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\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 $s<p$ and $s<1$
\begin{equation}
           \int |x-\eta|^s \frac{\alpha}{|x-\beta|^s}\, d\mu(x)
              \ge \left[ \Theta_s \left(  |\eta|  \right) \right]^s 
                        \int \frac{\alpha}{|x-\beta|^s}\, d\mu(x),
								\label{3.4}
\end{equation}
with $\Theta_s (\cdot)$ an increasing positive function on $\RR^+$, 
satisfying
\begin{equation}
            \lim_{\eta \to \infty} \frac{\Theta_s (\eta)}{\eta} = 1.
								\label{3.5}
\end{equation}
							\label{lemma3.1}
\end{lemma}

\noindent
{\bf{Proof:}} We note that the hypothesis on the density of the
measure ensures that it is $s$-regular for each $0 < s \leq 1$, so
the proof follows from Lemma 3.1 in \cite{am}. 


 The properties of the function $\Theta$ in the above lemma play
an essential role in the proof of Theorem \ref{thm2.1} 
via the following lemma.
\begin{lemma} Let $a_n, n \in \ZZ^\nu$ be a sequence satisfying the
conditions of theorem (\ref{thm2.1}) and let $\Theta_s(\cdot)$ be as in
the above lemma.  Then,
\begin{enumerate}
\item for E $\neq 0$, given any number $0 < u
<1$, there exists an
$R=R(E, s, u) \in \ZZ^+$ with 
\begin{equation}
   a_m \left[ \Theta_s \left( 
                \frac{|E|}{|a_m|}
                 \right) \right]
          > 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 $n=m$ is
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 (for $d_1$, $S$, $S_1$ 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 $0<s<1$.  Now taking a
subsequence $s_n \uparrow 1$ and taking $\omega \in \cap \Omega_{s_n}$,
we can conclude the theorem.

\section{Proof of Theorem \ref{thm2.2} and Corollary \ref{cor2.1}}
In this section we prove the existence of essential spectrum of the operators
$H_\omega$ outside the interval $[-2\nu, 2\nu]$. We start with the proof of Theorem \ref{thm2.2}

\noindent
{\bf Proof of Theorem \ref{thm2.2}:}
Let $\lambda \in \cap_{k \in \ZZ^+} (\{a_{kn}\}-\text{supp}\, \mu)$ and consider
 the "model-operator" $H(\lambda) = H_0 + \lambda \delta_0$. We prove that if $E \in \sigma (H (\lambda))$ then $E \in \sigma_{ess} (H_\omega)$.

As a direct conclusion from our definition of the $\{a_n\}-\text{supp}\, \mu$
 we know for all $\epsilon >0$ 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}). 


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