%%%%%%%%%%%%Latex2e file of mobility.tex \documentclass[a4paper,12pt]{article} \usepackage{amssymb} \usepackage{amsmath} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength{\topmargin}{-3mm} \setlength{\textwidth}{15.0cm} \setlength{\textheight}{23.0cm} \setlength{\oddsidemargin}{10mm} % \parindent0em \parskip1.2ex plus0.1ex minus0.1ex \topsep-0.3ex plus0.1ex minus0.1ex \itemsep0.3ex plus0.1ex minus0.1ex \parsep0.7ex plus0.1ex minus0.1ex % \renewcommand{\baselinestretch}{1.1} \renewcommand{\labelenumi}{{\rm(\roman{enumi})}} \renewcommand{\theenumi}{{\rm(\roman{enumi})}} % \newtheorem{thm}{Theorem}[section] \newtheorem{lemma}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}[thm]{Proposition} \newtheorem{defin}[thm]{Definition} \newtheorem{rem}[thm]{Remark} \newtheorem{assum}[thm]{Assumptions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\RR} {{\rm{I\!R}}} \newcommand{\R} {{\rm{I\!R}}} \newcommand{\Q} {{Q\!\!\!\!\!\:{\scriptstyle {\sf I}}\,\,}} \newcommand{\Qi} {{Q\!\!\!\!{\mbox{\scriptsize{\sf I}}}\;}} \newcommand{\N} {{\rm{I\!N}}} \newcommand{\ZZ} {{\rm{Z\!\!Z}}} \newcommand{\Z} {{\rm{Z\!\!Z}}} \newcommand{\W} {{\mathrm{I\!P}}} \newcommand{\E} {{\mathrm{I\!E}}} %\newcommand{\ZZ}{\mathbb{Z}} %\newcommand{\RR}{\mathbb{R}} \renewcommand{\phi} {\varphi} \renewcommand{\rho} {\varrho} \newcommand{\zd}{\ZZ^d} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \bibliographystyle{abbrv} \title{Anderson Model with decaying randomness : Mobility Edge} \author{W. Kirsch\thanks{E-mail: Werner.Kirsch@mathphys.ruhr-uni-bochum.de} \\ Institut f\"ur Mathematik\\ and Sonderforschungsbereich 237\\ Ruhr-Universit\"at Bochum, 44780 Bochum, Germany \and M. Krishna\thanks{E-mail:krishna@imsc.ernet.in} \\ Institute of Mathematical Sciences \\ Taramani, Chennai 600 113, India \and J. Obermeit\thanks{E-mail: joerg@mathphys.ruhr-uni-bochum.de} \\ Institut f\"ur Mathematik\\ and Sonderforschungsbereich 237\\ Ruhr-Universit\"at Bochum, 44780 Bochum, Germany } \date{} \maketitle \abstract{In this paper we consider the Anderson model with decaying randomness $a_nq_{\omega}(n)$, $a_n > 0, n \in \ZZ^{\nu}$ and $q_{\omega}(n)$, i.i.d random variables with an absolutely continuous distribution $\mu$. For a class of $\mu$ we show the following results on a set $\omega$ of full measure. (i) If $|a_n| \rightarrow 0$ as $|n| \rightarrow \infty$, then $\sigma_c(H_{\omega}) \subseteq [-2\nu, 2\nu]$ (ii) $\sigma(H_{\omega}) = \RR$. (iii) If $|a_n| \leq (|n|^{-1-\epsilon})$ for large $|n|$ and $\nu \geq 3$, the mobility edges are the two points $\{-2\nu, 2\nu\}$. %%%%%%%%%%%%%%%%%%section 0 \section{Introduction} The Anderson model (see below for a definition) is expected to have interesting spectral properties and Anderson \cite{An} gave the initial argument for the absence of diffusion in the model for large disorder. In particular in dimension higher than two, the model is expected to have both absolutely continuous and pure point spectra for small coupling constant. The case of dimension two is still unclear. The existence of pure point spectrum for large $\lambda$ or large energies has already been shown by Fr\"ohlich-Martinelli-Scoppola-Spencer \cite{fsms}, Delyon-Levy-Souillard \cite{dls} and Si\-mon-Wolff \cite{sw} based on the breakthrough obtained by Fr\"ohlich-Spencer \cite{fs} who obtained the necessary decay of the Green function of the Anderson operator. The result is further improved and simplified by Carmona-Klein-Martinelli \cite{ckm}, Dreifus-Klein \cite{dk} and a very simple proof of the decay estimates on the Green function is recently obtained by Aizenman-Molchanov \cite{am}, which was applied to the case of the Bethe lattice also by Aizenman \cite{ma}. In this context it is also expected that in higher dimension for small $\lambda$, there is absolutely continuous spectrum in an interval containing zero, the length of the interval depending upon $\lambda$, outside which it is expected that there is only pure point spectrum. The boundary points of this interval are called the mobility edges. In \cite{ma}, Aizenman studies the regime where point spectrum exists in the limit $\lambda \rightarrow 0$, thus getting upper bounds on the mobility edge as a function of $\lambda$. In one dimension Jacobi matrices with decaying random potentials were first studied by Simon \cite{bs1}, Delyon-Simon-Souillard \cite{dss} for Jacobi matrices and Kotani-Ushiroya \cite{kou} for Schr\"odinger operators. Recently there are several further interesting results in the one dimensional decaying randomness case exhibiting various types of spectra. we refere to Kislaev \cite{ki}, Last-Simon \cite{ls} and Kiselev-Last-Simon \cite{kls}, for example, and the literature therein for further references for Schr\"odinger/Jacobi operators. However, the existence of continuous spectrum (let alone the absolutely continuous spectrum ) is still elusive in the higher dimensional models, though in a very general context Simon \cite{bs2} showed that the operators having purely singular continuous spectrum form a dense $G_\delta$ in an appropriate metric space of operators. For the case of decaying randomness with sufficiently fast decay, one of us \cite{mk1} showed that there is absolutely continuous spectrum in the interval [-2$\nu$, 2$\nu$] for $\nu \geq 3$ and recently Klein \cite{ak} showed the existence of absolutely continuous spectrum for small $\lambda$ in the Bethe lattice. In this work we are mainly concerned with dimension $\nu > 1$ and decaying randomness and show that outside the interval [-2$\nu$,2$\nu$] there is only pure point spectrum, if any. We use the Aizenman-Molchanov technique \cite{am} (see also \cite{ma}, \cite{gr} and \cite{hu}) together with the Si\-mon-Wolff criterion for showing the absence of continuous spectrum. In particular we prove that on certain assumptions on the decaying random parameters $\pm 2 \nu$ are the mobility edges of the model, i.e. the transition points from pure point to absolutely continuous spectrum. We refer to the books of Cycon- Froese-Kirsch-Simon \cite{cfks}, Carmona-Lacroix \cite{cl}, Figotin-Pastur \cite{fp}, Reed-Simon \cite{rs} and Weidmann \cite{we} for the general introduction of the subject and for facts used without proof in this paper. {\bf Acknowledgment:} MK thanks the Ruhr-University Bochum for an invitation and Sonderforschungsbereich 237 for financial support, which made this work possible. %%%%%%%%%%%%%%%%%%%%%%section1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Model and Results} In the following we consider the Anderson model $H_\omega = H_0 + V_\omega$ on $l^2 (\ZZ^\nu)$ with \begin{equation} \left( H_0 u \right) \left(n \right) = \sum_{|i|=1} u \left( n+i\right) ~~~ \text{and} ~~~ V_\omega \left( n \right) = a_n q_n \left( \omega \right) , \label{2.1} \end{equation} where $\{q_n\}$ are independent, identically distributed (iid) random variables with an absolutely continuous probability distribution $\mu$. If not stated otherwise we will suppose $a_n > 0, n \in \ZZ^\nu$, and $a_n$ bounded. To state our results we have to introduce some additional properties on the probability measure $\mu$. \begin{defin} Let $\{a_n\}$ be a bounded, positive sequence on $\RR$. Then we say that $x \in \{a_n\}-{\text {supp}} \,\mu \equiv M$ if $\sum_n \mu (\frac{1}{a_n}(x-\epsilon, x + \epsilon))=\infty$ for all $\epsilon >0$. We call a probability measure $\mu$ {\bf asymptotically large} wrt. $a_n$ if $\{a_{kn}\}-{\text {supp}} \,\mu = \RR$ for all $k \in \ZZ^+$. $\mu$ is said to be asymptotically large at $+\infty$ (resp. $-\infty$) wrt. $a_n$ if $\{a_{kn}\}-{\text {supp}} \,\mu \supset [0, \infty)$ (resp. $(-\infty, 0]$) for all $k \in \ZZ^+$. \end{defin} For the application of the Aizenman-Molchanov method, we need the following definition (see \cite{am}). \begin{defin} A probability measure $\mu$, on the real line, is said to be $\tau$-regular (with $0 < \tau \le 1$) if, with some $v>0$ and $C < \infty$, \begin{equation} \mu \left( [z-\delta, z+\delta] \right) \le C |\delta|^\tau \mu \left( [z-v, z+v]\right) \label{2.2} \end{equation} for all $0 < \delta < 1$ and $z \in \RR$. \label{tauregular} \end{defin} \noindent {\bf Remark:} If $\mu$ is $\tau$-regular then $\mu$ is obviously $\tau^\prime$-regular with $0<\tau^\prime \le \tau$. In this paper we prove the following theorems. \begin{thm} Let $\nu \geq 1$ and consider the operator $H_{\omega}$ of (\ref{2.1}). Let $\mu$ be $1$-regular and $\int |x| \mu (dx) < \infty$. Furthermore assume that $a_n \to 0$ as $|n| \to \infty$. Then \begin{equation*} \sigma_c (H_\omega) \subseteq [-2\nu, 2\nu] \ \W-\text{a.s.} \end{equation*} \label{thm2.1} \end{thm} \noindent {\bf Remark:} \begin{enumerate} \item The assumptions on the measure are made to ensure that it is (1,1)-regular in the sense of Aizenman-Molchanov \cite{am}. The 1-regularity also ensures that $\text{sup}_x \mu(x-a,x+a) \leq C |a|$ required for the apriori bound on the average Green function. \item The above theorem does not assert the existence of continuous spectrum, though it asserts that any spectrum outside $[-2\nu, 2\nu]$ is pure point almost surely. In the case when $a_n$ and $\mu$ satisfy the conditions: $\sum_n |a_n|^{p}< \infty$ with \begin{equation*} p \equiv \sup \left\{ k \in \RR^+ , \int |x|^k \mu(dx) < \infty \right\} \end{equation*} there is no essential spectrum outside $[-2\nu, 2\nu]$, (see Delyon-Simon-Souillard \cite{dss}). \end{enumerate} The interesting case of the above theorem is contained in the following theorem. \begin{thm} Let $\nu \geq 1$ and let $H_{\omega}$ be the operator as in (\ref{2.1}) and $\mu$ be a probability measure with $M = \cap_{k \in \ZZ^+} ({a_{kn}}-\text{supp} \mu$). Then we have \begin{equation*} \bigcup_{\lambda \in M} \sigma (H_0 + \lambda \delta_0) \equiv S \subset \sigma_{\text{ess}} (H_\omega)\ \W-{\text{a.s.}}. \end{equation*} \label{thm2.2} \end{thm} As a corollary of Theorem \ref{thm2.2} we get \begin{cor} Let $\{a_n\}, \mu$ and $M$ be as in the above theorem. If in addition $M = \R^+$ (resp. $M = \R^-$) then we have \begin{equation} \sigma_{ess} (H_\omega) = [-2\nu, 2\nu] + \R^+ \quad (\text{resp }\sigma_{ess} (H_\omega) = [-2\nu, 2\nu] + \R^-). \label{2.2a} \end{equation} \label{cor2.1} \end{cor} Our final theorem is a corollary of the above two theorems, together with a theorem of Krishna \cite{mk1} which we quote here for completeness. \begin{thm} Consider the operator $H_\omega = H_0 + V_\omega$ on $l^2 (\Z^d), \nu \ge 3$ and $|a_n| \le |n|^{-\alpha}, \alpha >1$, as $|n| \to \infty$. Suppose that the corresponding probability measure satisfies the following conditions \begin{equation} \int x d\mu (x) = 0 \quad \quad \text{and} \quad \quad \int x^2 d \mu (x) = \sigma^2 < \infty, \label{2.2b} \end{equation} then we have \begin{equation} [-2\nu, 2\nu] \subseteq \sigma_{ac} (H_\omega). \label{2.2c} \end{equation} \label{thm2.2a} \end{thm} \begin{thm} Let $\nu \geq 3$ and $H_\omega$ be as in (\ref{2.1}) with $|a_n| \le C |n|^{-\alpha}$ as $|n| \to \infty$ for some $\alpha > 1$. Suppose that $\mu$ is a $1$-regular asyptotically large (with respect to $\{a_n\}$) probability measure with $\int |x|^2 \mu (dx) < \infty$. Then we have \begin{enumerate} \item $\sigma(H_\omega) = \RR, ~~ \sigma_{c}(H_{\omega}) \subseteq [-2\nu, 2\nu], ~~ \sigma_{ac}(H_\omega) = [-2\nu,2\nu]$ $\W$-a.s. \item The mobility edges are precisely the points $\{-2\nu, 2\nu\}$. \end{enumerate} \label{thm2.3} \end{thm} \noindent {\bf Remark:} We note that we cannot yet rule out the presence of singular continuous spectrum or the presence of point spectrum in $[-2\nu, 2\nu]$. \noindent {\bf {Example:}} Let $\mu = C \chi_{_{[d, \infty)}} (x) x^\beta dx$, $d>0$ be a probability measure. The asymptotically large condition at $+\infty$ is satisfied for the pair $a_n$,$\mu$ if $a_n \approx |n|^{-\alpha}, |n| \to \infty$ if $\beta < -3$ and $\alpha (\beta + 1) \le \nu$. Using this prototypical example one can construct a wide class of them. Before we actually get down to the proofs, let us explain the idea behind them. The Simon-Wolff criterion asserts that if $(a, b)$ is an interval and $H_\omega$ are operators as above, then the condition for absence of continuous spectrum of $H_\omega$ in $(a, b)$ is that the resolvent kernel $(H_\omega - E +i0)^{-1}(n,m)$ be square summable in $m$, for each $n \in \ZZ^\nu$ for almost every $(\omega, E)$. (Actually if this happens for a fixed $n$, it implies that the spectral measure of $H_\omega$ associated with the vector $\delta_n$ has no continuous component in $(a, b)$ for almost all $\omega$). The criterion does not guarantee the existence of spectrum, as for example the criterion is trivially valid if $(a, b)$ is in the resolvent set of $H_\omega$ for a set of $\omega$ of full measure. We use the Aizenman-Molchanov technique to verify the Si\-mon-Wolff criterion outside $[-2\nu, 2\nu]$ obtaining theorem (\ref{thm2.1}) to rule out the continuous component of the spectrum there. We independently show that there is spectrum outside the above interval by a probabilistic analysis using the conditions on the distribution, its support properties and the moment condition on the distribution for proving theorem \ref{thm2.2}. While both of the above are valid in more general situations, the limitations in theorem (\ref{thm2.3}) come from \cite{mk1}, where the existence of the absolutely continuous spectrum is shown for a restricted class of $a_n$. %%%%%%%%%%%%%%%%%%%%%%section2 \section{Proof of Theorem \ref{thm2.1} } The proof of Theorem \ref{thm2.1} will mainly follow the proof of Aizenman-Molchanov given in \cite{am}. Where necessary we shall introduce the changes that have to be made in order that Aizenman-Molchanov method works. Even though we consider the unbounded operators $H_\omega$, we do not need to worry about domain questions, since in most of the computations in the proofs, only the random variable $V_{\omega}(m)$ for some $m$ occurs and this is defined almost everywhere. For the proof of the theorem, we verify the Simon-Wolff criterion, namely: if for all $n \in \ZZ^\nu$, and Lebesgue-a.e. $E \in (a, b)$: \begin{equation} \lim_{\epsilon \to 0} \sum_{m \in \ZZ^\nu} |G_{E+i\epsilon} (n, m) |^2 < \infty. \label{3.1} \end{equation} for almost every realisation of $\{V_n\}_{n \in \ZZ^\nu}$, then almost surely the spectral measure of the operator $H_{\omega}$ associated with the vector $\delta_n$ has no continuous component in the interval $(a, b)$. Since the method works for all the vectors $\delta_n, n \in \ZZ^\nu$, it follows that there is no continuous component of the spectrum in the stated interval. (We note that unlike the case of $a_n \equiv ~ const.$, where ergodicity is used, here we have to argue for each $n$.) The need for looking at low moments of the integral kernel $|G_{z}(n,m)|$ is already sufficiently explained in Aizenman-Molchanov \cite{am} and Simon \cite{bs}, so we will not stress it further here. So, if for $0 < s < 1$, $\sum_{m \in \ZZ^\nu} \E ( |G_{E+i\epsilon} (n, m)|^s) <\infty$ for all $E \in (a, b)$ Fubini's theorem implies that for a.e. $(\omega, E)$ we have $\sum_{m \in \ZZ^\nu} |G_{E+i\epsilon} (n, m)|^s < C(\omega, E) < \infty$. Then, for a.e. $(\omega, E)$ we get \begin{align*} \sum_{m \in \ZZ^\nu} |G_{E+i\epsilon} (n, m)|^2 &=\sum_{m \in \ZZ^\nu}\left( |G_{E+i\epsilon} \left(n, m\right)|^s\right)^{\frac{2}{s}} \\ &\le \left(\sum_{m \in \ZZ^\nu} |G_{E+i\epsilon} \left(n, m\right)|^s\right)^{\frac{2}{s}} \\ &\le C(\omega, E)^{\frac{2}{s}} < \infty \end{align*} and therefore Simon-Wolff criterion holds for the interval $(a, b)$. In the following, for the application of Aizenman-Molchanov method, we need only a weaker condition on the measure, recall the definition \ref{tauregular} The following lemmas will provide the key tools for proving Theorem (\ref{thm2.1}). We start with a decoupling principle. \begin{lemma}(Aizenman-Molchanov) Let $\mu$ be a 1-regular, absolutely continuous (probability) measure on $\RR$. If \begin{equation} \int |x|^p \,d\mu(x) < \infty \label{3.3} \end{equation} then for $s u |E|, ~~ \forall m \in \ZZ^+ ~~ \text{with}~~ |m| > R. \label{3.6} \end{equation} \item For$|n-m|>2R$\begin{equation} \E \left( | G_{E+i0} \left( n, m\right) |^s \right) \le D_{R(n, m)} e^{-c \left( \frac{|m|-R}{2}\right)}, \label{3.7} \end{equation} where$D_{R(n, m)}$is a constant independent of E,$\epsilon$, but is polynomially bounded in$|n|$and$|m|$. \label{lemma3.2} \end{enumerate} \end{lemma} \noindent {\bf{Proof:}} The estimate in equation (\ref{3.6}) follows from the properties of$\Theta_s$of the previous lemma and the assumptions on$a_m$and$\mu$, since$E$is non-zero and$ s < 1.$To prove the rest, we set$z=E+i\epsilon$and consider the case$n = 0$. The proof is similar for any$n. We start with \begin{align*} \delta_{0, m} &= \langle \delta_0 , \left( H_\omega -z \right)^{-1} \left( H_\omega -z \right) \delta_m \rangle \\ &= \left( V_\omega \left(m \right) - z \right) \langle \delta_0 , \left( H_\omega - z \right)^{-1} \delta_m \rangle + \sum_{|i|=1} \langle \delta_0 , \left( H_\omega -z \right)^{-1} \delta_{m+i} \rangle , \end{align*} i.e. \begin{align*} \E \left( | V_\omega \left( m \right) - z|^s \, | G_z \left( 0, m \right) |^s \right) &\le \E \left( |\delta_{0,m} |^s \right) + \E \left( \sum_{|i|=1} |G_z \left(0, m+i \right) |^s \right) \\ &= \delta_{0, m} + \sum_{|i|=1} \E \left( |G_z \left( 0, m+i \right) |^s \right). \end{align*} By Lemma \ref{lemma3.1} we know that \begin{equation} \E \left( |V_\omega \left( m \right) - z|^s |G_z \left( 0, m \right) |^s \right) \ge |a_m|^s \left[ \Theta_s \left( \frac{|E|}{|a_m|}\right) \right]^s \E \left( |G_z \left( 0, m\right) |^s \right) \label{3.8} \end{equation} which implies \begin{equation} \begin{split} \E \left( |G_z \left( 0, m \right) |^s \right) &\le \frac{2\nu} {|a_m|^s \left[ \Theta_s \left( \frac{|E|}{|a_m|}\right) \right]^s} \left( \frac{1}{2\nu} \sum_{|i|=1} \E \left( |G_z \left( 0, m +i \right) |^s \right) \right) \\ &= {\underbrace{\left[ \frac{ \left(2\nu\right)^{1/s} } { |E| \left[ \frac{\Theta_s \left( \frac{|E|}{|a_m|} \right) } { \frac{|E|}{|a_m|}} \right] } \right]}_{\equiv \kappa_m}}^s \left( \frac{1}{2\nu} \sum_{|i|=1} \E \left( |G_z \left( 0, m +i \right) |^s \right) \right) \\ \label{3.9} \end{split} \end{equation} where\Theta_s (\cdot)$is as in Lemma \ref{lemma3.1}. We fix$0 < s < 1$,$E \in \RR \setminus [-(2\nu)^{1/s}, (2\nu)^{1/s}]$, define$\gamma_{E, \nu} \equiv \frac{(2\nu)^{1/s}}{|E|}$. Clearly$\gamma_{E,\nu} < 1.$Now we take,$u = \gamma_{E,\nu} (1 + \gamma(E,\nu))/2$in equation (\ref{3.6}), then from part (i) of the lemma we get the existence of an$R(E, \nu, \gamma(E,\nu))$such that \begin{equation*} \frac{\Theta_s \left( \frac{|E|}{|a_m|}\right)}{\frac{|E|}{|a_m|}} \ge \gamma_{E, \nu} + \frac{\left(1-\gamma_{E, \nu}\right)}{2}, ~~~ \forall ~~ m \in \ZZ^\nu ~~ \text{with} ~~ |m| > R. \end{equation*} We therefore get an upper bound for$\kappa_m$,$m \notin B_{2R} (0)$\begin{equation} \kappa_m = \frac{ \gamma_{E, \nu}} {\frac{\Theta_s \left( \frac{|E|}{|a_m|}\right)}{\frac{|E|}{|a_m|}}} \le \frac{\gamma_{E, \nu}} { \gamma(E,\nu) + \frac{\left(1-\gamma_{E, \nu}\right)}{2}} \equiv \gamma_0^{1/s} <1. \label{3.11} \end{equation} By repeating the above estimate$\frac{|m|-R}{2}$times, for$m \notin B_{2R} (0)$, we get \begin{equation} \E \left( |G_z \left( 0, m \right) |^s \right) \le \gamma_0^{\frac{|m|-R}{2}} \left[ \sup_{k \in B_{\frac{|m|-R}{2}}(m)} \E \left( |G_z \left( 0, k \right) |^s \right)\right]. \label{3.12} \end{equation} Suppose$\sup_k \E \left( |G_z \left( 0, k \right) |^s \right)$is polynomially bounded in m and R independent of$z$. Then setting$\gamma_0 = e^{-\ell}$, we finally have, \begin{equation} \begin{split} \sum_{m \in \ZZ^\nu} \E \left( |G_z \left( 0, m \right) |^s \right) &\le \sum_{|m| \le 2R} \E \left( |G_z \left( 0, m \right) |^s \right) + \sum_{|m| > 2R} \E \left( |G_z \left( 0, m \right) |^s \right) \\ &\le \sum_{|m| \le 2R} D_{0, m} + \sum_{|m| > 2R} e^{-\ell\left(\frac{|m|-R}{2} \right)} \\ &\le \left( 4R\right)^{\nu} D_{0, 2R}+ \sum_{|m| > 2R} e^{-\ell\left(\frac{|m|-R}{2} \right)} < \infty \label{3.13} \end{split} \end{equation} where$D_{0, \cdot}$denotes the polynomial bound. To prove the assumed polynomial boundedness of$ \E \left( |G_z \left( 0, m \right) |^s \right)$we need the following where we set$G_z(n,m) \equiv (H_{\omega} -z)^{-1}(n,m)$. Henceforth$\W(A)$will stand for the probability of the event$A$. \begin{lemma} Suppose$\mu$is a an absolutely continuous probability measure with bounded density and let$H_{\omega}$be as in (\ref{2.1}). Then for any pair of sites$n, m \in \ZZ^\nu$\begin{equation*} \W \left( |G_z \left( n, m\right) | \ge t \right) \le \left[\frac{C}{a_n} + \frac{C}{a_m}\right]\frac{1}{|t|}. \end{equation*} \label{lemma3.3} \end{lemma} \noindent {\bf{Proof (of Lemma \ref{lemma3.3}):}} We present here only the part of the proof of this lemma that requires an argument, the proof essentially follows Theorem II.1 in \cite{am} with$\lambda =1$. We note here that since the measure$\mu$has bounded density, we have the estimate$\sup_{u}\mu([u, u+t]) \leq C |t|$required by Theorem II.1 of \cite{am}. We present only the case of$n \neq m$, since the case$n=mis easy. We essentially need to calculate, \begin{align*} &\W \left( d_1 \left( \left( V_n, V_m \right), S \right) \le \frac{1}{t} \right) \\ & \quad \quad \le \W \left( |V_m -S_1\left(n\right)| \le \frac{\sqrt{2}}{t} \right) +\W \left( |V_n -S_2\left(m\right)| \le \frac{\sqrt{2}}{t} \right) \\ & \quad \quad=\W \left( |a_m q_m -S_1\left(n\right)| \le \frac{\sqrt{2}}{t} \right) +\W \left( |a_n q_n -S_2\left(m\right)| \le \frac{\sqrt{2}}{t} \right) \\ & \quad\quad =\int_{\frac{S_1}{a_m}-\frac{\sqrt{2}} {a_m t}}^{\frac{S_1}{a_m} +\frac{\sqrt{2}}{a_m t}} d \mu \,\left(q_n\right) +\int_{\frac{S_2}{a_n}-\frac{\sqrt{2}}{a_n t}}^{\frac{S_2}{a_n} +\frac{\sqrt{2}}{a_n t}} d \mu \,\left(q_m\right) \\ & \quad \quad = \mu \left( \left[\frac{S_1}{a_m}-\frac{\sqrt{2}}{a_m t}, \frac{S_1}{a_m}+\frac{\sqrt{2}}{a_m t} \right] \right) + \mu \left( \left[\frac{S_2}{a_n}-\frac{\sqrt{2}}{a_n t}, \frac{S_2}{a_n}+\frac{\sqrt{2}}{a_n t} \right] \right) \\ & \quad \quad \le \left( \frac{2\sqrt{2}}{a_n t} + \frac{2\sqrt{2}}{a_m t}\right) \end{align*} using notations (ford_1$,$S$,$S_1$and$S_2$) as given in Theorem II.1 in \cite{am}. With the help of Lemma \ref{lemma3.3} we finally prove the polynomial bound for$\E (|G_z (0, n)|^s). We get \begin{align*} \E \left( |G_z \left( n, m \right) |^s \right) &= \int_0^\infty \W \left( |G_z \left( n, m \right) |^s \ge t \right) dt \\ &= \int_0^\infty \W \left( |G_z \left( n, m \right) | \ge t^{1/s}\right) dt \\ &\le \int_0^\infty \min\left(1, \frac{2 \sqrt{2}}{a_n t^{1/s}}\right) dt +\int_0^\infty \min\left(1, \frac{2 \sqrt{2}}{a_m t^{1/s}}\right) dt \\ &\le C \left( a_n^{-1} + a_m^{-1} \right) \int_0^\infty \min\left(1, \frac{2 \sqrt{2}}{ t^{1/s}}\right) dt \\ &\equiv D_{n, m} < \infty. \end{align*} Thus,D_{n, m}$is polynomial bounded and independent of$z$, but not of$n$and$m$. From this the stated polynomial boundedness assumed after equation (\ref{3.12}) is clear. We showed that$\Omega_s = \left\{ \omega : \sigma_c(H_\omega) \subseteq [-(2\nu)^{1/s}, (2\nu)^{1/s}] \right\}$has probability 1 for each$00$that \begin{equation} \sum_{n \in k\ZZ^\nu} \mu \left( a_n^{-1} \left(\lambda-\epsilon, \lambda + \epsilon\right)\right) = \infty. \label{4.1} \end{equation} We define, \begin{equation} \begin{split} \Omega_{\epsilon, k} = \{ \omega \, |\ & \text{For$\infty$-many$n$we have: } q_n \in a_n^{-1} \left( \lambda-\epsilon, \lambda +\epsilon \right) \\ &\text{ and } q_j \in a_j^{-1} \left(-\epsilon, \epsilon\right) \ \text{for } 0 < |j-n| < k \}. \end{split} \label{4.2} \end{equation} We now cover the lattice$\ZZ^\nu$with cubes of side 2k centered at the points of$k\ZZ^\nu$for an arbitrary but fixed$k \in \ZZ^+$. (This covering is done to obtain a collection of independent events indexed by points in$k\ZZ^\nu$to apply the Borel-Cantelli lemma). An application of the Borel-Cantelli lemma implies that \begin{equation} \W \left( \Omega_{\epsilon, k}\right) =1. \label{4.3} \end{equation} Now let \begin{equation} \overline{\Omega}= \bigcap_{n} \Omega_{\frac{1}{n}, n}. \label{4.4} \end{equation} Then$\W(\overline{\Omega}) =1$, since$\overline{\Omega}$is a countable intersection of sets of measure$1$. Now, suppose$E \in \sigma (H(\lambda))$and$E \neq 0$. Then there exists a Weyl sequence$\psi_n$in$l^2 (\ZZ^\nu)$for$H(\lambda)$such that:$||\psi_n||=1$and$||H(\lambda)\psi_n - E \psi_n || < \frac{1}{n} $and$\text{supp}\, \psi_n$is compact. Then find cubes$\Lambda_{n} (\rho_n)$with \begin{equation*} |V_\omega (\rho_n) -E| < \frac{1}{n} ~~ and ~~ |V_\omega (k) | < \frac{1}{n} \ \ \ \ \ \ \ ,\ k \in \Lambda_{n}(\rho_n) \setminus \{\rho_n\} \end{equation*} where$\Lambda_{n}$is large enough so that$\text{supp}\, \psi_n \subset \Lambda_{n} (0)$. By$\Lambda_{n} (\rho_n)$we denote the cube of side$n$with center$\rho_n$. Now we define a new sequence$\phi_n (j)= \psi_n (j-\rho_n). Thus, \begin{align*} ||\left( H_0 + V_\omega \right) \phi_n - E \phi_n || &= || \left( H_0 +V_\omega \left(\cdot +\rho_n\right)\right)\psi_n - E \psi_n || \\ &\le || \left(H_0+ \lambda \delta_0 \right) \psi_n - E \psi_n|| \\ &\quad \quad + || \left( V_\omega \left( \cdot +\rho_n\right) - \lambda \delta_0 \left( \cdot \right) \right) \psi_n || \\ &\le \frac{1}{n} + \frac{1}{n} \\ \end{align*} Because of(\ref{4.4})$we are able to choose$\rho_n$and$\Lambda_{n} (\rho_n)$such that$\text{supp}\, \phi_n \cap \,\text{supp}\, \phi_k = \emptyset$,$k=1, \ldots, n-1$. E.g. choose$\rho_n$so large so that \begin{equation} \Lambda_{R_n} (\rho_n) \cap K_{n-1} = \emptyset \label{3.24} \end{equation} where$K_{n-1} = \bigcup_{j=1}^{n-1} \text{supp} \,\phi_j$. Therefore$\phi_n \overset{\text{w}}{\to} 0$. Thus$\phi_n$is a Weyl sequence for$E$for$H_\omega$, so$E \in \sigma(H_\omega)$. \noindent {\bf Proof of Corollary \ref{cor2.1}:} We just have to prove that if$\{a_n\}-\text{supp}\, \mu = \R^+$(resp.$\R^-$) then$\sigma_{ess} (H_\omega ) \supset \R^+$(resp.$\R^-$). Let$2\nu < E \in \R^+$. Then there exists a$\lambda \in \R^+$such that$E \in \sigma (H_0 + \lambda \delta_0)$. Furthermore, it is possible to find some$\lambda \in \R^+$with$E = \sup \sigma (H_0 + \lambda \delta_0)$. Now define$s_\lambda \equiv \sup \sigma (H_0 + \lambda \delta_0) = \sup_{\|\phi\|=1} \langle \phi, (H_0+ \lambda \delta_0) \phi \rangle$. By the min-max-principle it is easy to see, that$s_\lambda$is continuous in$\lambda$. Therefore it is sufficient to prove that$s_\lambda \to \infty $if$\lambda \to \infty$. We have \begin{equation} \begin{split} s_\lambda &= \sup_{\|\phi\|=1} \left( \langle \phi, H_0 \phi\rangle + \lambda \langle \phi, \delta_0 \phi \rangle \right) \\ &\ge \lambda \sup_{\|\phi\|=1} \langle \phi, \delta_0 \phi \rangle - 2\nu \\ &= \lambda -2\nu \to \infty \ \text{as } \lambda \to \infty. \end{split} \label{4.5} \end{equation} To prove the other statement we just have to take$-2 \nu > E \in \R^-$and to change$\sup $to$\inf\$ in (\ref{4.5}). \begin{thebibliography}{10} \bibitem{ma} M.~Aizenman. \newblock Localization at weak disorder: Some elementary bounds. \newblock {\em Rev. Math. Phys.}, 6:1163--1182, 1994. \bibitem{am} M.~Aizenman and S.~Molchanov. \newblock Localization at large disorder and at extreme energies: an elementary derivation. \newblock {\em Commun. Math. Phys.}, 157:245--278, 1993. \bibitem{An} P.~Anderson. \newblock Absence of diffusion in certain random lattices. \newblock {\em Phys. Rev.}, 109:1492--1505, 1958. \bibitem{ckm} R.~Carmona, A.~Klein, and F.~Martinelli. \newblock {Anderson} localization for {Bernoulli} and other singular potentials. \newblock {\em Commun. Math. Phys.}, 108:41--66, 1987. \bibitem{cl} R.~Carmona and J.~Lacroix. \newblock {\em Spectral theory of random {Schr\"odinger} operators}. \newblock Birkh\"auser Verlag, Boston, 1990. \bibitem{cfks} H.~Cycon, R.~Froese, W.~Kirsch, and B.~Simon. \newblock {\em Topics in the Theory of {Schr\"odinger} operators}. \newblock Springer-Verlag, Berlin, Heidelberg, New York, 1987. \bibitem{dls} F.~Delyon, Y.~Levy, and B.~Souillard. \newblock Anderson localization for multi dimensional systems at large disorder or low energy. \newblock {\em Commun. Math. Phys.}, 100:463--470, 1985. \bibitem{dss} F.~Delyon, B.~Simon, and B.~Souillard. \newblock From power pure point to continuous spectrum in disordered systems. \newblock {\em Ann. Institute H. Poincar\'e}, 42:283--309, 1985. \bibitem{dk} H.~v. Dreifus and A.~Klein. \newblock A new proof of localization in the {Anderson} tight binding model. \newblock {\em Commun. Math. Phys.}, 124:285--299, 1989. \bibitem{fp} A.~Figotin and L.~Pastur. \newblock {\em Spectral properties of disordered systems in the one body approximation}. \newblock Springer-Verlag, Berlin, Heidelberg, New York, 1991. \bibitem{fsms} J.~Fr\"ohlich, F.~Martinelli, E.~Scoppola, and T.~Spencer. \newblock Constructive proof of localization in the {Anderson} tight binding model. \newblock {\em Commun. Math. Phys.}, 101:21--46, 1985. \bibitem{fs} J.~Fr\"ohlich and T.~Spencer. \newblock Absence of diffusion in the {Anderson} tight binding model for large disorder or low energy. \newblock {\em Commun. Math. Phys.}, 88:151--184, 1983. \bibitem{gr} G.M.~Graf. \newblock {Anderson} localization and the space-time characteristic of continuum states. \newblock {\em J. Stat. Phys.}, 75:337--346, 1994. \bibitem{hu} D.~Hundertmark. \newblock On the time-dependent approach to {Anderson} localization. \newblock {\em Preprint}, 1997. \bibitem{ki} A.~Kiselev. \newblock Absolutely continuous spectrum of one dimensional {Schr\"odinger} operators and {Jacobi} matrices with slowly decreasing potentials. \newblock {\em Commun. Math. Phys.}, 179:377--400, 1996. \bibitem{kls} A.~Kiselev, Y.~Last and B.~Simon. \newblock Modified {Pr\"ufer} and {EFGP} transforms and the spectral analysis of one-dimensional {Schr\"odinger} operators. \newblock {\em Caltech Preprint}, 1997. \bibitem{ak} A.~Klein. \newblock Extended states in the {Anderson} model on the {Bethe} lattice. \newblock Preprint, Department of Mathematics, University of California, Irvine, 1994. \bibitem{kou} S.~Kotani and N.~Ushiroya. \newblock One-dimensional {Schr\"odinger} operators with random decaying potentials. \newblock {\em Commun. Math. Phys.}, 115:247--266, 1988. \bibitem{mk1} M.~Krishna. \newblock {Anderson} {Model} with decaying randomness - extended states. \newblock {\em Proc. Indian. Acad. Sci. ({Math}{Sci.})}, 100(4):285--294, 1990. \bibitem{ls} Y.~Last and B.~Simon. \newblock Eigenfunctions, transfer matrices and absolutely continuous spectrum of one dimensional {Schr\"odinger} operators. \newblock {\em Caltech Preprint}, 1996. \bibitem{rs} M.~Reed and B.~Simon. \newblock {\em Methods of modern {Mathematical} {Physics}: {Functional} Analysis}. \newblock Academic Press, New York, 1975. \bibitem{bs1} B.~Simon. \newblock Some {Jacobi} matrices with decaying potentials and dense point spectrum. \newblock {\em Comm. Math. Phys.}, 87:253--258, 1982. \bibitem{bs2} B.~Simon. \newblock Operators with singular continuous spectrum: {General} operators. \newblock {\em Ann. Math.}, 141:131--145, 1995. \bibitem{bs} B.~Simon. \newblock Spectral analysis of rank one perturbations and applications. \newblock In J.~Feldman, R.~Froese, and L.~Rosen, editors, {\em CRM Lecture Notes Vol. 8}, pages 109--149, Amer. Math. Soc., Providence, RI, 1995. \bibitem{sw} B.~Simon and T.~Wolff. \newblock Singular continuous spectrum under rank one perturbations and localization for random {Hamiltonians}. \newblock {\em Comm. Pure Appl. Math.}, 39:75--90, 1986. \bibitem{we} J.~Weidman. \newblock {\em Linear Operators in {Hilbert} spaces, GTM-68}. \newblock Springer-Verlag, Berlin, 1987. \end{thebibliography} \end{document}