Content-Type: multipart/mixed; boundary="-------------0005151549660" This is a multi-part message in MIME format. ---------------0005151549660 Content-Type: text/plain; name="00-226.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-226.comments" MSC: (2000 Revision) Primary 82B44; Secondary 34F05, 60H25 PACS Numbers: 03.65.-w, 72.15.Rn, 71.55.Jv, 71.20.-b, 73.23.-b, 72.10.Fk ---------------0005151549660 Content-Type: text/plain; name="00-226.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-226.keywords" Random Schr\"odinger operators, Lyapunov exponent, density of states ---------------0005151549660 Content-Type: application/x-tex; name="lyapunov-bound.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lyapunov-bound.tex" \documentclass[a4paper,11pt,centertags,psamsfonts]{amsart} \usepackage{times,amssymb} \renewcommand{\thefootnote}{\arabic{footnote}} %\renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \renewcommand{\thesection}{\arabic{section}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\E}{\mathbb{E}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\EE}{\mathsf{E}} \renewcommand{\P}{\mathbb{P}} \newcommand{\supp}{{\ensuremath{\rm supp}}} \newcommand{\tr}{\mathrm{tr}} \newcommand{\const}{\mathrm{const}} \newcommand{\sign}{\mbox{\rm sign}} \newcommand{\SL}{\mbox{\rm SL}} \newcommand{\GL}{\mbox{\rm GL}} \renewcommand{\Im}{\ensuremath{\rm Im}} \renewcommand{\Re}{\ensuremath{\rm Re}} \addtolength{\textwidth}{20mm} \addtolength{\textheight}{10mm} \addtolength{\marginparwidth}{9mm} \addtolength{\topmargin}{-10mm} \newtheorem{theorem}{Theorem}{\bf}{\it} \newtheorem{proposition}[theorem]{Proposition}{\bf}{\it} \newtheorem{corollary}[theorem]{Corollary}{\bf}{\it} \newtheorem{example}[theorem]{Example}{\it}{\rm} \newtheorem{lemma}[theorem]{Lemma}{\bf}{\it} \newtheorem{remark}{Remark}{\it}{\rm} \newtheorem{definition}[theorem]{Definition}{\bf}{\it} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title[Lyapunov Exponent and Integrated Density of States]{Global Bounds for the Lyapunov Exponent and the Integrated Density of States of Random Schr\"{o}dinger Operators in One Dimension} \author[V. Kostrykin and R. Schrader]{V. Kostrykin \and R. Schrader$^\ast$} \address{Vadim Kostrykin\\ Fraunhofer-Institut f\"{u}r Lasertechnik\\ Steinbachstra{\ss}e 15, D-52074\\ Aachen, Germany} \email{kostrykin@t-online.de, kostrykin@ilt.fhg.de} \address{Robert Schrader\\ Institut f\"{u}r Theoretische Physik\\ Freie Universit\"{a}t Berlin, Arnimallee 14\\ D-14195 Berlin, Germany} \email{schrader@physik.fu-berlin.de} \thanks{\textit{PACS Numbers}. 03.65.-w, 72.15.Rn, 71.55.Jv, 71.20.-b, 73.23.-b, 72.10.Fk} \thanks{$^\ast$ R.S. supported in part by DFG SFB 288 ``Differentialgeometrie und Quantenphysik''} \date{May 13, 2000} \keywords{Random Schr\"{o}dinger operators, Lyapunov exponent, density of states} \subjclass{(2000 Revision) Primary 82B44; Secondary 34F05, 60H25} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \begin{abstract} In this article we prove an upper bound for the Lyapunov exponent $\gamma(E)$ and a two-sided bound for the integrated density of states $N(E)$ at an arbitrary energy $E>0$ of random Schr\"{o}dinger operators in one dimension. These Schr\"{o}dinger operators are given by potentials of identical shape centered at every lattice site but with non-overlapping supports and with randomly varying coupling constants. Both types of bounds only involve scattering data for the single-site potential. They show in particular that both $\gamma(E)$ and $N(E)-\sqrt{E}/\pi$ decay at infinity at least like $1/\sqrt{E}$. As an example we consider the random Kronig-Penney model. \end{abstract} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this article we will consider random Schr\"{o}dinger operators $H(\omega)$ in $L^2(\R)$ of the form \begin{equation}\label{1.1} H(\omega)=H_0+V_\omega,\quad H_0=-\frac{d^2}{dx^2},\quad V_\omega=\sum_{j\in{\Z}} \alpha_j(\omega)f(\cdot-j), \end{equation} where $\{\alpha_j(\omega)\}_{j\in{\Z}}$ is a sequence of i.i.d.\ (independent, identically distributed) variables on a complete probability space $(\Omega,\cF,\P)$ having a common distribution measure $\kappa$ (i.e.\ $\P\{\alpha_j\in\Delta\}=\kappa(\Delta)$ for any Borel set $\Delta\subset\R$). In what follows we always suppose that $\kappa$ is supported on a compact interval and the single-site potential $f$ is integrable with support in the interval [-1/2,1/2]. Moreover, the random variables are assumed to form a stationary, metrically transitive random field, i.e.\ there are measure preserving ergodic transformations $\{T_j\}_{j\in\Z}$ such that $\alpha_j(T_k\omega)=\alpha_{j-k}(\omega)$ for all $\omega\in\Omega$. The spectral properties of the operator (\ref{1.1}) were studied in detail in \cite{KiMa,Delyon,KiKoSi,Kotani:Simon:87,Sims:Stolz:2000}. The results are most complete for the case when $f$ is the point interaction (see \cite{Albeverio:book}). The integrated density of states $N(E)$ and the Lyapunov exponent $\gamma(E)$ are important quantities associated with operators of the form \eqref{1.1} (see e.g.\ \cite{Carmona:Lacroix}). In particular, according to Ishii-Pastur-Kotani theorem \cite{Kotani} the set $\{E:\gamma(E)=0\}$ is the essential support of the absolute continuous part of the spectral measure for $H(\omega)$. The main idea of our approach is to approximate the operator \eqref{1.1} by means of the sequence \begin{equation*} H^{(n)}(\omega)=H_0+\sum_{j=-n}^{n} \alpha_j(\omega)f(\cdot-j) \end{equation*} with unchanged $H_0$, which converges to $H(\omega)$ in the strong resolvent sense. This differs from the usual approach where one puts the whole system in a box, which then tends to infinity (see e.g.\ \cite{Carmona:Lacroix}). In \cite{KS1} (see also \cite{Kostrykin:Schrader:99c}) we used this approximation to invoke scattering theory for the study the spectral properties of the limiting operator \eqref{1.1}. Some other applications of scattering theory to the study of spectral properties of such type Schr\"{o}dinger operators in one dimension can be found in \cite{KiKoSi} and \cite{Sims:Stolz:2000}. One of the important ingredients of our approach developed in \cite{KS1} is the Lifshitz-Krein spectral shift function. The spectral shift function naturally replaces the eigenvalue counting function usually used to construct the density of states for the operator \eqref{1.1}. The celebrated Birman-Krein theorem (see e.g.\ \cite{Birman:Yafaev}) relates the spectral shift function to scattering theory. In fact, up to a factor $-\pi^{-1}$ it may be identified with the scattering phase for the pair ($H^{(n)}(\omega)$, $H_0$), i.e. $\xi^{(n)}(E;\omega)=-\pi^{-1}\delta^{(n)}(E;\omega)$ when $E>0$, \begin{displaymath} \delta^{(n)}(E;\omega)=\frac{1}{2i}\log\det S^{(n)}(E;\omega)= \frac{1}{2i}\log\det \left(\begin{array}{lr} T_\omega^{(n)}(E) & R_\omega^{(n)}(E) \\ L_\omega^{(n)}(E) & T_\omega^{(n)}(E) \end{array}\right). \end{displaymath} Here $|T^{(n)}(E)|^2$ and $|R^{(n)}(E)|^2=|L^{(n)}(E)|^2$ have the meaning of transmission and reflection coefficients, respectively, such that $|T^{(n)}(E)|^2+|R^{(n)}(E)|^2=1$. For $E<0$ the quantity $\xi^{(n)}(E;\omega)$ equals minus the counting function for $H^{(n)}(\omega)$. In particular in \cite{KS1} we proved the almost sure existence of the limit \begin{equation}\label{xi} \xi(E)=\lim_{n\rightarrow\infty}\frac{\xi^{(n)}(E;\omega)}{2n+1}, \end{equation} which we called the spectral shift density. Also we proved the equality $\xi(E)=N_0(E)-N(E)$, where $N(E)$ and $N_0(E)=\pi^{-1}[\max(0,E)]^{1/2}$ are the integrated density of states of the Hamiltonians $H(\omega)$ and $H_0$ respectively. This result also extends to higher dimension in the continuous \cite{Kostrykin:Schrader:99d} and discrete \cite{Chahrour} cases. Also we showed that almost surely the Lyapunov exponent $\gamma(E)$ at energy $E>0$ is given as \begin{equation}\label{1.3} \gamma(E)=-\lim_{n\rightarrow\infty}\frac{\log |T^{(n)}(E;\omega)|}{2n+1}, \end{equation} where $T^{(n)}(E,\omega)$ is the transmission amplitude for the pair of Hamiltonians ($H^{(n)}(\omega)$, $H_{0}$) at energy $E$. We recall that $\gamma(E)$ is defined as the upper Lyapunov exponent for the fundamental matrix at energy $E$ of the Schr\"{o}dinger operator $H(\omega)$. The connection between the Lyapunov exponent and the transmission coefficient $|T_\omega^{(n)}(E)|$ was recognized long ago \cite{Lifshitz:Gredeskul:Pastur:82,Lifshitz:Gredeskul:Pastur:88}. A complete proof has appeared in \cite{KS1}. We note that the theory of the spectral shift function was also recently used to show that the integrated density of states is independent of the choice of boundary conditions \cite{Nakamura:2000} on the sides of a large box, in which the system is put. The conditions on the random variables $\alpha_j$ and the single-site potential $f$ stated above are slightly weaker than those in \cite{KS1}. However the results of \cite{KS1} which will be used below remain valid also in this more general case. The aim of the present paper is to prove \emph{global} bounds for the Lyapunov exponent and the integrated density of states, i.e.\ bounds which hold for all $E>0$ and describe the correct asymptotic behavior in the limit $E\rightarrow\infty$. These results are formulated as Theorems \ref{theorem1} and \ref{thm:xi} below. To the best of our knowledge the first article to look for the asymptotic behavior of $\gamma(E)$ and $N(E)$ in the limit $E\rightarrow\infty$ is \cite{APW}. The best known estimate for the integrated density of states is due to Kirsch and Martinelli \cite[Corollary 3.1]{KiMa2}. This bound however does not reproduce the correct asymptotic behavior of $N(E)$ in the large energy limit. Another estimate, which is due to Pastur and Figotin (see \cite[Sec.\ V.11.B]{Pastur:Figotin}), is valid for an $\R$-metrically transitive random field. Since our potential $V_\omega(x)$ is a $\Z$-metrically transitive field this estimate does not apply directly to the present situation. Our two-sided estimate leads to the bound \eqref{IDS} below which is very close to that of Pastur and Figotin. In what follows $C$ will denote a finite positive generic constant varying with the context, but which depends only on $f$ and $\kappa$. We are indebted to Leonid Pastur for reading the preliminary version of this article. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Lyapunov exponent} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We recall that the scattering matrix $S(E)$ for a pair of Hamiltonians ($H$, $H_0$) on $L^2(\R)$ at fixed energy $E\geq 0$ is a $2\times 2$ unitary matrix \begin{equation}\label{Smatrix.def} S(E)=\left(\begin{array}{cc} T(E) & R(E) \\ L(E) & T(E) \end{array} \right), \end{equation} where $L(E)$ and $R(E)$ denote the left and right reflection amplitudes respectively. The transmission amplitude $T(E)$ can vanish only for $E=0$ (see \cite{Faddeev,Deift:Trubowitz}). To any S-matrix \eqref{Smatrix.def} we associate the unimodular matrix \begin{equation*} \Lambda(E)=\begin{pmatrix} \frac{1}{T(E)} & -\frac{R(E)}{T(E)}\\[2mm] \frac{L(E)}{T(E)} & \frac{1}{\ \overline{T(E)}\ } \end{pmatrix}. \end{equation*} Let $T_\alpha(E)$, $R_\alpha(E)$, $L_\alpha(E)$ be the elements of the S-matrix at energy $E$ for the pair of operators ($H_0+\alpha f$, $H_0$) and $\Lambda_{\alpha}(E)$ the corresponding $\Lambda$-matrix. Also let $\widetilde{\Lambda}_{\alpha}(E)=U_{E}^{1/2}\Lambda_{\alpha}(E)U^{1/2}_{E}$ with \begin{equation*} U_E=\begin{pmatrix} e^{i\sqrt{E}} & 0 \\ 0 & e^{-i\sqrt{E}} \end{pmatrix}. \end{equation*} Explicitly we have \begin{equation*} \widetilde{\Lambda}_{\alpha}(E)=\begin{pmatrix} \frac{e^{i\sqrt{E}}}{T_\alpha(E)} & -\frac{R_\alpha(E)}{T_\alpha(E)}\\[2mm] \frac{L_\alpha(E)}{T_\alpha(E)} & \frac{e^{-i\sqrt{E}}}{\ \overline{T(E)}\ } \end{pmatrix}. \end{equation*} Consider the matrix \begin{displaymath} A(E)=\E\left\{\widetilde{\Lambda}_{\alpha(\omega)}(E)^{\dagger} \widetilde{\Lambda}_{\alpha(\omega)}(E) \right\}=\int \widetilde{\Lambda}_{\alpha}(E)^{\dagger} \widetilde{\Lambda}_{\alpha}(E)d\kappa(\alpha)\geq 0, \end{displaymath} where for brevity we write $\alpha(\omega)$ instead of $\alpha_j(\omega)$ with some $j\in\Z$. Let $\beta_{+}(E)$ be the largest eigenvalue of $A(E)$ and $\beta_{-}(E)$ the smallest. It will turn out below that $\beta_{+}(E)\ge 1$. Set $\widetilde{\gamma}(E)=(\log \beta_{+}(E))/2\geq 0$. The first main result of the present article is \begin{theorem}\label{theorem1} Given the Hamiltonian \eqref{1.1} and the distribution $\kappa$ for the coupling constant $\alpha$, for all $E>0$ the resulting Lyapunov exponent satisfies the upper bound \begin{equation}\label{1.4} \gamma(E)\le\widetilde{\gamma}(E). \end{equation} In particular $\gamma(E)$ decays at least like $1/\sqrt{E}$ at infinity. \end{theorem} \begin{proof} Let $\Lambda^{(n)}(E;\omega)$ denote the $\Lambda$-matrix for the pair ($H^{(n)}(\omega)$, $H_{0}$), which by the factorization property can be represented in the form \begin{equation}\label{Lambda.U} \Lambda^{(n)}(E;\omega)=U_E^{-n-1/2}\prod_{j=-n}^n \widetilde{\Lambda}_{\alpha_j(\omega)}(E)\cdot U_E^{-n-1/2}. \end{equation} In fact, this factorization property is a consequence of the multiplicativity property of the fundamental matrix (see \cite{KS1} for a proof and for references to earlier work). A short calculation gives \begin{equation}\label{1.5} |T^{(n)}(E;\omega)|^{-2}=\frac{1}{4} \tr\left(\Lambda^{(n)}(E;\omega)^{\dagger}\Lambda^{(n)}(E;\omega)\right) +\frac{1}{2}. \end{equation} With $\E$ denoting the expectation with respect to the measure $\P$, by Jensen's inequality and \eqref{1.5} we therefore have the estimate \begin{eqnarray}\label{1.6} \lefteqn{e^{-2\E\{\log |T^{(n)}(E;\omega)|\}}\le \E\left\{|T^{(n)}(E;\omega)|^{-2}\right\}}\nonumber\\ &=& \frac{1}{4} \E\left\{\tr\left(\Lambda^{(n)}(E;\omega)^{\dagger} \Lambda^{(n)}(E;\omega)\right)\right\}+\frac{1}{2}. \end{eqnarray} From the factorization property \eqref{Lambda.U} it follows that \begin{equation}\label{1.7} \tr\left(\Lambda^{(n)}(E;\omega)^{\dagger}\Lambda^{(n)}(E;\omega)\right) =\tr\left(\prod_{j=n}^{-n} \widetilde{\Lambda}_{\alpha_j(\omega)}(E)^{\dagger}\prod_{j=-n}^n \widetilde{\Lambda}_{\alpha_j(\omega)}(E)\right). \end{equation} We will now make use of the fact that the $\alpha_{k}(\omega)$ are i.i.d.\ random variables. For this purpose define the $2\times 2$ matrices $A_{j}(E)\geq 0$ recursively by $A_{0}=I$ and \begin{equation}\label{1.8} A_{j}(E)=\int \widetilde{\Lambda}_{\alpha}(E)^{\dagger}A_{j-1}(E) \widetilde{\Lambda}_{\alpha}(E)d\kappa(\alpha), \end{equation} such that in particular $A(E)=A_{1}(E)$. Now it is easy to see that \begin{eqnarray}\label{1.10} \lefteqn{\E\left\{\tr\left(\Lambda^{(n)}(E;\omega)^{\dagger} \Lambda^{(n)}(E;\omega)\right)\right\}}\nonumber\\ &=& \tr\left(\E\left(\Lambda^{(n)}(E;\omega)^{\dagger} \Lambda^{(n)}(E;\omega)\right)\right) =A_{2n+1}(E). \end{eqnarray} We now use the fact that the operator inequality $0\le A\le A^{\prime}$ implies $0\le \tr\,A\le\tr\,A^{\prime}$ and $B^{\dagger}AB\le B^{\dagger}A^{\prime}B$ for all $B$. In particular we have $A(E)\le \beta_{+}(E)\,I$ from which we obtain the recursive estimates $A_{j}(E)\le \beta_{+}(E)A_{j-1}(E)\le\cdots\le\beta_{+}(E)^{j}\, I$ and hence \begin{equation}\label{1.11} \E\left\{\tr\left(\Lambda^{(n)}(E;\omega)^{\dagger} \Lambda^{(n)}(E;\omega)\right)\right\}\le 2\beta_{+}(E)^{2n+1}. \end{equation} We remark that with the same arguments one proves the lower bound \begin{equation*} 2\beta_{-}(E)^{2n+1}\le\E\left(\tr\left(\Lambda^{(n)}(E;\omega)^{\dagger} \Lambda^{(n)}(E;\omega)\right)\right). \end{equation*} The relation \eqref{1.3}, the estimate \eqref{1.11} combined with \eqref{1.6} and Fatou's lemma imply now \begin{eqnarray*} \gamma(E) &\leq & \frac{1}{2}\lim_{n\rightarrow\infty} \frac{\log \E\left\{|T^{(n)}(E;\omega)|^{-2}\right\}}{2n+1}\\ &\leq & \frac{1}{2}\lim_{n\rightarrow\infty}\frac{\log\left(\beta_+(E)^{2n+1}/2 +1/2\right)}{2n+1}=\frac{1}{2}\log \beta_+(E), \end{eqnarray*} which proves the claim \eqref{1.4}. To establish the last claim of the theorem we recall the following well known estimates (see e.g.\ \cite{Faddeev,Deift:Trubowitz}) \begin{equation}\label{est} |T_{\alpha}(E)-1|+|R_{\alpha}(E)|\le C\frac{1}{\sqrt{E}} \end{equation} valid for all large $E>0$ uniformly for all $\alpha$ in the (compact) support of $\kappa$ for fixed $f$. Using the estimate \eqref{est} in \eqref{1.12} gives the estimate $\beta_{+}(E)\le 1+C/\sqrt{E}$ for all large $E$. Since $\widetilde{\gamma}(E)=(\log\beta_{+}(E))/2$, this concludes the proof of the theorem. \end{proof} Since $\gamma(E)\ge 0$, we obviously have the inequality $\beta_{+}(E)\ge 1$ for almost all $E$. We will give now a direct independent proof of this fact and simultaneously obtain an expression for $\beta_{+}(E)$. The matrix $A(E)$ may be written in the form \begin{displaymath} A(E)=\begin{pmatrix} a(E)& b(E) \\ \overline{b(E)}& a(E) \end{pmatrix} \end{displaymath} with \begin{eqnarray}\label{1.12} a(E)&=&\int\left(\frac{2}{|T_{\alpha}(E)|^{2}}-1\right)d\kappa(\alpha)\\ b(E)&=&-e^{i\sqrt{E}}\int\frac{R_{\alpha}(E)}{T_{\alpha}(E)^{2}} d\kappa(\alpha). \end{eqnarray} This gives the two eigenvalues of $A(E)$ in the form \begin{equation} \label{1.13} \beta_{\pm}(E)=a(E)\pm |b(E)| \end{equation} Obviously $a(E)\ge 1$ and hence $\beta_{+}(E)\ge 1$. In fact, $a(E)=1$ is possible if and only if $R_{\alpha}(E)=0$ for almost all $\alpha$ in the support of $\kappa$. Then also $b(E)=0$ and $\beta_{+}(E)=1$. Actually (if $\supp\ \kappa$ has at least one non-isolated point) we do not believe there are nontrivial $f$ and $E$ for which this holds but in any case for such $E$'s the Lyapunov exponent vanishes as is easily verified (see also \cite{KS1}), so this is a trivial confirmation of estimate \eqref{1.4} in this case. In the remaining case we trivially have $\beta_{+}(E)>1$. As an example we consider the random Kronig - Penney model which is formally obtained from $H(\omega)$ by replacing $f$ with the Dirac $\delta$-function at the origin. Then we have (correcting for a misprint on page 232 of \cite{KS1}) \begin{eqnarray} \label{1.14} T_{\alpha}(E)&=&\left(1+i\frac{\alpha}{2\sqrt{E}}\right)^{-1}\\ R_{\alpha}(E)&=& -i\frac{\alpha}{2\sqrt{E}}\left(1+i\frac{\alpha}{2\sqrt{E}}\right)^{-1} \end{eqnarray} and our method still applies. This gives \begin{eqnarray} \label{1.15} a(E)&=&1+\frac{\langle\alpha^{2}\rangle}{4E}\\ b(E)&=&i\frac{\langle\alpha\rangle}{2\sqrt{E}}-\frac{\langle\alpha^{2}\rangle}{4E}. \end{eqnarray} Here for brevity by $\langle\;\rangle$ we denote the mean with respect to the probability measure $\P$ such that \begin{equation*} \langle\alpha\rangle = \E\{\alpha(\omega)\} = \int\alpha d\kappa(\alpha),\qquad \langle\alpha^2\rangle = \E\{\alpha(\omega)^2\} = \int\alpha^2 d\kappa(\alpha). \end{equation*} In particular \eqref{1.15} gives \begin{equation} \label{1.16} \beta_{+}(E)=1+\frac{\langle\alpha^{2}\rangle}{4E}+\frac{1}{2\sqrt{E}} \left(\frac{\langle\alpha^{2}\rangle}{4E}+\langle\alpha\rangle^{2}\right)^{1/2}. \end{equation} So also in this case $\gamma(E)$ decays at least like $1/\sqrt{E}$ as $E\rightarrow\infty$ and at least like $1/E$ if the mean $\langle\alpha\rangle$ of $\alpha$ vanishes, i.e.\ if on average the coupling constant is zero. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The integrated density of states} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We denote by $\xi_\alpha(E)$ the spectral shift function for the pair $(H_0+\alpha f, H_0)$. The second main result of this article is given by \begin{theorem}\label{thm:xi} For all $E>0$ the spectral shift density $\xi(E)$ for the operator \eqref{1.1} satisfies the following two-side bound \begin{equation}\label{esti} \E\{\xi_{\alpha(\omega)}(E)\}-r(E)\leq \xi(E)\leq \E\{\xi_{\alpha(\omega)}(E)\}+r(E), \end{equation} where \begin{equation*} r(E)=\min\left\{\frac{1}{2}, \frac{1}{\pi}\E\left\{\frac{|R_{\alpha(\omega)}(E)}{1-|R_{\alpha(\omega)}(E)|}\right\}\right\}. \end{equation*} In particular $\E\{\xi_{\alpha(\omega)}(E)\}$ and $r(E)$ decays at least like $1/\sqrt{E}$ at infinity. \end{theorem} \textit{Remarks}. 1. One can easily prove the following estimate \begin{equation*} \E\{\xi_{\alpha(\omega)}(E)\}-1\leq \xi(E)\leq \E\{\xi_{\alpha(\omega)}(E)\}+1, \end{equation*} which is valid for all $E\in\R$. 2. By the monotonicity of the spectral shift function with respect to perturbation $\xi(E)\geq 0$ if $\supp\ \kappa\subset\R_+$ and $\xi(E)\leq 0$ if $\supp\ \kappa\subset\R_-$ for almost all $E>0$. 3. For large $E>0$ by \eqref{est} \begin{equation*} r(E)=\min\left\{\frac{1}{2}, \frac{1}{\pi}\E\left\{\frac{|R_{\alpha(\omega)}(E)}{1-|R_{\alpha(\omega)}(E)|}\right\}\right\} =\frac{1}{\pi}\E\left\{\frac{|R_{\alpha(\omega)}(E)}{1-|R_{\alpha(\omega)}(E)|}\right\} \leq \frac{C}{\sqrt{E}}. \end{equation*} 4. In \cite{KS1} we proved the relation $\xi(E)=N_0(E)-N(E)=\sqrt{E}/\pi-N(E)$, where $N_0(E)$ is the integrated density of states for the free operator $H_0$. Theorem \ref{thm:xi} then gives the following two-sided bound for the integrated density of states \begin{equation}\label{IDS} \frac{\sqrt{E}}{\pi}-\E\{\xi_{\alpha(\omega)}(E)\}-r(E)\leq N(E) \leq \frac{\sqrt{E}}{\pi}-\E\{\xi_{\alpha(\omega)}(E)\}+r(E),\quad E>0. \end{equation} There are some other upper bounds on the integrated density of states. A well-known result is a one-sided bound due to Kirsch and Martinelli \cite[Corollary 3.1]{KiMa2}, \begin{equation*} N(E)\leq \frac{C}{\sqrt{\eta}}\ \E\left\{\int_{-1/2}^{1/2}(E+\eta-V_\omega(x))_+ dx \right\} \end{equation*} for any $\eta>0$ and all $E\in\R$. This bound however does not reproduce the correct asymptotic behavior of $N(E)$ in the large energy limit. 5. The bounds \eqref{1.4} and \eqref{esti} are of interest in the context of the Thouless formula (see e.g.\ \cite{Pastur:Figotin}) \begin{equation}\label{Thouless} \gamma(E)-\gamma_0(E)=-\int_\R \log|E-E^\prime|\ d\xi(E^\prime),\qquad E\in\R, \end{equation} where $\gamma_0(E)=[\max(0,-E)]^{1/2}$ is the Lyapunov exponent for $H_0$. The Thouless formula in the form \eqref{Thouless} can be viewed as a subtracted dispersion relation (see e.g.\ \cite{KS1}). \begin{proof} In \cite{KS1} we proved (see Theorem 3.3 there and its proof) that for any two potentials $V_1$ and $V_2$ with (compact) disjoint supports one has \begin{equation*} \xi(E;H_0+V_1+V_2,H_0) = \xi(E;H_0+V_1,H_0)+\xi(E;H_0+V_2,H_0)+\xi_{12}(E) \end{equation*} with \begin{equation*} \xi_{12}(E)=-\frac{1}{2\pi i}\log\frac{1-R_1(E)\ L_2(E)}{1-\overline{R_1(E)}\ \overline{L_2(E)}}, \end{equation*} where $R_k(E)$ and $L_k(E)$ are the right and left reflection coefficients for the Schr\"{o}dinger equation with the potential $V_k$, $k=1,2$. Actually Theorem 3.3 in \cite{KS1} states that $|\xi_{12}(E)|\leq 1/2$ for all $E\geq 0$. Now we improve on this estimate. As in \cite{KS1} we set \begin{displaymath} L_k(E)=a_k(E)e^{i\delta_k^{(L)}},\quad R_k(E)=a_k(E)e^{i\delta_k^{(R)}},\quad k=1,2 \end{displaymath} with $0\leq a_k(E)\leq 1$. Moreover $a_k(E)=1$ only when $T_k(E)=0$, which we recall can happen only if $E=0$. Therefore \begin{eqnarray*} &&\log\frac{1-R_1(E)\ L_2(E)}{1-\overline{R_1(E)}\ \overline{L_2(E)}}\\ &&=\log\frac{1-a_1(E)a_2(E)e^{i(\delta_1^{(R)}+\delta_2^{(L)})}} {1-a_1(E)a_2(E)e^{-i(\delta_1^{(R)}+\delta_2^{(L)})}}\\ &&=-2i\arctan\frac{a_1(E)a_2(E)\sin(\delta_1^{(R)} +\delta_2^{(L)})}{1-a_1(E)a_2(E)\cos(\delta_1^{(R)}+\delta_2^{(L)})}. \end{eqnarray*} By means of the inequality $|\arctan x| \leq |x|$ we immediately obtain \begin{equation}\label{ineq} |\xi_{12}(E)|\leq \min\left\{\frac{1}{2}, \frac{1}{\pi}\frac{a_1(E) a_2(E)}{1-a_1(E) a_2(E)}\right\}. \end{equation} Since $0\leq a_k(E)<1$ we can replace $a_1(E) a_2(E)(1-a_1(E) a_2(E))^{-1}$ either by $a_1(E)(1-a_1(E))^{-1}$ or by $a_2(E)(1-a_2(E))^{-1}$. Now let us consider the operator $H^{(n)}(\omega)$ for finite $n$. Applying the inequality \eqref{ineq} we obtain \begin{eqnarray*} \lefteqn{\left|\xi^{(n)}(E;\omega)-\xi_{\alpha_n(\omega)}(E)-\xi_{\alpha_{-n}(\omega)}(E) -\xi^{(n-1)}(E;\omega) \right|}\\ &\leq& \min\left\{\frac{1}{2}, \frac{1}{\pi}\frac{|R_{\alpha_n(\omega)}(E)|} {1-|R_{\alpha_n(\omega)}(E)|} \right\}+ \min\left\{\frac{1}{2}, \frac{1}{\pi}\frac{|R_{\alpha_{-n}(\omega)}(E)|} {1-|R_{\alpha_{-n}(\omega)}(E)|} \right\} \end{eqnarray*} Repeating this procedure recursively we obtain \begin{displaymath} \left|\xi^{(n)}(E;\omega)-\sum_{j=-n}^{n} \xi_{\alpha_j(\omega)}(E)\right|\leq \sum_{j=-n}^n \min\left\{\frac{1}{2}, \frac{1}{\pi}\frac{|R_{\alpha_j(\omega)}(E)|} {1-|R_{\alpha_j(\omega)}(E)|}\right\}. \end{displaymath} From the existence of the spectral shift density \eqref{xi} by the Birkhoff ergodic theorem it follows that \begin{equation*} \left|\xi(E)-\E\left\{\xi_{\alpha(\omega)}(E)\right\}\right|\leq \E\left\{\min\left\{\frac{1}{2}, \frac{1}{\pi}\frac{|R_{\alpha(\omega)}(E)|} {1-|R_{\alpha(\omega)}(E)|}\right\} \right\}. \end{equation*} From the obvious inequality \begin{equation*} \E\left\{\min\left\{\frac{1}{2}, \frac{1}{\pi}\frac{|R_{\alpha(\omega)}(E)|} {1-|R_{\alpha(\omega)}(E)|}\right\} \right\}\leq \min\left\{\frac{1}{2},\frac{1}{\pi}\E\left\{\frac{|R_{\alpha(\omega)}(E)|} {1-|R_{\alpha(\omega)}(E)|}\right\} \right\} \end{equation*} the bound \eqref{esti} follows. For large $E$ we have the following asymptotics \cite{Deift:Trubowitz} uniformly in $\alpha$ on compact sets: \begin{eqnarray*} R_{\alpha}(E) &=& \frac{\alpha}{2i\sqrt{E}}\int_\R e^{2i\sqrt{E}t}f(t)dt+O(E^{-1}),\\ L_{\alpha}(E) &=& \frac{\alpha}{2i\sqrt{E}}\int_\R e^{-2i\sqrt{E}t}f(t)dt+O(E^{-1}) \end{eqnarray*} such that $R_{\alpha}(E)=O(1/\sqrt{E})$ and $L_{\alpha}(E)=O(1/\sqrt{E})$. If the single-site potential $f$ has $p$ derivatives in $L^1(\R)$ then $L_{\alpha}(E)=O(E^{-(p+1)/2})$ and $R_{\alpha}(E)=O(E^{-(p+1)/2})$ as $E\rightarrow\infty$ \cite{Deift:Trubowitz}. The estimate $\E\{\xi_{\alpha(\omega)}(E)\}=O(1/\sqrt{E})$ is Proposition \ref{prop3} below. \end{proof} As an example we consider again the random Kronig-Penney model. The single-site spectral shift function is given in this case by \begin{equation*} \xi_\alpha(E)=\frac{1}{\pi}\arctan\left(\frac{\alpha}{2\sqrt{E}}\right),\quad E>0. \end{equation*} Therefore \begin{equation*} \E\left\{\xi_{\alpha(\omega)}(E)\right\}= \frac{1}{\pi}\int_\R \arctan\left(\frac{\alpha}{2\sqrt{E}}\right)d\kappa(\alpha) \end{equation*} and thus \begin{equation*} \left|\E\left\{\xi_{\alpha(\omega)}(E)\right\}\right|\leq \frac{\langle|\alpha|\rangle}{2\pi\sqrt{E}}. \end{equation*} Using the explicit expression for the reflection amplitude one can easily show that \begin{equation*} \frac{\langle|\alpha|\rangle}{2\sqrt{E}}+\frac{\langle\alpha^2\rangle}{4E}\leq \E\left\{\frac{|R_{\alpha(\omega)}(E)|}{1-|R_{\alpha(\omega)}(E)|}\right\} \leq \frac{\langle|\alpha|\rangle}{2\sqrt{E}}+\frac{\langle\alpha^2\rangle}{2E}. \end{equation*} We complete this section with an estimate on $\E\{\xi_{\alpha(\omega)}(E)\}$ in the general case. We will prove \begin{proposition}\label{prop3} There is a constant $c>0$ independent of $E$, $f$, and $\kappa$ such that for all $E>0$ \begin{equation*} \left|\E\left\{\xi_{\alpha(\omega)}(E)\right\}\right|\leq \frac{C}{2\sqrt{E}} \E\left\{|\alpha(\omega)|^{1/2}\right\}^2 \int_{-1/2}^{1/2}|f(x)|dx. \end{equation*} \end{proposition} Let $l^{1/2}(L^1)$ denote the Birman-Solomyak class of measurable functions $V$ for which \begin{equation*} \|V\|_{l^{1/2}(L^1)}=\left[\sum_{j=-\infty}^\infty \left(\int_{j-1/2}^{j+1/2}|V(x)|dx \right)^{1/2} \right]^{2}<\infty. \end{equation*} The claim of the proposition immediately follows from the following \begin{lemma} Let $V\in l^{1/2}(L^1)$. There is a constant $c_1$ independent of $V$ and $E$ such that \begin{equation*} |\xi(E;H_0+V,H_0)|\leq \frac{c_1}{2\sqrt{E}}\|V\|_{l^{1/2}(L^1)} \end{equation*} for all $E>0$. \end{lemma} \begin{proof} As proved in \cite{Sobolev} there is a constant $c_2>0$ independent of $E$ and $V$ such that \begin{equation*} |\xi(E; H_0+V,H_0)|\leq C_1 \|V^{1/2} R_0(E+i0)|V|^{1/2}\|_{\cJ_1}, \end{equation*} where $V^{1/2}=\sign V\ |V|^{1/2}$, $R_0(z)=(H_0-z)^{-1}$, and $\|\cdot\|_{\cJ_1}$ denotes the trace class norm (see e.g.\ \cite{Simon:79a}). From the proof of Proposition 5.6 in \cite{Simon:79a} it follows that \begin{equation*} \|V^{1/2} R_0(E+i0)|V|^{1/2}\|_{\cJ_1}\leq \frac{c_3}{\sqrt{E}}\|V\|_{l^{1/2}(L^1)} \end{equation*} for all $E>0$. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{70} \bibitem{Albeverio:book} S. Albeverio, F. Gesztesy, R. H\o egh-Krohn and H. 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