Content-Type: multipart/mixed; boundary="-------------0412260057408" This is a multi-part message in MIME format. ---------------0412260057408 Content-Type: text/plain; name="04-425.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-425.keywords" Schr\"odinger operators, bound states, essential spectrum, absolutely continuous spectrum, Lieb-Thirring Inequalities ---------------0412260057408 Content-Type: application/x-tex; name="Damanik_Remling.TEX" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Damanik_Remling.TEX" \documentclass{amsart} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{amssymb} \newtheorem{Theorem}{Theorem}[section] \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Lemma}[Theorem]{Lemma} \newtheorem{Corollary}[Theorem]{Corollary} \theoremstyle{definition} \newtheorem{Definition}{Definition}[section] \numberwithin{equation}{section} \newcommand{\Z}{{\mathbb Z}} \newcommand{\R}{{\mathbb R}} \newcommand{\C}{{\mathbb C}} \newcommand{\N}{{\mathbb N}} \newcommand{\T}{{\mathbb T}} \newcommand{\bbR}{{\mathbb{R}}} \newcommand{\bbD}{{\mathbb{D}}} \newcommand{\bbP}{{\mathbb{P}}} \newcommand{\bbZ}{{\mathbb{Z}}} \newcommand{\bbC}{{\mathbb{C}}} \newcommand{\bbQ}{{\mathbb{Q}}} \newcommand{\bbS}{{\mathbb{S}}} \begin{document} \title{Schr\"odinger Operators with Many Bound States} \author{David Damanik and Christian Remling} \address{Mathematics 253--37\\ California Institute of Technology\\ Pasadena, CA 91125} \email{damanik@caltech.edu} \urladdr{math.caltech.edu/people/damanik.html} \address{Universit\"at Osnabr\"uck\\ Fachbereich Mathematik\\ 49069 Osnabr\"uck\\ Germany} \email{cremling@mathematik.uni-osnabrueck.de} \urladdr{www.mathematik.uni-osnabrueck.de/staff/phpages/remlingc.rdf.html} \date{\today} \thanks{2000 {\it Mathematics Subject Classification.} Primary 34L15 81Q10} \keywords{Schr\"odinger operator, bound states} \begin{abstract} Consider the Schr\"odinger operators $H_{\pm}=-d^2/dx^2\pm V(x)$. We present a method for estimating the potential in terms of the negative eigenvalues of these operators. Among the applications are inverse Lieb-Thirring inequalities and several sharp results concerning the spectral properties of $H_{\pm}$. \end{abstract} \maketitle \section{Introduction} We are interested in Schr\"odinger equations \begin{equation} \label{se} -y''(x) + V(x)y(x)=Ey(x) \end{equation} and the associated self-adjoint operators $H_+=-d^2/dx^2+V(x)$ on $L_2(0,\infty)$. The potential $V$ is assumed to be locally integrable on $[0,\infty)$. One also needs a boundary condition of the following form: \begin{equation} \label{bc} y(0)\cos\alpha - y'(0)\sin\alpha = 0 . \end{equation} The basic question we would like to address is the following: \textit{What is the influence of the structure of the discrete spectrum on the potential $V\!$ and on the spectral properties of $H_+$?} It is then necessary to consider $H_+$ and $H_-=-d^2/dx^2-V(x)$ simultaneously because otherwise sign definite potentials provide counterexamples to any possible positive result one might imagine. So our basic assumption is the following: \begin{equation} H_{\pm} \text{ are bounded below and } \sigma_{{\rm ess}}(H_{\pm}) \subset [0,\infty). \tag{$\Sigma_{{\rm ess}}$} \end{equation} Assuming ($\Sigma_{{\rm ess}}$), we can list the negative eigenvalues of $H_+$ and $H_-$ together as $-E_1\le -E_2 \le \ldots$, with $E_n>0$. The list is either finite (or even empty) or $E_n\to 0$. The $E_n$'s of course depend on the boundary condition \eqref{bc}. However, we will usually be interested in situations where $\sum E_n^p<\infty$, with $p\ge 0$, and, by the interlacing property, this condition is independent of $\alpha$. We deal with the one-dimensional case only in this paper, but our methods are not limited to this situation. We plan to explore higher-dimensional operators in a future project. Our original motivation for this work came from the following result, which completely clarifies the situation when $\{E_n\}$ is a finite set. \begin{Theorem}[Damanik-Killip \cite{dk} (see also \cite{dks})] \label{TDK} Assume {\rm (}$\Sigma_{{\rm ess}}${\rm )}. Moreover, assume that $\{ E_n\}$ is finite. Then $\sigma_{{\rm ess}}=[0,\infty)$, and the spectrum is purely absolutely continuous on $[0,\infty)$ for any boundary condition at $x=0$. \end{Theorem} Here the statements refer to $H_+$, say. Of course, since $-V$ satisfies the same hypotheses as $V\!$, we automatically obtain the same assertions for $H_-$ as well, so this distinction is actually irrelevant. We keep this convention, however, because it will help to slightly simplify the formulation of our results. In \cite{dk}, it is also assumed that $V\in\ell_{\infty}(L_2)$, that is, $\sup_{x\ge 0} \int_x^{x+1} V^2(t)\, dt <\infty$. Our treatment below, especially the material from Sect.~3, will show that this technical assumption is unnecessary. The aim of this paper is to develop tools for handling arbitrary discrete spectra $\{E_n \}$, not necessarily finite. At the heart of the matter is a new method for estimating the potential in terms of the $E_n$'s in general situations. We defer the exact description of this to Sect.~2 and limit ourselves to a few general remarks in this introduction (refer to Theorem~\ref{TWQ} below for the full picture). The most important aspect of our method is this: The rate of convergence with which $E_n$ tends to zero determines the \textit{geometry} of the situation. More precisely, we obtain intervals $I_n$ whose lengths obey the scaling relation $|I_n| \sim E_n^{-1/2}$. We will also write $V$ as $W'+W^2$ plus a remainder, with $\|W\|_{L_2(I_n)} \lesssim |I_n|^{-1/2}$. This representation of $V$ is natural in this context because $-d^2/dx^2+q(x)$ with Dirichlet boundary conditions ($y=0$) has no negative spectrum if and only if $q=w'+w^2$ for some $w$. More importantly, it is also very useful in the applications we want to make in this paper. Note also that any relation between the potential and the eigenvalues must respect the invariance of the problem under the rescaling $V(x)\to g^2V(gx)$, $E\to g^2 E$. To give a more specific impression of what we can do with our techniques, we mention the following: \begin{Corollary} \label{C1.2} Assume {\rm (}$\Sigma_{{\rm ess}}${\rm )}. Moreover, assume that $\sum E_n^{1/2}<\infty$. Then there exists $V_0\in L_1(0,\infty)$ so that $H_+ + V_0$ with Dirichlet boundary conditions has no negative spectrum. \end{Corollary} This is indeed immediate from Theorem~\ref{TWQ}, which says that $V-(W'+W^2)\in L_1$ for a suitable $W$ in this situation. So small eigenvalues can be removed by a small perturbation. The exponent $1/2$ in the hypothesis is not essential. For instance, it is also true that if, more generally, $\sum E_n^p<\infty$ with $p\ge 1/2$, then there exists $V_0\in\ell_{2p}(L_1)$, that is, \[ \sum_{n=0}^{\infty} \left( \int_n^{n+1} |V_0(x)|\, dx \right)^{2p} < \infty, \] so that $H_+ +V_0\ge 0$. In some instances, our basic problem of obtaining information on the potential from a knowledge of its eigenvalues can be attacked with completely different tools, called \textit{sum rules} (aka trace formulae). While this approach is elegant and leads to very satisfactory results where it works, it is indirect and less systematic and one is restricted to those combinations of the $E_n$'s that happen to show up in the sum rules one can produce. See, for instance, \cite{ks,Kup,LNS,NPVY,sz} for recent work on sum rules. The converse problem, that is, the problem of estimating the eigenvalues in terms of the potential, is classical and has received considerable attention over the years. We mention, in particular, the topic of \textit{Lieb-Thirring inequalities} (see, e.g., \cite{LW} for further information on this). For sign-definite potentials, we obtain \textit{inverse} Lieb-Thirring inequalities as a by-product of our general method; see Theorem~\ref{TILT} below. We now turn to discussing the consequences of our method concerning the spectral properties of $H_{\pm}$. Taking related results into account (see \cite[Theorem~1]{dhks} and Theorem~\ref{TDK} above), the following does not come as a surprise. \begin{Theorem} \label{Tess} Assume {\rm (}$\Sigma_{{\rm ess}}${\rm )}. Then $\sigma_{{\rm ess}}=[0,\infty)$. \end{Theorem} It is now natural to inquire about the structure of the spectrum on $(0,\infty)$. \begin{Theorem} \label{TDe-K} Assume {\rm (}$\Sigma_{{\rm ess}}${\rm )}. Moreover, assume that $\sum E_n^{1/2}<\infty$. Then there exists absolutely continuous spectrum essentially supported by $(0,\infty)$. \end{Theorem} Killip and Simon have proved earlier the discrete analog of this. The essential ingredient in their analysis is a sum rule, and they in fact establish the so-called Szeg\H{o} condition. See \cite{ks} for these statements, especially Theorems 3 and 7; compare also \cite{sz}. Under a slightly stronger assumption, we will also prove that on large sets of energies $E$, the solutions asymptotically look like plane waves. By this we mean that \begin{equation} \label{y-asymp} y(x)= e^{i\sqrt{E} x} + o(1) \quad\quad (x\to\infty). \end{equation} We let $S$ be the exceptional set where we do \textit{not} have solutions of this asymptotic form. So we define \[ S= \{ E>0: \text{ no solution of \eqref{se} satisfies \eqref{y-asymp}} \} . \] Note that if $E\in (0,\infty)\setminus S$, then the complex conjugate of the solution $y$ from \eqref{y-asymp} is a linearly independent solution of the same equation, so we have complete control over the solution space for such $E$'s. In particular, there is no subordinate solution then, so that the singular part of the spectral measure on $(0,\infty)$ must be supported by $S$ for any boundary condition \eqref{bc}. \begin{Theorem} \label{TC-K} Assume {\rm (}$\Sigma_{{\rm ess}}${\rm )}. Moreover, assume that $\sum E_n^p<\infty$ for some $p<1/2$. Then $|S|=0$. \end{Theorem} In particular, this again implies that $H_+$ has absolutely continuous spectrum essentially supported by $(0,\infty)$. We obtain a more detailed statement here, giving asymptotic formulae for the solutions, but are unable to treat the borderline case $p=1/2$. The situation is completely analogous to the known results on operators with $L_q$ potentials $V\!$. This is no coincidence, because the techniques are the same: Theorem~\ref{TC-K} crucially depends on work of Christ and Kiselev \cite{CK,CK2}, and the proof of Theorem~\ref{TDe-K} follows ideas of Deift and Killip \cite{DeK}. Perhaps somewhat surprisingly, the statement of Theorem~\ref{TC-K} can be sharpened if $p$ can be taken smaller than $1/4$. \begin{Theorem} \label{Tdim} Assume {\rm (}$\Sigma_{{\rm ess}}${\rm )}. Moreover, assume that $\sum E_n^p < \infty$, with $0\le p< 1/4$. Then $\dim S\le 4p$. \end{Theorem} Note that the corresponding statement on $L_q$ potentials is false: There are potentials $V\in \bigcap_{q>1} L_q$ with $\dim S=1$ (see \cite[Theorem 4.2b)]{Remtams}). As explained above, Theorem~\ref{Tdim} implies that the singular part of the spectral measure is supported on a set of dimension $\le 4p$. A related consequence is the fact that the spectrum is \textit{purely} absolutely continuous on $[0,\infty)$ for all boundary conditions not from an exceptional set $B\subset [0,\pi)$, where again $\dim B\le 4p$ (see \cite[Theorem 5.1]{Remdim} for this conclusion). If, on the other hand, $\sum E_n^p <\infty$ with $1/4\le p < 1/2$, no strengthening of the statement of Theorem~\ref{TC-K} is obtained, in spite of the stronger hypothesis. The following theorem shows that no such improvement is possible: \begin{Theorem} \label{Tdimopt} Let $e_n>0$ be a non-increasing sequence with $\sum e_n^{1/4}=\infty$. Then there exists a potential $V\!$ so that {\rm (}$\Sigma_{{\rm ess}}${\rm )} holds, $E_n\le e_n$, and $\dim S=1$. \end{Theorem} Thus the bound from Theorem~\ref{Tdim} is correct at the extreme values $p=0$ and $p=1/4$. We make the obvious conjecture that it is optimal throughout its range of validity. The examples used in the proof of Theorem~\ref{Tdimopt} also show that given $e_n$'s with $\sum e_n^{1/2} = \infty$, there exists a potential so that $E_n\le e_n$ and $\sigma_{{\rm ac}}=\emptyset$. Hence Theorem~\ref{TDe-K} is optimal, too, and Theorem~\ref{TC-K} fails to address only the borderline value $p=1/2$. In this context, the work of Muscalu, Tao, and Thiele \cite{MTT,MTT2} is also relevant. One of the main difficulties, when estimating $V$ in terms of the $E_n$'s, comes from the fact that $V$ can take both signs. It is therefore interesting to compare the above results with the situation for sign-definite potentials, say $V\le 0$. Then only $H_+$ can have negative eigenvalues. In this situation, the control on $V\!$ exerted by the eigenvalues gets more explicit. \begin{Theorem} \label{TILT} Assume {\rm (}$\Sigma_{{\rm ess}}${\rm )}. Moreover, assume that $V\le 0$, and consider Neumann boundary conditions {\rm (}$\alpha=\pi/2$ in \eqref{bc}{\rm )}. {\rm a)} For $00$, there exists a constant $C_p(E_0)$ so that \[ \sum_{n=0}^{\infty} \left( \int_n^{n+1} |V(x)|\, dx \right)^{2p} \le C_p(E_0) \sum E_n^{p} , \] provided that $E_1\le E_0$. \end{Theorem} This is reassuring, but we emphasize again that these inequalities do not really catch the essence of our method. As outlined above, the behavior of $E_n$ governs the geometry of the situation, and this part of the information gets lost when we pass to global bounds as in Theorem~\ref{TILT}. This effect is also responsible for the additional assumption that $\sup E_n \le E_0$ from part b): The intervals $(n,n+1)$ are not adapted to the underlying geometry. The need for such a restriction is also apparent from the fact that part b) is not invariant under the rescaling $V(x)\to g^2V(gx)$, $E\to g^2E$. The estimates from part a) might be called \textit{inverse Lieb-Thirring inequalities.} Lieb-Thirring inequalities are bounds of the type of part a) but with the opposite sign. They hold for $p\ge 1/2$. In particular, for $p=1/2$, we have inequalities in both directions, so $\int |V|$ and $\sum E_n^{1/2}$ are comparable. This is not a new result; on the contrary, $p=1/2$ is essentially a sum rule, and the best constant is known (\cite{GGM}, see also \cite{Schm}). It is clear that we cannot have inverse Lieb-Thirring inequalities for $p>1/2$ because $V$ can have local singularities so that $V \notin L_q$ for any $q>1$. It is also important to work with \textit{Neumann} boundary conditions as there are non-zero potentials $V \le 0$ with no Dirichlet eigenvalues. The whole-line analog of Theorem~\ref{TILT} also holds and is perhaps more natural for precisely this reason. As for the spectral properties, Theorem~\ref{TILT} has the following consequences: \begin{Corollary} \label{C1.1} Assume {\rm (}$\Sigma_{{\rm ess}}${\rm )}. Moreover, assume that $V\le 0$. {\rm a)} If $\sum E_n^{1/2}<\infty$, then the spectrum is purely absolutely continuous on $(0,\infty)$ for all boundary conditions. {\rm b)} If $\sum E_n<\infty$, then there is absolutely continuous spectrum essentially supported by $(0,\infty)$. {\rm c)} If $\sum E_n^p<\infty$ for some $p<1$, then the solutions satisfy WKB-type asymptotic formulae for Lebesgue almost all energies $E>0$. \end{Corollary} Part a) follows because Theorem~\ref{TILT}a) says that $V\in L_1$. As pointed out above, this part of the corollary has been known before. Rybkin \cite[Theorem~1]{Ryb} has proved that the assertion of part b) holds if $V \in \ell_2(L_1)$, that is, if \[ \sum \left( \int_n^{n+1} |V(x)|\, dx \right)^2 < \infty . \] This is the ultimate form of a well-known theorem of Deift and Killip \cite{DeK} which states that there is absolutely continuous spectrum essentially supported by $(0,\infty)$ if $V\in L_1+L_2$. So part b) of the corollary follows from Theorem~\ref{TILT}b). The asymptotic formula alluded to in part c) reads \[ y(x,E) = \exp \left( i\sqrt{E} x - \frac{i}{2\sqrt{E}}\int_0^x V(t)\, dt \right) + o(1)\quad\quad (x\to\infty) . \] Christ and Kiselev \cite{CK} prove that this holds at almost all energies if $V\in\ell_p(L_1)$ for some $p<2$, so Theorem~\ref{TILT}b) also implies part c) of the corollary. As above, these results are complemented by the following: \begin{Theorem} \label{TL1opt} Let $e_n>0$ be a non-increasing sequence with $\sum e_n=\infty$. Then there exists a potential $V\le 0$ so that {\rm (}$\Sigma_{{\rm ess}}${\rm )} holds, $E_n\le e_n$, and $\sigma_{{\rm ac}}=\emptyset$. \end{Theorem} We organize this paper in the obvious way: Section~2 gives a detailed discussion of our general method. The subsequent sections are concerned with the applications of this to the spectral theory of $H_{\pm}$, in the order suggested by this introduction. In the final section, we present the examples announced in Theorems~\ref{Tdimopt} and \ref{TL1opt}. \medskip \noindent\textit{Acknowledgments.} It is a pleasure to thank Rowan Killip and Barry Simon for useful conversations. C.\ R.\ would like to express his gratitude for the hospitality of Caltech, where this work was begun. \section{A Method for Estimating $V$} As in the previous section, let $H_{\pm}=-d^2/dx^2 \pm V(x)$. We write $H_{\sigma}$ if we work with one of these operators, but do not want to specify which one. Boundary conditions (where necessary) will \textit{always} be Dirichlet boundary conditions ($y=0$). The following theorem may be viewed as the principal result of this paper. Things become slightly easier in the whole-line setting because we avoid the somewhat artificial technical problems associated with the effect that a boundary condition can screen part of the potential. So we discuss this case first. The modifications needed to handle half-line problems will be described after having completed the treatment of the whole-line case. \begin{Theorem} \label{TWQ} Consider $H_{\pm}$ on $L_2(\mathbb R)$. Assume {\rm (}$\Sigma_{{\rm ess}}${\rm )}. Then there exist a partition of $\mathbb R$ into intervals $J_n^{(k)}$ with disjoint interiors and a decomposition $V=W'+Q$ with the following properties: {\rm a)} {\rm (basic properties of $W,Q$)} $W$ is absolutely continuous. If $\sum E_n^{1/2}<\infty$, then $W\in L_2(\mathbb R)$ and $Q\in L_1(\mathbb R)$. {\rm b)} {\rm(geometry of the intervals)} The indices $k,n$ vary over the following sets: $n\in\mathbb N$ and $k\in\mathbb Z$, $-N_n-1