is called the spectral measure. In this way, each $V$ gives rise to a spectral measure $d\rho$. In fact, the correspondence is one-to-one: Gel'fand-Levitan \cite{GLev} (see also Simon \cite{Sim271}) found an inverse procedure to go from $d\rho$ to $V$\!. Similarly, given a Jacobi matrix, $a_n >0$, $b_n\in\bbR$: \begin{equation} \lb{1.4} J= \begin{pmatrix} b_1 & a_1 & 0 & 0 & \cdots \\ a_1 & b_2 & a_2 & 0 & \cdots \\ 0 & a_2 & b_3 & a_3 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix} \end{equation} on $\ell^2 (\bbZ_+)$, we define $d\mu$ to be the measure associated to the vector $\delta_1$ by the spectral theorem. That is, \begin{equation} \lb{1.5} m(z) \equiv\langle \delta_1, (J-z)^{-1} \delta_1\rangle =\int \f{d\mu(E)}{E-z} \end{equation} In this setting, the inverse procedure dates back to Jacobi, Chebychev, Markov, and Stieltjes. It is easy to describe: By applying Gram-Schmidt to $\{1, E, E^2, \ldots \}$ in $L^2(d \mu)$, we obtain the orthonormal polynomials $p_n(E)$. These obey the three-term recursion relation \begin{equation} \lb{1.7} E p_n (E) =a_{n+1} p_{n+1}(E) + b_{n+1} p_n(E) + a_n p_{n-1}(E) \end{equation} Alternatively, one can obtain $a_n,b_n$ from a continued fraction expansion of $m$ (\cite{Stie,WallBk}). The main subject of spectral theory is to find relations between general properties of the spectral measures $d\rho$ or $d\mu$ and of the differential/difference equation parameters $V$ and $a_n,b_n$. Clearly, the gems of the subject are ones that provide necessary and sufficient conditions, that is, a one-to-one correspondence between some explicit family of measures and some explicit set of parameters. In this note, we announce three such results (one involving asymptotics of orthogonal polynomials rather than the measures) whose details will appear elsewhere \cite{KS2,Jost1,Jost2}. In the context of orthogonal polynomials on the unit circle \cite{SimOPUC}, Verblunsky's form \cite{V36} of Szeg\H{o}'s theorem \cite{Sz15,Sz20,Sz21} is such a one-to-one correspondence between a measure and the recurrence coefficients for its orthogonal polynomials. Baxter's theorem \cite{Bax,Bax2} and Ibragimov's theorem \cite{Ib,GoIb} can be viewed as other examples. Our work here is related to and motivated by the more recent result of Killip-Simon \cite{KS}: \begin{theorem}[\cite{KS}]\lb{T1.1} $J-J_0$ is Hilbert-Schmidt, that is \begin{equation} \lb{1.a} \sum_{n=1}^\infty (a_n -1)^2 + b_n^2 <\infty \end{equation} if and only if the spectral measure $d\mu$ obeys \begin{SL} \item[{\rm{(i)}}] {\rm{(Blumenthal-Weyl)}} $\supp (d\mu) = [-2,2] \cup \{E_j^+\}_{j=1}^{N_+} \cup \{E_j^-\}_{j=1}^{N_-}$ with $E_1^+ > E_2^+ > \cdots > 2$ and $E_1^- < E_2^- < \cdots < -2$ with $\lim_{j\to\infty} E_j^\pm =\pm2$ if $N_\pm =\infty$. \item[{\rm{(ii)}}] {\rm{(Normalization)}} $\mu$ is a probability measure. \item[{\rm{(iii)}}] {\rm{(Lieb-Thirring Bound)}} \begin{equation}\lb{1.b} \sum_{\pm, j} (\abs{E_j^\pm} -2)^{3/2} <\infty \end{equation} \item[{\rm{(iv)}}] {\rm{(Quasi-Szeg\H{o} Condition)}} Let $d\mu_{\ac}(E)=f(E)\, dE$. Then \begin{equation} \lb{1.c} \int_{-2}^2 \log [f(E)] \sqrt{4-E^2}\, dE >-\infty \end{equation} \end{SL} \end{theorem} Our first result is the analog of this theorem for Schr\"odinger operators. This is discussed in Section~\ref{s2}. Our second result concerns Szeg\H{o} asymptotics for orthogonal polynomials. In 1922, Szeg\H{o} \cite{Sz22} proved that if $d\mu =f(E)\,dE$ where $f$ is supported on $[-2,2]$ and \begin{equation} \lb{1.8} \int \log [f(E)] \f{dE}{\sqrt{4-E^2}} > -\infty \end{equation} then \begin{equation} \lb{1.9} \lim_{n\to\infty}\, z^n p_n (z + z^{-1}) \end{equation} exists and is nonzero (and finite) for all $z\in\bbD = \{ z \in \bbC : |z| < 1 \}$. There is work by Gon\v{c}ar \cite{Gon}, Nevai \cite{Nev79}, and Nikishin \cite{Nik} that allow point masses outside $[-2,2]$. The following summarizes more recent results on this subject by Peherstorfer-Yuditskii \cite{PY}, Killip-Simon \cite{KS}, and Simon-Zlato\v{s} \cite{SZ}: \begin{theorem}\lb{T1.2} Suppose $d\mu =f(E)\, dE + d\mu_\s$ with $\supp(d\mu_\sing) \cup\supp(f)\subset [-2,2]$ and \begin{equation} \lb{1.10} \sum_{j,\pm}\, (\abs{E_j^\pm}-2)^{1/2} <\infty \end{equation} Then the following are equivalent: \begin{SL} \item[{\rm{(i)}}] $\inf (a_1 \dots a_n) >0$ \item[{\rm{(ii)}}] All of the following: \begin{SL} \item[{\rm{(a)}}] \begin{equation} \lb{1.d} \sum_{n=1}^\infty\, \abs{a_n -1}^2 + \abs{b_n}^2 <\infty \end{equation} \item[{\rm{(b)}}] $\lim_{n\to\infty} a_n \dots a_1$ exists and is finite and nonzero. \item[{\rm{(c)}}] $\lim_{n\to\infty} \sum_{j=1}^n b_j$ exists. \end{SL} \item[{\rm{(iii)}}] \begin{equation} \lb{1.e} \int_{-2}^2 \log [f(E)] \f{dE}{\sqrt{4-E^2}} >-\infty \end{equation} \end{SL} Moreover, if these hold, then the limit \eqref{1.9} exists and is finite for all $z\in\bbD$ and is nonzero if $z + z^{-1} \notin\{E_j^\pm\}$. \end{theorem} Because \eqref{1.10} is required a priori here, this result is not a necessary and sufficient condition with only parameter information on one side and only spectral on the other. In Section~\ref{s3}, we will discuss a necessary and sufficient condition for the asymptotics \eqref{1.9} to hold, thereby closing a chapter that began in 1922. Finally, in Section~\ref{s4}, we discuss necessary and sufficient conditions on the measure for the $a$'s and $b$'s to obey \begin{equation} \lb{1.11} \limsup\, (\abs{a_n-1} + \abs{b_n})^{1/2n} \leq R^{-1} \end{equation} for some $R>1$. Namely, $d\mu$ must give specified weight to those eigenvalues $E_j$ with $|E_j| < R+R^{-1}$ and the Jost function must admit an analytic continuation to the disk $\{z : |z|0$ for a.e.~$E >0$. There is related work when $-\frac{d^2}{dx^2}+V\geq 0$ in Sylvester-Winebrenner \cite{SW} and Denisov \cite{DenJDE}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Szeg\H{o} Asymptotics} \lb{s3} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The proofs of the results in this section will appear in \cite{Jost1}. For the study of Szeg\H{o} asymptotics, it is useful to map $\bbD=\{z : \abs{z}<1\}$ to $\bbC\backslash [-2,2]$ by $z\to E=z+{z}^{-1}$. Our main result on this issue uses the following conditions: \begin{alignat}{2} &\sum_{n=1}^\infty \, \abs{a_n-1}^2 + \abs{b_n}^2 <\infty \lb{3.1} \\ &\lim_{N\to\infty}\, \sum_{n=1}^N \log (a_n) \qquad&& \text{exists (and is finite)} \lb{3.2} \\ &\lim_{N\to\infty} \, \sum_{n=1}^N b_n \qquad&& \text{exists (and is finite)} \lb{3.3} \end{alignat} \begin{theorem}\lb{T3.1} If for some $\veps >0$, $z^n p_n(z+z^{-1})$ converges uniformly on compact subsets of $\{z:0<|z|<\varepsilon\}$ to a non-zero value, then \eqref{3.1}--\eqref{3.3} hold. Conversely, if \eqref{3.1}--\eqref{3.3} hold, then $z^n p_n(z+z^{-1})$ converges uniformly on compact subsets of $\bbD$ and has a non-zero limit for those $z\neq0$ where $z+z^{-1}$ is not an eigenvalue of J. % and all $z\in\bbC$ with $0<\abs{z} %<\veps$, the limit \eqref{1.b} exists and is nonzero. Then %\eqref{3.1}--\eqref{3.3} hold. Conversely, if %\eqref{3.1}--\eqref{3.3} holds, then for all $z\in\bbD$, the limit %\eqref{1.b} exists and is nonzero if $z+z^{-1}$ is not an %eigenvalue of J. \end{theorem} {\it Remarks.} 1. By Theorem~\ref{T1.1}, \eqref{3.1} implies only the quasi-Szeg\H{o} condition \eqref{1.c} whereas all prior discussions of Szeg\H{o} asymptotics have assumed the stronger Szeg\H{o} condition \eqref{1.e}. We have examples in \cite{Jost1} where \eqref{3.1}--\eqref{3.3} hold and $\sum (\abs{E_n^\pm}-2)^{1/2}=\infty$ which, by \cite{SZ}, implies that \eqref{1.e} fails, so we have examples where Szeg\H{o} asymptotics hold, although the Szeg\H{o} condition fails. \smallskip 2. The first step in the proof is to show that for fixed $z\in\bbD$, Szeg\H{o} asymptotics hold if and only if there is a solution with Jost asymptotics, that is, for which \begin{equation} \lb{3.4} \lim z^{-n} u_n(z) \end{equation} exists and is non-zero. \smallskip 3. We have two constructions of the Jost solution when \eqref{3.1}--\eqref{3.3} hold: one using the nonlocal step-by-step sum rule of \cite{Sim288} and the other using perturbation determinants \cite{KS}. In either case, one makes a renormalization: In the first approach, one renormalizes Blaschke products and Poisson-Fatou representations, and, in the second case, one uses renormalized determinants for Hilbert-Schmidt operators. \smallskip While these are the first results we know for Szeg\H{o}/Jost asymptotics for Jacobi matrices with only $L^2$ conditions, Hartman \cite{Hart} and Hartman-Wintner \cite{HW} (see also Eastham \cite[Ch.~1]{East}) have found Jost asymptotics for Schr\"odinger operators with $V\in L^2$ with conditionally convergent integral. \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Jacobi Parameters With Exponential Decay} \lb{s4} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The proofs of the results in this section will appear in \cite{Jost2}. If $m$ is given by \eqref{1.5}, we define $M(z)$ by \begin{equation} \lb{4.1} M(z) =-m(z+z^{-1}) \end{equation} Suppose $M(z)$ is the $M$-function of a Jacobi matrix and that $M(z)$ has an analytic continuation to a neighborhood of $\bar\bbD$ with the only poles in $\bar\bbD$ lying in $\bar\bbD\cap\bbR$ and all such poles are simple. Then we can define \begin{equation} \lb{4.2} u(z) =\prod_{k=1}^N b(z,z_k) \exp\biggl( \int \biggl( \f{e^{i\theta}+z}{e^{i\theta}-z}\biggr) \log \biggl( \f{\sin\theta}{\Ima M(e^{i\theta})}\biggr)\, \f{d\theta}{4\pi}\biggr) \end{equation} where $\{z_k\}_{k=1}^\infty$ are the poles of $M$ in $\bbD$. This agrees with the Jost function from scattering theory (see \cite{KS}), so we call it by this name. Given $M$ and the Jost function, $u$, suppose $u$ is analytic in $\{z : \abs{z}1$ and all $\veps >0$ if and only if \begin{SL} \item[{\rm{(i)}}] $M$ is meromorphic on $\{z : \abs{z}