Content-Type: multipart/mixed; boundary="-------------0605080200283" This is a multi-part message in MIME format. ---------------0605080200283 Content-Type: text/plain; name="06-149.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-149.keywords" Anderson localization, localization center ---------------0605080200283 Content-Type: application/x-tex; name="Repulsion11.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Repulsion11.tex" %#format LaTeX \ifx\documentclass\undefined \documentstyle[12pt]{article} \else \documentclass[12pt]{article} \fi \sloppy %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Adaptation of spaces in eqnarray \makeatletter \renewcommand{\theequation}{\thesection.\arabic{equation}} \@addtoreset{equation}{section} \def\eqnarray{% \stepcounter{equation}% \let\@currentlabel=\theequation \global\@eqnswtrue \global\@eqcnt\z@ \tabskip\@centering \let\\=\@eqncr $$\halign to \displaywidth\bgroup\@eqnsel\hskip\@centering $\displaystyle\tabskip\z@{##}$&\global\@eqcnt\@ne \hfil$\displaystyle{{}##{}}$\hfil &\global\@eqcnt\tw@$\displaystyle\tabskip\z@{##}$\hfil \tabskip\@centering&\llap{##}\tabskip\z@\cr} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Taken From Mathsing \def\bbbr{{\rm I\!R}} %reelle Zahlen \def\bbbn{{\rm I\!N}} %natuerliche Zahlen \def\bbbp{{\rm I\!P}} \def\bbbe{{\rm I\!E}} \def\bbbz{{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}} {\hbox{$\sf\textstyle Z\kern-0.4em Z$}} {\hbox{$\sf\scriptstyle Z\kern-0.3em Z$}} {\hbox{$\sf\scriptscriptstyle Z\kern-0.2em Z$}}}} % \def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}} % \def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle \rm C$}\hbox{\raise 0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm C$}\hbox{\raise 0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\raise 0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\raise 0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}}} % \def\B{\bf B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % eqnum \makeatletter \renewcommand{\theequation}{% \thesection.\arabic{equation}} \@addtoreset{equation}{section} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{remark}{Remark}[section] \newtheorem{definition}{Definition}[section] \newsavebox{\toy} \savebox{\toy}{\framebox[0.65em]{\rule{0cm}{1ex}}} \newcommand{\QED}{\usebox{\toy}} \def\nlni{\par\ifvmode\removelastskip\fi\vskip\baselineskip\noindent} \newenvironment{proof}{\nlni\begingroup\it Proof.\rm}{ \endgroup\vskip\baselineskip} \newcommand{\supp}{\mathop{\mathrm{supp}}\nolimits} \begin{document} %%%%%%% DOUBLE SPACED %%%%%%%% \setlength{\baselineskip}{15pt} % \title{ The repulsion between localization centers in the Anderson model } \author{Fumihiko Nakano \thanks{Faculty of Science, Department of Mathematics and Information Science, Kochi University, 2-5-1, Akebonomachi, Kochi, 780-8520, Japan. % e-mail : nakano@math.kochi-u.ac.jp}} \date{} \maketitle %%%%%%% ABSTRACT %%%%%%%%%%%%% \begin{abstract} In this note we show that, a simple combination of deep results in the theory of random Schr\"odinger operators yields a quantitative estimate of the fact that the localization centers become far apart, as corresponding energies are close together. \end{abstract} Mathematics Subject Classification (2000): 82B44, 81Q10 %%%%% INTRODUCTION %%%%% \section{Introduction} % In this paper, we consider a simple random system and its spectral region where the Anderson localization holds (i.e., we have dense point spectrum with exponentially decaying eigenfunctions). % We study the ``center" of these localized eigenfunctions and prove a repulsive property on the distribution of those in relation to their corresponding energies, that is, % ``as eigenvalues get closer, the corresponding localization centers become far apart" $\cdots (*)$. % A naive explanation which supports the observation $(*)$ is : (1) Suppose we have two eigenvalues $E_1, E_2$ with their corresponding localization centers $x_1, x_2$ satisfy $| x_1 - x_2 | \sim L$. % Because eigenfunctions are exponentially localized, we can find a finite box $\Lambda$ of size in the order of $L$ surrounding $x_1, x_2$, so that $H_{\Lambda}$ has two eigenvalues close to $E_1, E_2$. % If the density of state is finite, eigenvalues of $H_{\Lambda}$ would arrange in the order of $| \Lambda |^{-1}$ so that we could have $| E_1 - E_2 | \ge | \Lambda |^{-1} \sim L^{-d}$. % (2) (Carmona-Lacroix \cite[p.338]{CL}) Because locally we have no repulsion between eigenvalues(Molchanov, Minami \cite{Molchanov, Minami}), the overlap between the eigenfunctions should be small as the corresponding energies get closer. The phenomenon $(*)$ has been observed in numerical calculations \cite[p.338]{CL}. % \cite{Molchanov} proves $(*)$ in a special model on one space dimension, with a complicated statement. % On the other hand, the observation $(*)$ was used by Mott in the study of the ac-conductivity of random systems whose mathematical study is done recently (Kirsch-Lenoble-Pastur, Klein-Lenoble-M\"uller \cite{KLP, KLM}). % The purpose of this paper is to obtain a quantitative statement of $(*)$ which holds for almost surely. % To study the same property for the averaged quantity would require more elaborate analysis as is done in \cite{KLM}. % Our model is the standard tight binding Hamiltonian with random potential on ${\bf Z}^d$ as was treated by \cite{Minami}. % \[ (H \varphi)(x) := \sum_{|y-x| =1} \varphi (y) + \lambda V_{\omega} \varphi (x), \quad \varphi \in l^2 ({\bf Z}^d) \] % where $\lambda > 0$ is the coupling constant and $\{ V_{\omega} (x) \}_{x \in {\bf Z}^d}$ is independent, identically distributed real-valued random variables on a probability space $(\Omega, {\cal F}, {\bf P})$ whose common distribution is assumed to have a bounded density $\rho$. % Under this assumption, the following facts are well-known. % (1) $\sigma (H) = \Sigma := [-2d, 2d] + \mbox{supp } \rho$, a.s. (Kunz-Souillard \cite{Kunz-Souillard}), % (2) We can find a bounded interval $I (\subset \Sigma)$ such that the spectrum of $H$ in $I$ is almost surely pure point with exponentially decaying eigenfunctions (Anderson localization). % $I$ can be taken, for instance (Fr\"olich-Spencer, von Dreifus-Klein, Aizenman \cite{FS, vonDK, Aizenman}), (i) (high disorder) $I = \Sigma$ if $\lambda \gg 1$, (ii) (extreme energy) away from the origin, (iii) (weak disorder) away from the spectrum of the free Laplacian if $| \lambda | \ll 1$, and (iv) band edges. % Before stating our results, we define some notations. \\ % \noindent {\bf Definition}\\ % {\it (1) $\Lambda_L (x) = \{ y \in {\bf Z}^d : | y_j - x_j | \le \frac L2, j=1, 2, \cdots, d \}$ is the finite box in ${\bf Z}^d$ of length $L > 0$ with its center $x = (x_1, \cdots, x_d) \in {\bf Z}^d$. % For simplicity, $\Lambda_L := \Lambda_L (0)$. % $\partial \Lambda := \{ y \in \Lambda : \exists z \notin \Lambda, |y - z | = 1\}$ be the boundary of the box $\Lambda$. % $H_{\Lambda}$ is the restriction of $H$ on $\Lambda$ with Dirichlet boundary condition. % $G_{\Lambda}(E; x,y) := \langle \delta_x, (H_{\Lambda}- E)^{-1} \delta_y \rangle_{l^2({\bf Z}^d)}$ is the matrix element of the resolvent $(H_{\Lambda}- E)^{-1}$ % where $\delta_x(z) = 1 (z=x), 0 (z\ne x)$ and $\langle \cdot, \cdot \rangle_{l^2({\bf Z}^d)}$ is the inner-product in $l^2({\bf Z}^d)$. % $|\Lambda| = \sharp \Lambda$ is the volume of a box $\Lambda (\subset {\bf Z}^d)$ and $| I | = b - a$ is the width of an interval $I = (a,b) (\subset {\bf R})$. % $\chi_{\Lambda}$ is the characteristic function of a box $\Lambda$. \\ % (2) We say the box $\Lambda_L (x)$ is $(\gamma, E)$-regular iff $E \notin \sigma (H_{\Lambda_L (x)})$ and for any $y \in \partial \Lambda_L (x)$, % \[ | G_{\Lambda_L (x)} (E; x, y) | \le e^{- \gamma L/2}. \] % (3) For $\phi \in l^2 ({\bf Z}^d)$, let $X({\phi})$ be the set of its localization centers given by % \[ X({\phi}) := \left\{ x \in {\bf Z}^d : | \phi (x) | = \max_{y \in {\bf Z}^d} | \phi (y) | \right\}. \] % This notion is due to Germinet-De Bi\`evre \cite{BG}. % Since $\phi \in l^2 ({\bf Z}^d)$, $X({\phi})$ is a finite set. % Moreover, for the set $\{ E_j (\omega) \}_{j \ge 1}$ of eigenvalues of $H$ counting multiplicity, we take the corresponding eigenfunctions $\{ \phi_j (\omega) \}_{j \ge 1}$ and let $X(E_j(\omega)) := X(\phi_j (\omega))$. \\ % (4) For a finite box $\Lambda$ and $\phi \in l^2({\bf Z}^d)$, we say $\phi$ is localized in $\Lambda$ iff $X({\phi}) \cap \Lambda \ne \emptyset$. % For an eigenvalue $E_j (\omega)$ of $H$, we say $E_j (\omega)$ is localized in $\Lambda$ iff $X(E_j(\omega)) \cap \Lambda \ne \emptyset$}. \\ Let $I (\subset \Sigma)$ be the bounded interval where the initial length scale estimate of the multiscale analysis holds : \\ % %%%%% \noindent {\bf Assumption} \\ % {\it We have an interval $I (\subset \Sigma)$ with % \[ {\bf P}\left( \mbox{for $\forall E \in I$, $\Lambda_{L_0}$ is $(\gamma, E)$-regular} \right) \ge 1 - L_0^{-p} \] % for some $\gamma > 0$, $p > 2d$ and some $L_0 > 0$ sufficiently large. }\\ % \noindent $I$ can be taken in regions mentioned in the paragraph preceding Definition in this section. % This condition, together with Wegner's estimate, guarantee to apply the multiscale analysis, and from which the following fact is deduced \cite{vonDK} : % we can find $\alpha = \alpha(p,d)$, $1 < \alpha < 2$ such that, putting % \[ \Lambda_k (x) = \Lambda_{L_k} (x), \quad L_k = L_{k-1}^{\alpha}, \quad 1 < \alpha < 2, \quad x \in {\bf Z}^d, \] % we have % \begin{equation} {\bf P} \left\{ \mbox{For all $E \in I$ either $\Lambda_k(x)$ or $\Lambda_k(y)$ is $(\gamma, E)$-regular} \right\} \ge 1 - L_k^{-2 p} \label{MSA} \end{equation} % for any $x, y$ with $|x - y | > L_k$. % The first result implies the distribution of localization centers are ``thin" in ${\bf Z}^d$. % \begin{theorem} {\bf (localization centers are thin)}\\ % Let $d_k = | \Lambda_k |^{-1} k^{-2}$, $E \in I$, $J_k = (E - \frac {d_k}{2}, E + \frac {d_k}{2}) (\subset {\bf R})$, $k=1, 2, \cdots$. % Then for a.e. $\omega$, we can find $k_0 = k_0 (\omega)$ such that, if $k \ge k_0$, there are no eigenvalues of $H$ in $J_k$ localized in $\Lambda_k$. % \label{localized centers are thin} \end{theorem} % \begin{remark} Theorem \ref{localized centers are thin} states that, for any $E \in I$, localization center run away from the origin, as the corresponding eigenvalue approaches $E$. % It implies the distribution of the localization centers is thin, while the eigenvalues are dense in $I$. % A naive explanation of this fact is : we have infinite number of eigenvalues near $E$ while the number of states is proportional to the volume, by the finiteness of the density of states. \end{remark} % \begin{remark} As the density of states obeys the Lifschitz tail asymptotics on the bottom of the spectrum, another estimate is obtained there : % $d_k$ can be replaced by $d_L = (a, a + \frac {1}{| \Lambda_L |^{2/d}})$, $a = \inf \sigma (H)$ (Simon \cite{Simon-Lifschitztail}). \end{remark} % \begin{remark} Theorem \ref{localized centers are thin} holds true for random Hamiltonians where the multiscale analysis is applicable and Wegner's estimate holds. % \end{remark} % \begin{theorem} {\bf (localization centers are repulsive)}\\ % Let $d_k = | \Lambda_k |^{-2}k^{-2}$, $k =1, 2, \cdots$. % For a.e. $\omega$, the following event occurs. % For any $x \in {\bf Z}^d$, there exists $k_0 = k_0(\omega, x)$ such that for $k \ge k_0(\omega, x)$ and any interval $J \subset I$ with $|J| \le d_k$, there is at most one eigenvalue of $H$ in $J$ localized in $\Lambda_k (x)$. % \label{localized centers are repulsive} \end{theorem} % \begin{remark} % Theorem \ref{localized centers are repulsive} implies : % for a.e. $\omega$ and for any eigenvalue $E = E_j (\omega) \in I$ of $H$, % we can find $k_1 = k_1 (\omega,E_j(\omega))$ such that, for any $k \ge k_1$, we have no eigenvalues of $H$ in $J_k = (E_j(\omega) - \frac {d_k}{2}, E_j(\omega) + \frac {d_k}{2})$ localized in $\Lambda_k(x)$ for any $x \in X(E_j(\omega))$ except $E_j(\omega)$ itself. % Hence this theorem roughly states that, for a localization center $x$ with energy $E$, any other localization center must be away from $x$ at least in the distance of $L_k/2$, if the corresponding eigenvalues are within the distance of $| \Lambda_k |^{-2}$ from $E$. % And this happens simultaneously for all eigenvalues in $I$ almost surely. \end{remark} % \begin{remark} Theorem \ref{localized centers are repulsive} also proves that eigenvalues in $I$ are simple almost surely. % Indeed, this is done by Klein-Molchanov \cite{KM} by the argument similar to ours, but without relying on the multiscale analysis. \end{remark} % \begin{remark} One of the essential ingredients of the proof of Theorem \ref{localized centers are repulsive} is the estimate of the probability to have more than two eigenvalues on a given interval obtained by Minami \cite[Lemma 2]{Minami}, which is also essential to prove the absence of repulsion of eigenvalues, and known to hold in the Anderson model only so far. % Hence something different could be expected for the acoustic type operator, in which the level repulsion is known to occur (Grenkova-Molchanov-Sudarev \cite{Grenkova}). \end{remark} % \begin{remark} This result concerns the distribution of the localization centers which holds for almost surely. % If one is interested in the fluctuation of those, by proceeding along the ideas in \cite{Minami}, one could expect the Poisson-type behavior also for localization centers, under a suitable scaling. % To verify this observation would be an interesting problem\footnote{The author would like to thank Rowan Killip for pointing this out. } \cite{Nakano}. \end{remark} % In the following section, we prove Theorem \ref{localized centers are thin}, \ref{localized centers are repulsive}, along the naive argument given at the beginning of this section, which is done by making use of the machinery developed by Germinet-De Bi\`evre, Damanik-Stollmann, Klein-Molchanov and Minami \cite{BG, DS, KM, Minami} : % (i) If we have an (resp. at least two) eigenvalue in an interval $J (\subset I)$, the corresponding eigenfunction is exponentially small outside a box $\Lambda$ surrounding its localization center \cite{BG, DS}. % Then $H_{\Lambda}$ has an (resp. at least two) eigenvalue in $J$ \cite{KM}. % (ii) We estimate the event that $H_{\Lambda}$ has an (resp. at least two) eigenvalue in $J$ by Wegner's (resp. Minami's) estimate. % Then the usual Borel-Cantelli argument gives the assertion. % In Appendix, we collect lemmas used in these proofs borrowed from \cite{BG, DS, KM, Minami}. % \section{Proof of Theorems} % We first set % \begin{eqnarray*} E_j &=& \Bigl\{ \omega \in \Omega : \mbox{ For some $E \in I$ and some $x, y \in \Lambda_{3 L_{j+1}}$ with $\Lambda_j (x) \cap \Lambda_j (y) = \emptyset$, } \\ &&\qquad\qquad \mbox{ $\Lambda_j (x)$, $\Lambda_j(y)$ are both $(\gamma, E)$-singular} \Bigr\} \\ % \Omega_k &=& \bigcap_{j \ge k} E_j^c. \end{eqnarray*} % Then by (\ref{MSA}), % \begin{equation} {\bf P} (\Omega_k) \ge 1 - C_1(\alpha, d, p) L_k^{2d \alpha - 2 p} \label{Omega} \end{equation} % for some $C_1= C_1(\alpha, d, p)$. % In what follows, we take and fix any $0 < \gamma' < \gamma$, and $k_0 (\alpha, d, \gamma)$, $k_1(\alpha, d, \gamma, \gamma')$, $L_0(\gamma')$, and $C_2$ are positive constants given in Appendix. % \begin{lemma} {\bf \mbox{}}\\ % Let $J=(a,b)( \subset I)$ and let $\epsilon_{L_k} = C_2 e^{-\gamma' L_k/2}$. % If $k \ge k_0(\alpha, d, \gamma) \vee k_1(\alpha, d, \gamma, \gamma')$, $L_k \ge L_0(\gamma')$, the following estimates hold. % \label{whole Hamiltonian} % \begin{eqnarray*} (1) \quad && {\bf P} \left( \mbox{we have an eigenvalue of $H$ in $J$ localized in $\Lambda_k$} \right) \\ && \qquad \le \| \rho \|_{\infty} ( | J |+ 2\epsilon_{L_k} ) | \Lambda_{3k} | + C_1(\alpha, d, p) L_k^{2d \alpha -2 p}. \\ % (2) \quad && {\bf P} \left( \mbox{we have at least 2 eigenvalues of $H$ in $J$ localized in $\Lambda_k$} \right) \\ % && \qquad \le \pi^2 \| \rho \|^2_{\infty} | \Lambda_{3 L_k} |^2 ( | J | + 2\epsilon_{L_k} )^2 + C_1(\alpha, d, p) L_k^{2d \alpha -2 p}. \end{eqnarray*} % \end{lemma} % \begin{proof} (1) Let % \[ A_k := \left\{ \omega \in \Omega : \mbox{we have an eigenvalue of $H$ in $J$ localized in $\Lambda_k$} \right\}. \] % Let $\omega \in A_k \cap \Omega_k$. % Then we have an eigenvalue $E \in J$ localized in $\Lambda_k$ and since $j \ge k_0 \vee k_1$, $L_k \ge L_0$, the corresponding eigenfunction $\phi$ satisfies % \[ \| ( 1- \chi_{3 L_k} ) \phi \| \le e^{- \gamma' L_k/2} \] % by Lemma \ref{decay estimate2}. % Then the argument of the proof of Lemma \ref{approximation} shows that $H_{\Lambda_{3 L_k}}$ has an eigenvalue in $(a - \epsilon_{L_k}, b + \epsilon_{L_k})$. % By Wegner's estimate: \footnote{$\{ E_j (\Lambda) \}_{j=1}^{| \Lambda |}$ is the set of eigenvalues of $H_{\Lambda}$} % \[ {\bf P}(\sharp \{ E_j (\Lambda) \in J \} \ge 1) \le \| \rho \|_{\infty} | \Lambda | \cdot | J |, \] % we have % $ {\bf P} (A_k \cap \Omega_k) \le \| \rho \|_{\infty} | \Lambda_{ 3 L_k } | ( | J |+ 2\epsilon_{L_k} ). $ \\ % (2) Let % \[ B_k := \left\{ \omega \in \Omega : \mbox{we have at least 2 eigenvalues of $H$ in $J$ localized in $\Lambda_k$} \right\} \] % By the same argument in the proof of Lemma \ref{whole Hamiltonian}(1), if $\omega \in B_k \cap \Omega_k$, $H_{\Lambda_{3 L_k}}$ has at least two eigenvalues in $(a - \epsilon_{L_k}, b + \epsilon_{L_k})$. % By Minami's estimate \cite[Lemma 2]{Minami}, \cite[Appendix]{KM}: % \[ {\bf P}\left( \{ \sharp \{ E_j (\Lambda) \in J \} \ge 2 \} \right) \le \pi^2 \| \rho \|^2_{\infty} | \Lambda |^2 | J |^2, \] % we have % $ {\bf P} (B_k \cap \Omega_k) \le \pi^2 \| \rho \|^2_{\infty} \| \Lambda_{3 L_k} |^2 ( | J |+ 2\epsilon_{L_k} )^2. $ % \QED \end{proof} % % \noindent {\it Proof of Theorem \ref{localized centers are thin}} \\ % We consider the following event. % \[ A_k = \left\{ \omega \in \Omega : \mbox{ We have an eigenvalue of $H$ in $J_k$ localized in $\Lambda_k$ } \right\}. \] % By Lemma \ref{whole Hamiltonian}, if $k \ge k_0(\alpha, d, \gamma) \vee k_1(\alpha, d, \gamma, \gamma')$ and $L_k \ge L_0(\gamma')$, we have % \[ {\bf P}(A_k) \le \| \rho \|_{\infty} | \Lambda_{3 L_k} | ( d_k + 2\epsilon_{ L_k} ) + C_1 (\alpha, d, p) L_k^{2d(\alpha-1) - 2 p} \] % % Since $d_k +2 \epsilon_{L_k} \le 2 d_k$ for sufficiently large $k$, $\sum_k {\bf P} (A_k) < \infty$. % The Borel-Cantelli argument then proves the assertion of Theorem \ref{localized centers are thin}. \QED\\ %%%%% %%%%% \noindent {\it Proof of Theorem \ref{localized centers are repulsive}} \\ % We consider the following events. % \begin{eqnarray*} B_k (x, J) &=& \left\{ \omega\in \Omega : \mbox{at least two eigenvalues of $H$ in $J$ localized in $\Lambda_k(x)$} \right\}, \\ % B_k (x) &=& \Bigl\{ \omega\in \Omega : \mbox{at least two eigenvalues of $H$ in $J$} \\ % &&\qquad\qquad \mbox{ for some $J(\subset I)$ with $|J| \le d_k$ in $\Lambda_k(x)$} \Bigr\}. \end{eqnarray*} % Suppose $| J | \le 2 d_k$, $J \subset I$. % Then by the argument in the proof of Lemma \ref{whole Hamiltonian}(2), % \[ {\bf P}(B_k(x, J) \cap \Omega_k) \le \pi^2 \| \rho \|_{\infty}^2 ( 3 d_k )^2 \cdot | \Lambda_{3 L_k} |^2 \] % for $k$ sufficiently large. % Here we use the argument in \cite[Lemma 2]{KM} and cover the interval $I$ by those $J(i, d_k)$, $i = 1, 2, \cdots, N_k$ of width $2 d_k$ such that the left end of $J(i+1, d_k)$ coincides with the mid point of $J(i, d_k)$. % Then $N_k \le \frac {| I |}{2 d_k} \cdot 2 = \frac {| I |}{d_k}$ and any interval (in $I$) of width less than $d_k$ is contained by some $J(i, d_k)$. % Hence % \[ {\bf P} (B_k (x) \cap \Omega_k) \le \pi^2 \| \rho \|_{\infty}^2 ( 3 d_k )^2 \cdot \frac {|I|}{d_k} \cdot | \Lambda_{3 L_k} |^2 \] % for large $k$ and therefore $\sum_k {\bf P}(B_k(x)) < \infty$ by (\ref{Omega}). % By the Borel-Cantelli lemma, $\Omega(x) = \liminf_{k \to \infty} B_k^c(x)$ satisfies ${\bf P}(\Omega(x)) = 1$ and for $\omega \in \Omega(x)$ we can find $k_0 = k_0(\omega,x)$ such that for any $k \ge k_0(\omega, x)$ and for any interval $J (\subset I)$ with $| J | \le d_k$, we have at most one eigenvalue of $H$ in $J$ localized in $\Lambda_k (x)$. % For $\omega \in \Omega' = \bigcap_{x\in {\bf Z}^d} \Omega(x)$, the event described in the statement of Theorem \ref{localized centers are repulsive} occurs. % \QED %%%%% %%%%% \section{Appendix} % In this section, we state Lemmas used in section 2, which are borrowed from \cite{BG, DS, KM, Minami}. % The following lemma is \cite[Lemma 3.5]{BG}. % \begin{lemma} {\bf (\cite[Lemma 3.5]{BG})}\\ % We can find a constant $k_0 = k_0 (\alpha, d, \gamma)$ such that, if $k \ge k_0$ and $\phi \in l^2 ({\bf Z}^d)$ satisfies $H \phi = E \phi$, then $\Lambda_{L_k} (x_{\phi})$ is $(\gamma, E)$-singular. % \label{localized centers live in bad box} \end{lemma} % In what follows, we take and fix any $\gamma'$ with $0 < \gamma' < \gamma$. % \begin{lemma} {\bf \mbox{}}\\ % We can find a constant $k_1 = k_1 (\alpha, d, \gamma, \gamma')$ such that, if $k\ge k_0(\alpha, d, \gamma) \vee k_1 (\alpha, d, \gamma, \gamma')$, $\omega \in \Omega_k$ and if $\phi \in l^2({\bf Z}^d)$ satisfies $H \phi = E \phi$, $\| \phi \| = 1$ and localized in $\Lambda_k$, % \label{decay estimate2} % \[ \| ( 1 - \chi_{\Lambda_{3 L_k}}) \phi \| \le e^{- \gamma' L_k/2}, \quad \gamma' < \gamma. \] % \end{lemma} % Lemma \ref{decay estimate2} is proved along the argument in \cite[Step 3, Theorem 3.1]{DS}.\\ % \noindent {\it Sketch of proof } % We divide $\Lambda_{3 L_k}^c$ into annulus : % $ \Lambda_{3 L_k}^c = \bigcup_{i \ge k} M_i, \; M_i = \Lambda_{3 L_{i+1}} \setminus \Lambda_{3 L_i}, \; i \ge k. $ % Then we have % $ \| (1 - \chi_{3 L_k}) \phi \|^2 \le \sum_{i = k}^{\infty} \| \chi_{M_i} \phi \|^2 % \le \sum_{ i = k }^{\infty} \sum_{x \in M_i} | \phi (x) |^2. $ % Since $\Lambda_{L_i} (x_{\phi}) \cap \Lambda_{L_i}(x) = \emptyset$ for any $x \in M_i$, $\Lambda_{L_i}(x)$ is $(\gamma, E)$-regular by Lemma \ref{localized centers live in bad box}. % \QED % % \begin{lemma} {\bf \mbox{}}\\ % We can find positive constants $C_2$, $L_0 = L_0 (\gamma')$ with the following property. % If $\varphi_1, \varphi_2$ satisfy % \begin{eqnarray*} && H \varphi_j = E_j \varphi_j, \quad \| \varphi_j \| = 1, \quad E_j \in J = (a,b), \quad j=1,2, \\ % && \| ( 1 - \chi_{\Lambda_L}) \varphi_j \| \le e^{ - \gamma' L/2}, \quad j = 1,2, \quad L \ge L_0, \end{eqnarray*} % then $H_{\Lambda_L}$ has at least two eigenvalues in $(a - \epsilon_L, b + \epsilon_L)$ where $\epsilon_L = C_2 e^{-\gamma' L/2}$. % \label{approximation} \end{lemma} % The proof is found in \cite{KM} : we orthonormalize $\varphi_j^{\Lambda} = \chi_{\Lambda} \varphi_j$, $j=1,2$ and estimate from below the trace of the spectral projection of $H$ corresponding to the interval $(a- \epsilon_L, b+\epsilon_L)$.\\ % \noindent {\bf Acknowledgement } The author would like to thank Professors Abel Klein, Rowan Killip, Nariyuki Minami and a referee for their discussions and comments. %%%%% REFERENCES %%%%%%%%%%%%%%%%%%%%% % \small \begin{thebibliography}{99} % \bibitem{Aizenman} Aizenman, M., : Localization at Weak Disorder: Some Elementary Bounds, Rev. 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