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\begin{document} \title{Dynamical Localization for Discrete and
Continuous Random Schr\"odinger Operators} \author{F. Germinet\\
UFR de Math\'ematiques et LPTMC\\ Universit\'e Paris VII - Denis
Diderot \\75251 Paris Cedex 05 France\\e-mail:
germinet@math.jussieu.fr \and S. De Bi\`evre\\ UFR de
Math\'ematiques et URA GAT\\Universit\'e des Sciences et
Technologies de Lille\\ 59655 Villeneuve d'Ascq Cedex
France\\e-mail: debievre@gat.univ-lille1.fr} \date{1997}
\maketitle
\begin{abstract} We show for a large class of random Schr\"odinger
operators $H_\omega$ on $\ell^2(\Z^\nu)$ and on $L^2(\R^\nu)$ that
dynamical localization holds, {\em i.e.} that, with probability
one, for a suitable energy interval $I$ and for $q$ a positive
real, $$ \sup_t r^q(t) \equiv \sup_t \ <\infty. $$ Here $\psi$ is a function of
sufficiently rapid decrease, $\psi_t=e^{-iH_\omega t} \psi$ and
$P_I(H_\omega)$ is the spectral projector of $H_\omega$
corresponding to the interval $I$. The result is obtained through
the control of the decay of the eigenfunctions of $H_\omega$ and
covers, in the discrete case, the Anderson tight-binding model
with Bernouilli potential (dimension $\nu=1$) or singular
potential ($\nu>1$), and in the continuous case Anderson as well
as random Landau Hamiltonians. \end{abstract}
\section{Introduction} \setcounter{equation}{0}
We show for a large class of random Schr\"odinger operators
$H_\omega$ on $\ell^2(\Z^\nu)$ and on $L^2(\R^\nu)$ that dynamical
localization holds, {\em i.e.} that, with probability one, for a
suitable energy interval $I$ and $q>0$, $$ \sup_t r^q(t) \equiv
\sup_t \
<\infty. $$ Here $\psi$ is a function of sufficiently rapid
decrease, $\psi_t=e^{-iH_\omega t} \psi$, $P_I(H_\omega)$ is the
spectral projector of $H_\omega$ corresponding to the interval
$I$. The result covers all random Schr\"odinger operators for
which exponential localization has been proved, including
operators with Bernouilli potentials in dimension $1$ and random
Landau Hamiltonians, for example.
The strategy of the proof is as follows. First recall that
exponential localization, {\em i.e.} pure point spectrum and
exponentially decaying eigenfunctions, is by now a well
established property of random Schr\"odinger operators in many
situations. On the other hand, it is also known that exponential
localization does not systematically entail dynamical localization
\cite{iv}. The authors of \cite{iv} point out that, to obtain
dynamical localization, some control is needed on the location and
the size of the boxes outside of which the eigenfunctions
``effectively'' decrease exponentially. This is precisely what is
achieved for random Schr\"odinger operators in the present paper
(Theorem~\ref{th2} and Theorem~\ref{thcont}). Our proof here uses
the ideas of Von Dreifus and Klein \cite{124}, and in particular
those of the proof of their Theorem 2.3. We proceed as follows:
once exponential localization has been proved, and using the fact
that the spectrum is now known to be discrete, one can exploit the
result of the multi-scale analysis a second time to get better
(and sufficient) control on the eigenfunction decay. We first deal
with the discrete case (Sections 2 and 3). In section 2 we start
by proving (along the lines of \cite{iv}) a sufficient condition
(see (\ref{eq:wsule})) on the eigenfunctions of a Hamiltonian $H$
which implies dynamical localization. In section \ref{discrete} we
give the proof of the announced result for the discrete Anderson
model.
The continuous case is dealt with in sections 4 and 5. Exponential
localization for Schr\"odinger operators has recently been
carried over to the continuum by Combes and Hislop \cite{combhis}
and by Klopp \cite{klopp1}. The case of random Landau Hamiltonians
is dealt with by Combes, Hislop and Barbaroux \cite{bacombhis1}
\cite{combhisL}, by Wang \cite{wang} and by Dorlas, Macris and
Pul\'e \cite{dormacpul}. All those papers use an adaptation to the
continuous case of the multi-scale analysis originally developed
for discrete Schr\"odinger operators (\cite{88} \cite{101}
\cite{124} or see \cite{carlac}). This reduces the proof of
exponential localization to the verification of two hypotheses: a
Wegner estimate and an estimate allowing the ``initialization" of
the multi-scale analysis. Our central result here
(Theorem~\ref{thcont}) shows that those two hypotheses actually
imply dynamical localization. We give some applications in
section 5.
To put our results in perspective, we recall that Del Rio,
Jitomirskaya, Last and Simon \cite{iv} used bounds of Aizenman
\cite{aiz} to give a simple proof (avoiding the multi-scale
analysis) of dynamical localization
for the discrete Anderson model with a potential with bounded
density. But the bounds of \cite{aiz} do not seem to carry over to
the continuous case, nor to Bernouilli and other singular
potentials in the discrete case. To deal with these cases, we
were therefore obliged to return to the (rather painful)
multi-scale analysis.
A further application of our results to the random dimer model
\cite{dunlap} will be given elsewhere \cite{dbgdimer}.
%%%%%%%%%%%%% SULE and LOCALIZATION %%%%%%%%%%%%%%
\section{Eigenfunction decay and
dynamical localization} \label{dynloc} \setcounter{equation}{0}
In this section we give, for a class of self-adjoint operators $H$
with pure point point spectrum, defined on either $l^2(\Z^\nu)$ or
$L^2(\R^\nu)$,
a sufficient condition on the eigenfunctions (see
(\ref{eq:wsule})) that guarantees dynamical localization. Our
strategy for proving dynamical localization for random
Schr\"odinger operators is then to prove that a property much
stronger than this condition is indeed satisfied (Theorem
\ref{th2}, Theorem \ref{thcont}).
Let $H_0$ be the following operator on $L^2(\R^\nu)$: $$
H_0=H_1\oplus H_2, $$ where $H_1=p_1^2+p_2^2$,
$p_1=\partial_{x_1}+Bx_2/2$, $p_2=\partial_{x_2}-Bx_1/2$, $B\geq
0$, and $H_2=\sum_{3}^{\nu} p_i^2$, $p_i=\partial_{x_i}$. One can
also write $H_0=(P-A)^2$, where $A$ is the vector potential
$B/2(x_2,-x_1)$, written in the symmetric gauge, associated to the
constant magnetic field $\overrightarrow{B}=B\vec{e}_{x_3}$.
%%%%%%%%%% Theorem of section 2 %%%%
\begin{th} \label{th1} Let
$H$ be a self-adjoint operator on $\Hh=l^2(\Z^\nu)$ or
$L^2(\R^\nu)$ with pure point spectrum on some interval
$I\subset\R$. Let $\varphi_n$ be its eigenfunctions with
corresponding eigenvalues $E_n\in I$. In the case
$\Hh=L^2(\R^\nu)$, suppose that $I$ is compact and that $H$ has
the form $H_0+V$, $H_0$ as described above, $V\in
L^{\infty}(\R^\nu)$. Suppose moreover that \begin{equation}
\exists \gamma>0, \gamma'\in]0,\gamma/2[\; and\; sites\, (x_n)\,
s.t.\,
\forall\;n, \;|\varphi_n(x)|0$ and $\psi\in \Hh$
decaying exponentially at a rate $\theta>2\gamma'$. Then there
exists a constant $C_\psi=C_\psi(I,\gamma,\gamma',\theta,q)$ such
that: \begin{equation} \forall\,t\geq 0,\,\,||\,|X|^{q/2}
e^{-iHt}P_I(H)\psi||^2 \leq C_\psi.
\label{eq:dynloc} \end{equation} \end{th}
This simple result relies on ideas of section 7 in \cite{iv}.
Note that (\ref{eq:wsule}) says roughly that the eigenfunctions
are localized inside boxes of size $|x_n|/2$ around ``centers"
$x_n$. This is stronger than exponential localization of $H$ on
$I$, which only means that \begin{equation} \exists \gamma>0
\mbox{ such that } \forall n,\; \exists C_n>0,\; |\varphi_n(x)|
\leq C_n e^{-\gamma |x|}, \label{eq:loc} \end{equation} but weaker
than what the authors of \cite{iv} called SULE (Semi-Uniformly
Localized Eigenfunctions).
We choose to present the proof of this theorem in the continuous
case, since the first part of the proof (Lemma \ref{lem1}) is a
little bit more technical in this situation. In order to prove
Theorem \ref{th1} we need control on the growth of the $|x_n|$ in
$n$, which is given by the following preliminary lemma:
\begin{lem} \label{lem1} Let H be as in the proposition and
$\delta>0$. Then one can order the $|x_n|$ in increasing order,
and there exists a constant $C=(4\,\max(B,1))^{-1}$ such that for
$n$ large enough (depending on $\delta$): $$ |x_n|\geq
Cn^{1/(\nu+\delta)}. $$ \end{lem}
\noindent {\bf Proof of Lemma \ref{lem1}:} Essentially we follow
the ideas of the proof of Theorem 7.1 of \cite{iv}. We recall that
the energies $E_n$ that we consider belong to $I$. Let $\delta>0$
be given and let $\delta'>0$ so that $\delta>\delta'(\nu-1)$. Let
$L>0$ be given, define $J=[0,L^{\delta'}]$, and write
$\chi_{2L}(x)$ for the characteristic function of the ball of
radius $2L$ and centered at 0.
Suppose that $|x_n|}
\hspace{.5in} \nonumber \\
& \leq & ||(1-\chi_{2L}(X))\varphi_n||_{L^2}
||\chi_{2L}(X)\varphi_n||_{L^2}
\nonumber \\
& \leq & C_\gamma e^{(\gamma'+\gamma)|x_n|}
\left|\left|(1-\chi_{2L}(X))
e^{-\gamma|x|}\right|\right|_{L^2} \nonumber \\
& \leq & C_1 e^{-(\gamma-\gamma')L}. \label{eq:cc1}
\end{eqnarray} Secondly, using $1-\chi_J(y)\leq y
L^{-\delta'},\,y\geq 0$, and $H\varphi_n=E_n\varphi_n$:
\begin{eqnarray} <\varphi_n,(1-\chi_J(H_0))\varphi_n>
& \leq & \left(<\varphi_n,H\varphi_n>+
||V||_{\infty}\right)L^{-\delta'}
\nonumber \\
& \leq & C(I,||V||_{\infty})L^{-\delta'}. \label{eq:cc2}
\end{eqnarray} So, using (\ref{eq:cc1}) and (\ref{eq:cc2}),
\begin{eqnarray} \lefteqn{tr(\chi_{2L}(X)\chi_J(H_0)\chi_{2L}(X))}
\hspace{.1in} \label{eq:tr}\\
& \geq & \sum_{n|\,E_n\in I,\,|x_n|\leq L}
<\varphi_n,\chi_{2L}(X)\chi_J(H_0)\chi_{2L}(X)\varphi_n>
\nonumber \\
& \geq & \sharp \{n|\,E_n\in I,\,|x_n|\leq L\}
\left(1-3C_1 e^{-(\gamma-\gamma')L}-
C(I,||V||_{\infty})L^{-\delta'}\right)
\nonumber \\
& \geq & \frac{1}{2} \sharp \{n|\,E_n\in I,\,|x_n|\leq L\}\mbox{
for }
L\geq
L_0=L_0(\gamma,\gamma',I,||V||_{\infty},\delta'). \label{eq:cc3}
\end{eqnarray}
The next step is then to bound the trace class norm of the
operator $Q=\chi_{2L}(X)\chi_J(H_0)\chi_{2L}(X)$. Let's study the
case $B\neq 0$. Since $\{u_1^2+u_2^2\leq L\} \subset \{u_1^2\leq
L,u_2^2\leq L\}$, and denoting by $\chi_{2L}^{(d)}(x)$ the
characteristic function of the $d$-dimensional ball of radius $2L$
centered at 0, remark that \begin{eqnarray} tr (Q) & \leq & tr
\left(\chi_{2L}(X)(\chi_J(H_1)\otimes\chi_J(H_2))\chi_{2L}(X)\right)
\nonumber \\
& = & tr \left((\chi_{2L}^{(2)}(X)
\chi_J(H_1)\chi_{2L}^{(2)}(X))
\otimes (\chi_{2L}^{(\nu-2)}(X)
\chi_J(H_2)\chi_{2L}^{(\nu-2)}(X)) \right)
\nonumber \\
& \equiv & tr \left(Q_1\otimes Q_2\right). \label{eq:q1q2}
\end{eqnarray} The operator $Q_2$ is the product of two
Hilbert-Schmidt operators with respective kernel
$\chi_{2L}^{(\nu-2)}(x)\left({\cal F}^{-1}g\right)(x-y)$ and
$\chi_J(x)\left({\cal F}^{-1}\chi_{2L}^{(\nu-2)}\right)(x-y)$
\cite{rs}, where ${\cal F}^{-1}g$ denotes the inverse Fourier
transform of $g(x)=\chi_J\circ s(x)$, with
$s(x_1,...,x_{\nu-2})=\sum_1^{\nu-2} x_i^2$. So, denoting by
$||C||_1$ the trace class norm of an operator $C$ defined on
$\Hh$, one has: \begin{eqnarray} ||Q_2||_1& \leq &
||\chi_{2L}(X)\chi_J(H_2)||_{HS}||\chi_J(H_2)\chi_{2L}(X)||_{HS}
\nonumber \\
& \leq & ||(\chi_{2L}^{(\nu-2)}(x)||^2_{L^2} ||\chi_J\circ
s(x)||^2_{L^2}
\leq (4L)^{\nu-2}L^{\delta'(\nu-2)}. \label{eq:q2}
\end{eqnarray} We turn now to the magnetic part $Q_1$. It is well
known that the spectrum of our $H_1$ consists of eigenvalues
$(2n+1)B$, $n=0,1,2,...$. The corresponding projectors are
operators with kernel \cite{kunz}, \begin{equation}
P_n(x,x')=e^{i\frac{B}{2} x\wedge x'}
p_n\left(\sqrt{B}(x-x')\right),\label{eq:Pn} \end{equation} where
$$ p_n((x_1,x_2))=\frac{\textstyle 1}{\textstyle 2\pi}
\int_{\R} e^{ikx_1} h_n(k+x_2/2)h_n(k-x_2/2) dk,
$$ and $h_n(k)$ are the normalised Hermite functions. Note that we
gave the expression of $P_n$ in the symmetric gauge, and not in
the Landau gauge as in \cite{kunz}. Write now
$P_J=\sum_{n|\,E_n\in J} P_n$, the projector corresponding to the
eigenvalues belonging to $J$. Then
$Q_1=\left(\chi_{2L}^{(2)}(X)P_J\right)\left(P_J\chi_{2L}^{(2)}(X)\right)$,
which are two Hilbert-Schmidt operators with the same norm
(because of (\ref{eq:Pn})). And one has, for each $n$,
\begin{eqnarray*} \lefteqn{||\chi_{2L}^{(2)}(X)P_n||^2_{HS}}
\hspace{.2in}\\
& = & \int_x \chi_{2L}^{(2)}(x)\int_{x'}
\left|p_n\left(\sqrt{B}(x-x')\right)\right|^2 dx
dx' \\
& \leq & (4BL)^2 \int_{x_2}
\left|\left|{\cal
F}\left(h_n(.+\frac{x_2}{2})h_n(.-\frac{x_2}{2})\right) (x_1)
\right|\right|^2_{L^2} dx_2 \\
& = & (4BL)^2 \int_{x_2,k} \left|h_n(k+\frac{x_2}{2})\right|^2
\left|h_n(k-\frac{x_2}{2})\right|^2
dx_2\, dk \\
& = & (4BL)^2 ||h_n||_{L_2}^4 \end{eqnarray*} So, since
$||h_n||_{L^2}=1$, and using $\sharp \{n\geq 0, (2n+1)B\leq
L^{\delta'}\} \leq L^{\delta'}/2B$ if $L^{\delta'}\geq B$, one has
$$ ||\chi_{2L}^{(2)}(X)P_J||^2_{HS} \leq 8B L^{2+\delta'}. $$ And
then, using (\ref{eq:q1q2}) and (\ref{eq:q2}), \begin{equation} tr
(Q) \leq C L^{\nu+\delta}, \label{eq:trq} \end{equation} taking
$\delta>\delta'(\nu-1)$, and $C=4\,\max(B,1)$. Note here that if
$B=0$ then the free Hamiltonian $H_0$ has the form $\sum_1^\nu
\partial_i^2$. Hence the analysis made previously for $Q_2$ is
valid for such a $Q$ and (\ref{eq:trq}) holds.
In order to finish the argument, note that, together with
(\ref{eq:cc3}) and (\ref{eq:q1q2}), (\ref{eq:trq}) tells us that,
for $L\geq L_0$, $N(L) \equiv\sharp \{n|,\;E_n\in I, |x_n|\leq
L\}$ is finite. Order then the eigenfunctions in such a way that
$|x_n|$ increases. So, $N(|x_n|)=n$, and if $n\geq N_0\equiv
N(L_0)$, one has, with $L=|x_n|$: \begin{equation}
|x_n|\geq C n^{1/(\nu+\delta)}
\mbox{, for }n>N_0. \label{eq:xn}
\end{equation}
The main difference between the continuous and discrete cases
comes from this Lemma, in the sense that on $L^2(\R^\nu)$ one has
to control the behaviour of the eigenfunctions in the momentum
variables: that was achieved through the use of $\chi_J(H_0)$. In
the discrete case, (\ref{eq:trq}) is replaced by the trivial
equality $tr(\chi_{2L})=(4L+1)^\nu$. This also explains why no
specific form for $H$ is needed on $l^2(\Z^\nu)$. It is easy,
then, to rewrite the proof of Lemma \ref{lem1} in this case (see
also \cite{iv}). \fin
\noindent {\bf Proof of Theorem \ref{th1}:} Let $\psi\in\Hh$ such
that, for some constant $C(\psi)>0$ and $\theta>2\gamma'$,
$|\psi(x)|0$, $t>0$. Let
$0<\epsilon< \inf(\gamma/2-\gamma',\theta/2-\gamma')$. Then
\begin{eqnarray*} \lefteqn{||X^{q/2}P_I(H)e^{-iHt}\psi||^2} \\
\leq & \bd\sum_{n|\,E_n\in I}\ed & |\!<\varphi_n,\psi>\!|^2
||X^{q/2}\varphi_n||^2 \\
\leq & \bd\sum_{n|\,E_n\in I}\ed & \left(C(\psi)C_\gamma\right)^2
e^{-4\epsilon|x_n|} \\
& & \hskip-2mm \int dx\int dy \left(|y|^q
e^{-2\theta|x|}e^{-2(\gamma-\gamma'-\epsilon)|x-y|}
e^{2(\gamma'+\epsilon)(2|x_n|-|x-x_n|-|y-x_n|)} \right)\\
\leq &&\hskip-12mm C_3 \left(\sum_{n|\,E_n\in I}
e^{-4\epsilon|x_n|}\right)
\int \left(\int |y|^q
e^{-2(\gamma-2(\gamma'-\epsilon))|x-y|}dy\right)
e^{4(\gamma'+\epsilon)|x|-2\theta|x|}dx.
\end{eqnarray*} This last line is uniformly bounded by a constant
$C_\psi(\gamma,\gamma',I,\theta,\epsilon)$, according to Lemma
\ref{lem1} and because of the choice of $\epsilon$. \fin
\vskip2mm
%%%%%%%%%%%%%% DISCRETE CASE %%%%%%%%%%%%%
\section{The discrete Anderson model}
\label{discrete} \setcounter{equation}{0} We consider the
self-adjoint operator $H_\omega$ $$ H_\omega = -\Delta +
V_\omega, $$ where $\Delta$ is the discrete Laplacian on
$\ell^2(\Z^\nu)$ and $V_\omega$ ($\omega\in\Omega$) is a random
potential, the $(V_\omega(x))_{x\in\Z^\nu}$ being i.i.d. random
variables. Their common probability measure $\mu$ is assumed to be
non degenerate, {\em i.e.} not concentrated on a single point. The
conditions that we impose on $\mu$ are: \begin{equation} \left.
\begin{array}{ll} &\mbox{if } \nu=1: \exists\;\eta>0\; ,\;
\int_{\R} \mid\! v\!\mid^\eta d\mu(v)<
\infty ; \\ &\mbox{if } \nu\geq 2: \mu\mbox{ is
}\alpha\mbox{-H\"older continuous}. \end{array} \right.
\label{eq:mu} \end{equation} \noindent Let us recall how the
disorder $\delta(\mu)$ of a $\alpha$-H\"older continuous measure
$\mu$ is defined: \[ \delta(\mu)^{-1} = \inf_{\tau>0}
\sup_{|b-a|<\tau} |b-a|^{-\alpha}\mu([a,b]). \]
\begin{th} \label{th2} Let $H_\omega$ be the Anderson Hamiltonian,
and $\mu$ the common probability measure of the $V_\omega(x)$,
$x\in\Z^\nu$, not concentrated in a single point and satisfying
condition~(\ref{eq:mu}). Let $I$ be a compact interval and
$\epsilon>0$. \begin{itemize} \item If $\nu=1$, define $\Gamma
\equiv \gamma(I) \equiv inf\{\gamma(E),\; E\in I\}$, where
$\gamma(E)$ is the Lyapu\-nov exponent at energy $E$. Suppose
$\Gamma=\gamma(I)>0$. \item If $\nu>1$, pick $\Gamma>0$. Suppose
the disorder $\delta (\mu)$ is taken sufficiently high.
\end{itemize} \vskip-1mm \noindent Then, $\P$ almost surely,
$H_\omega$ has pure point spectrum on $I$, and there exist centers
$\x$ associated to the eigenfunctions $\phin$ with energy $\En\in
I$ such that: $\forall\;\gamma_0\in]0,\Gamma[$, there exists a
constant $C(\omega,\epsilon,\gamma_0)$ such that \begin{equation}
\forall x\in\Z^\nu,\;\;|\phin(x)| \leq C(\omega,\epsilon,\gamma_0)
e^{\gamma_0|\x|^\epsilon} e^{-\gamma_0|x-\x|}.\label{eq:ssule}
\end{equation} \end{th}
Evidently, one can also write a ``low energy" version of this
theorem. As an immediate consequence of Theorem \ref{th1} and
Theorem \ref{th2} we have: \begin{cor} \label{corth2} Let
$H_\omega$ be as in Theorem~\ref{th2}, and $P_I(H_\omega)$ the
spectral projection on the compact interval $I$. Then for $q>0$
and $\psi\in \ell^2(\Z^\nu)$ decaying exponentially with rate
$\theta>0$, $||\,|X|^q P_I(H_\omega)e^{-iH_\omega t}\psi||^2$ is
bounded uniformly in $t$ almost surely. \end{cor}
Comparing (\ref{eq:ssule}) to (\ref{eq:loc}) and to
(\ref{eq:wsule}), one notices that now the
size of the boxes in which the eigenfunctions ``live" can grow at
most as $|\x|^\epsilon$. One expects this can be improved to a
polynomial bound (but no more - see \cite{iv}). Supposing $\mu$
has a bounded density with compact support, the polynomial bound
follows from the proof of Theorem 7.6 in \cite{iv}.
The proof of Theorem \ref{th2} is based on the ideas of
\cite{124}, and in particular on the proof of Theorem 2.3 in
\cite{124}. The strategy is the following: since the hypotheses of
Theorem \ref{th2} imply exponential localization, we know that
there exist ``centers" $\x$ where the eigenfunction $\phin$ is
maximal, and one can then exploit the result of the multi-scale
analysis a second time to improve the control of the decay of the
eigenfunctions. As already pointed out, this proof has the
advantage of yielding the result for singular potentials and in
particular for Bernouilli potentials in dimension $1$. In
addition, the proof extends to continuous random Schr\"odinger
operators, as shown in sections \ref{cont} and \ref{appl}.
To make this paper self-contained, we start by recalling the
elements from \cite{124} that we need. First of all,
$\Lambda_L(x)$ denotes the cube of side $L/2$ centered in $x$ and
$\partial\Lambda_L(x)$ its boundary. $H_{\Lambda_L(x),\omega}$ is
the restriction of the operator $H_\omega$ to the cube
$\Lambda_L(x)$ with Dirichlet boundary conditions, and
$G_{\Lambda_L(x)}(E,.,.)$ is its resolvent. Given $L_0>1$,
$\alpha\in]1,2[$, we define $L_k$ ($k\in\N$) recursively via
$L_{k+1}=L_k^\alpha$. Given in addition an integer $b\geq 2$, we
define $$
A_{k+1}(x_o)=\Lambda_{2bL_{k+1}}(x_o)\backslash\Lambda_{2L_k}(x_o).
$$ Note that we do not indicate the dependence of $L_k$ and $A_k$
on $L_0,\, \alpha$ and $b$ since these quantities will at any rate
be fixed later on. We further need the following definition:
%%%%%%%%%%%% regular box in discrete case %%%%%%%%%
\begin{de} \label{defreg} Let $\gamma>0$ and an
energy $E\in\R$ be given. A cube $\Lambda_L(x)$ is said to be
$(\gamma,E)$-regular if $E\not\in\sigma(H_{\Lambda_L(x)})$ and if
for all $y\in\partial\Lambda_L(x),$ $$
|G_{\Lambda_L(x)}(E,x,y)|\leq e^{-\gamma L/2}. $$ Otherwise
$\Lambda_L(x)$ will be called $(\gamma,E)$-singular. \end{de} Note
that this definition is $\omega$-dependent, but we follow the
usual practice by not indicating this. For
$x_o\in\Z^\nu,\;E_k(x_o)$ is defined to be the following set: \[
\{\omega|\,\exists\,E\in I,\;\exists\; x\in A_{k+1}(x_o),\;
\Lambda_{L_k}(x_o)\: and\: \Lambda_{L_k}(x)\: are\:
(\gamma,E)\!-\!singular\}. \] Finally, we recall a well known
identity. Let $x\in\Z^\nu$, $E\not\in\sigma(H_{\Lambda_L(x)})$,
and $\varphi\in \ell^2(\Z^\nu)$ so that $H\varphi=E\varphi$ be
given, then: \b
\varphi(x)=\sum_{(y,y')\in\partial\Lambda_L(x)}G_{\Lambda_L(x)}(E,x,y)\varphi(y').
\label{eq:recouv} \e Here (with some abuse of notation)
$(y,y')\in\partial\Lambda_{L}$ means $y$ and $y'$ are nearest
neighbours with $y\in\Lambda_L(x)$ and $y'\not\in\Lambda_L(x)$.
$\partial\Lambda_{L_k}^{+}(x)$ will denote the points $y'$ just
outside $\Lambda_{L_k}(x)$.
\vskip1mm In order to prove Theorem \ref{th2}, we start with the
following three lemmas:
%%%%%%%%%%%%%%%%%%%%%% lemma 1 %%%%%%%%%%%%%
\begin{lem}
\label{lem124} Let $p>\nu$ and $\alpha\in]1,2-2\nu/(p+2\nu)[$ and
$b>1$ be given. Assume the hypotheses of Theorem \ref{th2} are
satisfied. Then for any $\gamma\in]0,\Gamma[$ there exists
$L_0=L_0(p,\nu,\gamma,b,\delta(\mu))>1$ such that: $$
\forall\;k\in\N,\; \forall\;x\in\Z^\nu,\;\;\P(E_k(x))
\leq\frac{(2bL_{k+1}+1)^\nu}{(L_k)^p}. $$ \end{lem}
%%%%%%%%%%%%%%%%%%%%%% lemma 2 %%%%%%%%%%%%%
\begin{lem}
\label{lem-max} Let $\gamma>0$ be fixed. There exists a constant
$L_*(\nu,\gamma)$ so that, if $H$ is a Schr\"odinger operator and
$\varphi\in \ell^2(\Z^\nu)$ an eigenvector of $H$ with eigenvalue
$E$, and if $x_*\!\in\!\Z^\nu$ satisfies $|\varphi(x_*)|=
sup\{|\varphi(x)|,\;x\!\in\!\Z^\nu\}$, then $\Lambda_L(x_*)$ is
$(\gamma,E)$-singular, for all $L\geq L_*(\nu,\gamma)$. \end{lem}
%%%%%%%%%%%%%%%%%%%%%% lemma 3 %%%%%%%%%%%%%
\begin{lem}
\label{lem3} If $\nu,\; s_\nu$, and $\gamma$ are some positive
constants, then $\forall\; \eta\in]0,1[$ there exists
$L(\eta,\;\gamma,\;\nu)$ such that, if $L\geq
L(\eta,\;\gamma,\;\nu)$: \begin{equation} \forall\;x,\; x_o \in
\Z^\nu, \; \left( s_\nu L^{\nu-1} e^{-\gamma L/2}
\right)^{\frac{|x-x_o|}{L/2+1}} \leq
e^{-\gamma\eta|x-x_o|}.
\label{eq:k3} \end{equation}
\end{lem}
The first lemma follows immediately from the Appendix and from
Theorem 2.2 of Von Dreifus and Klein \cite{124} and constitutes
the core of the proof of exponential localization in \cite{124}.
We will prove an analog of it for continuous random Schr\"odinger
operators in the Appendix.
\vskip1mm The second lemma says roughly that if $\varphi$ is an
eigenvector of $H$ with eigenvalue $E$, then $E$ must be close to
the spectrum of $H_{\Lambda_L(x)}$ provided $L$ is big enough and
$\Lambda_L(x)$ is centered on a maximum of $|\phin|$. It is quite
simple, but central for what follows.
\vskip1mm Obviously the third lemma doesn't need a proof. We have
stated it separately in order to make clear later on that
$L(\eta,\gamma,\nu)$ only depends on the model parameters and not
on the particular eigenfunction we consider. Note that
$L(\eta,\gamma,\nu)$ behaves like $(1/\gamma)$ at a positive power.
\vskip2mm \noindent {\bf Proof of Lemma \ref{lem-max}:} Let
$\varphi$ be as in the lemma: $\varphi\in \ell^2(\Z^\nu)$, so
$x_*$ exists. Suppose that $\Lambda_L(x_*)$ is
$(\gamma,E)-regular$, and apply the identity (\ref{eq:recouv}) at
the point $x_*$. Then for some $y'\in\partial\Lambda^+_L(x)$:
\begin{eqnarray} |\phin(x_*)| & \leq & s_\nu L^{\nu-1}e^{-\gamma
L/2} |\phin(y')| \nonumber \\
& \leq & s_\nu L^{\nu-1}e^{-\gamma L/2} |\phin(x_*)|,
\label{eq:maj*} \end{eqnarray} where $s_\nu$ is a constant
depending only on the dimension. Now let $L_*(\gamma,\;\nu)$ be a
positive real such that $s_\nu L^{\nu-1}e^{-\gamma L/2}<1$ for
$L\geq L_*$. Then, for such $L$, (\ref{eq:maj*}) is impossible,
and $\Lambda_{L}(x_*)$ cannot be $(\gamma,E)$-regular any more,
that is: $\Lambda_{L}(x_*)$ is $(\gamma,E)$-singular for $L\geq
L_*(\gamma,\nu)$. \fin
\vskip2mm \noindent {\bf Proof of Theorem \ref{th2}:} Under the
hypotheses of the theorem, $H_\omega$ has $\P-a.s.$ exponential
localization on $I$ (see \cite{108} and \cite{124}). This means
that there exists $\Omega_0\subset\Omega$, $\mu(\Omega_0)=1$ so
that for all $\omega\in\Omega_0$, $\sigma_c(H_\omega)\cap
I=\emptyset$ and for all eigenvalue $\En\in I$, the corresponding
eigenfunction $\phin$ is $\ell^2$ and satisfies (\ref{eq:loc}).
The aim is therefore to control the constant $C_{n,\omega}$ of
(\ref{eq:loc}) and more precisely to show that $\x$ can be chosen
so that this $C_{n,\omega}$ grows slower than an exponential in
$|\x|^\epsilon$. In order to prove Theorem \ref{th2}, we wish to
use equation (\ref{eq:recouv}) repeatedly, and on a scale $L_k$
for suitably large $k$, to estimate the value of $\phin(x)$ when
$x$ belongs to $A_{k+1}(\x)$ for suitably chosen $\x$. To do this,
one has to work ``outward" from $x\in A_{k+1}(\x)$ to the boundary
of $A_{k+1}(\x)$, making sure that the boxes of size $L_k$ to
which one applies (\ref{eq:recouv}) are regular.
After these preliminaries, let's start the proof properly
speaking, which will consist of three steps. Firstly, let
$I,\;\Gamma>0$ and $\gamma_0$ be as in the theorem. Pick
$\epsilon>0$ and $\gamma\in]\gamma_0,\Gamma[$.
%%%%%%%%%%%%%%%%%%%%%%% the Borel-Cantelli step %%%%%
\vskip2mm
\noindent \underline{First step}: Let $p>\nu $,
$\alpha\in]1,2-2\nu/(p+2\nu)[$ and $b>1$ be given. With the $L_k$,
$k\geq 0$, defined in Lemma \ref{lem124}, consider $$ F_k= \bd
\bigcup_{|x_o|\leq (L_{k+1})^{1/\epsilon}} \ed E_k(x_o). $$
\noindent Lemma \ref{lem124} then implies that for some constant
$C(\epsilon,\nu,b)$ $$ \P(F_k) \leq C(\epsilon,\nu,b)
L_k^{-p+\nu\alpha(1+1/\epsilon)}. $$ Hence, since $p$ can be
chosen larger than $2\nu(1+1/\epsilon)$, one has
$\sum_{k=0}^\infty \P(F_k)<\infty$. The Borel-Cantelli lemma then
implies that: \[ \P\left( \lim_{m\rightarrow\infty} \bigcup_{k\geq
m} F_k\right) = 0, \] so that the set $$ \Omega_1 =
\{\omega\in\Omega | \exists\;
\tilde{k}_1=\tilde{k}_1(\omega,\epsilon,p,\gamma)\; such\; that
\;\forall\; k\geq \tilde{k}_1,\; \omega\not\in F_k\} $$ has full
measure. This ends the probabilistic part of the proof. Note that
the choice of $\epsilon$ puts a lower bound on $p$. This in turn
forces the disorder to be high {\em via} Lemma~\ref{lem124}.
%%%%%%%%%%%%%%%%%%%%%%%% second step %%%%%%%%%%
\vskip2mm
\noindent \underline{Second step}: Now pick an $\omega$ in
$(\Omega_0\cap\Omega_1)$, which will be kept fixed throughout the
rest of the proof. Let $E_{n,\omega}\in I$, $\phin$ its
eigenfunction, and let $\x$ be a point where $|\phin(x)|$ is
maximal. Note that such a point exists since $\omega\in\Omega_0$
and therefore $\phin\in \ell^2(\Z^\nu)$. Let \begin{equation} k_1=
k_1(\omega, \epsilon,p,\gamma,\x) = max(\tilde{k}_1,
k_0(\epsilon,\x)), \label{eq:k_1} \end{equation}
where for all $y\in\Z^\nu$, the integer $k_0(\epsilon,y)$ is
defined as follows: \begin{equation} k_0(\epsilon,y) = min\{k\geq
0\; such\; that \;|y|^\epsilon\gamma_0$ and choose the integer $b$ introduced
at the beginning of the proof so that $b>(1+\rho)/(1-\rho)$. Then
set $$ \tilde{A}_{k+1}(\x) =
\Lambda_{[2bL_{k+1}/(1+\rho)]}(\x)\backslash
\Lambda_{2L_k/(1-\rho)}(\x) \subset A_{k+1}(\x). $$ Note that if
$x\!\in\!\tilde{A}_{k+1}(\x)$ then $d(x,\partial A_{k+1}(\x))\geq
\rho|x-\x|$. Hence, repeating (\ref{eq:recouv}) $\rho|x-\x|$
times, one has that for all $k\geq k_2$ and for all
$x\in\tilde{A}_{k+1}(\x)$, $$ |\phin(x)|\leq \left( s_\nu
L_k^{\nu-1} e^{-\gamma L_k/2}
\right)^{\frac{\rho|x-x_n|}{L_k/2+1}}, $$ \noindent or, applying
Lemma \ref{lem3}, and choosing $\eta\in]0,1[$ such that
$\gamma_0=\rho\eta\gamma$ we conclude that there exists an integer
$k_3= max(\tilde{k}_3,k_0(\epsilon,\x))$ where $\tilde{k}_3$
depends again not on $n$, such that: \begin{equation} \forall\,
k\geq k_3,\, and\,\forall\,x\in\tilde{A}_{k+1}(\x),
\;\;|\phin(x)|\leq e^{-\gamma_0 |x-x_n|}. \label{eq:maj1}
\end{equation}
But now note that for all $x\in\Z^\nu$, and provided
$|x-\x|>L_0/(1-\rho)$ there exists a $k$ so that
$x\in\tilde{A}_{k+1}(\x)$. This means that there exists an integer
$\kb=max(\tilde{k}_4,k_0(\epsilon,\x))$, $\tilde{k}_4$ depending
once again not on $n$, such that (\ref{eq:maj1}) holds for all
$x\in\Z^\nu$ satisfying $|x-\x|>L_{\kb}$. Hence, using that
$|\phin(x)|\leq 1$ for all $x\in\Z^\nu$: \begin{equation} \forall
x\in\Z^\nu,\; |\phin(x)|\leq C(\omega,\epsilon,\gamma_0)
e^{\gamma_0 L_{\kb}} e^{-\gamma_0|x-\x|}. \label{eq:maj2}
\end{equation}
So far, we have only proved that the eigenfunctions decay
exponentially, but we are now in a position to control the
$n$-dependence of the constant $e^{\gamma_0 L_{\kb}}$ as follows.
Note that the only $n$-dependence of $\kb$ comes from
$k_0(\epsilon,\x)$. Suppose $sup \{|\x|, \En\in I\}<\infty$, then
$\kb$ can be chosen $n$-independently, so that we actually obtain
a uniform localization (called ULE in \cite{iv}), and {\em a
fortiori} condition~(\ref{eq:wsule}) of Theorem~\ref{th1}. But
Lemma \ref{lem1} contradicts this first possibility. So, in fact,
$sup \{|\x|, \En\in I\}=\infty$, and, for $n$ sufficiently large,
one has: $$ \kb = k_0(\epsilon,\x) $$ {\em i.e.}, with
(\ref{eq:k_0}), $$ L_{\kb}\leq |\x|^\epsilon. $$ Inserting this in
(\ref{eq:maj2}) yields the announced result. \fin
%%%%%%%%%%%%%%%%% The continuous case %%%%%%%%%%%%%%
\section{The continuous case} \label{cont} \setcounter{equation}{0}
In this section, our goal is to obtain the analog of Theorem
\ref{th2} and Corollary \ref{corth2} for continuous random
Schr\"odinger operators. The result is stated in Theorem
\ref{thcont} below. In section \ref{appl}, we will present some
models where the hypotheses of this theorem are satisfied.
We consider random Schr\"odinger operators on $L^2(\R^\nu)$ of the
following type ($\nu\geq 1$): \begin{equation} H_\omega = H_0 +
\bd\sum_{i\in\Z^\nu}\ed \lambda_i(\omega) u(x-i). \label{eq:H}
\end{equation} \noindent Here
i) $H_0=(i\nabla-A)^2+V_{per}$, where $A$ is a vector potential of
a constant magnetic field \overrightarrow{B}=rot(A), and $V_{per}$
a periodic potential.
ii) The variables $\lambda_i(\omega)$, $i\in\Z^\nu$ are
independent and identically distributed, with common distribution
$\mu$.
iii) The function $u(x)$ belongs to $C^2_0(\R^\nu)$, with $supp\;
u\subset[-R,R]^\nu$.
\vskip1mm To state the hypotheses, we need to recall some
notations and simple facts. We introduce $|x|= \max \{|x_i|,\;
i=1,...,\nu\}$, and denote by $\Lambda_L(x)$ the cube $$
\Lambda_L(x) = \{y\in\R^\nu |\, |y-x|0$ being fixed (independently of $L$),
$\tilde{\Lambda}_L(x)$ is the subset $$ \tilde{\Lambda}_L(x) =
\{y\in\Lambda_L(x)\, such\, that\, L/2-\delta<|x-y|0$ and an energy $E\in\R$ be given.
A cube $\Lambda_L(x)$ is said to be $(\gamma,E)$-regular if
$E\not\in\sigma(H_{\Lambda_L(x)})$ and if: $$
||\chi_{4\delta,x}R_{\Lambda_L(x)}(E)W_{L,x}||\leq e^{-\gamma
L/2}. $$ Otherwise $\Lambda_L(x)$ will be called
$(\gamma,E)$-singular. \end{de}
We now state the result. Given a compact interval $I$, and reals
$\gamma_0>0,\;p>\nu$, $L_0,\tilde{L}>1$, we introduce
\noindent {\bf Hypothesis [H1]($\gamma_0,I,p,L_0$)}: $$
\P(\forall\,E\in I,\, \Lambda_{L_0}\, is\,
(\gamma_0,E)-regular)>1-\frac{1}{L_0^p}. $$
\noindent {\bf Hypothesis [H2]($I,\tilde{L}$)} (Wegner): {\em
there exists $C_W$ so that for all $\tilde{I}\subset
I_1\equiv\{E\,|\, d(E,I)<1\}$ and $L>\tilde{L}$, $$
\E(Tr(E_{\Lambda_L(0)}(\tilde{I})))\tilde{L}$, $0<\eta<1$
and $E\in I$, \begin{equation}
\P(d(E,\sigma(H_{\Lambda_L(\omega)}))<\eta) < C_W|\Lambda_L|\eta.
\label{eq:wegner} \end{equation} This is the so-called ``Wegner
Estimate''. We will need both [H2] and (\ref{eq:wegner}) for the
proof of Proposition~\ref{propcont}.
%%%%%%%%%%%%%%%% Theorem thcont %%%%%%%%%%%%%%%
\begin{th} \label{thcont} Let
$\epsilon>0$. Suppose that for some interval $I$ and reals
$\gamma_0>0$, $p>2\nu(1+1/\epsilon)$, the hypotheses
[H1]($\gamma_0,I,p,L_0$) and [H2]($I,\tilde{L}$) hold for
$L_0>\tilde{L}$ large enough. Then with probability one there
exist points $\x$, associated to eigenfunctions $\phin$ with
energy $\En\in I$, so that: $\forall\,\gamma\in]0,\gamma_0[$ and
for some constant
$C_\omega=C(\omega,\epsilon,\gamma,\gamma_0,I,L_0)$, one has, for
all $x\in\R^\nu$, \begin{equation} |\phin(x)| \leq C_\omega
e^{\gamma|\x|^\epsilon} e^{-\gamma|x-\x|}. \label{eq:ssulecont}
\end{equation} Moreover, if $q>0$ and $\psi\in
L^2(\R^\nu)$ decays exponentially with mass $\theta>0$, then, with
probability 1, there exists a constant $C_{\psi,\omega}$ such
that, $$ ||\,|X|^q P_I(H(\omega))e^{-iH(\omega)t}\psi||^2\leq
C_{\psi,\omega}. $$ \end{th}
Analysing the proofs of the previous two sections, one sees that
the only missing ingredient for the proof of Theorem~\ref{thcont}
is an analog of Lemma~\ref{lem124}, stated as
Proposition~\ref{propcont} below. Indeed, the arguments of
sections \ref{dynloc} and \ref{discrete} are readily transcribed
to the continuous case, provided one makes the following
adaptation. First, one replaces equation (\ref{eq:recouv}) by the
following equality: if $\varphi\in L^2(R^\nu)$ satisfies
$H\varphi=E\varphi$ for some $E$, then for
$\Lambda_L(x)\subset\R^\nu$ so that
$E\not\in\sigma(H_{\Lambda_L(x)})$: $$
\chi_{4\delta,x}\varphi=\chi_{4\delta,x}R_{\Lambda_L(x)}(E)W_{L,x}\varphi.
$$ Secondly, one defines $x_*(\varphi)$, the analog of $x_*$ in
Lemma~\ref{lem-max}, as follows. If $\varphi$ belongs to
$L^2(\R^\nu)$ and $H\varphi=E\varphi$, then \[ \sup_{y\in
4\delta\Z^\nu} \left\{ \int_{\Lambda_{4\delta}(y)} |\varphi(x)|^2
dx\right\} = \int_{\Lambda_{4\delta}(x_*(\varphi))} |\varphi(x)|^2
dx. \]
So, writing $\x\equiv x_*(\phin)$, and redefining the annular
$A_{k+1}(\x)$ as
$\Lambda_{2bL_{k+1}}(x)\backslash\Lambda_{2L_k+2R}(\x)$, one
obtains the bound written in (\ref{eq:ssulecont}), but for
$||\chi_{4\delta,x}\phin||$ rather than for $|\phin(x)|$. Then to
get the pointwise estimate (\ref{eq:ssulecont}) apply Theorem 2.4
of \cite{cfks}, or decompose $||e^{\gamma|\x|^\epsilon}
e^{-\gamma|x-\x|}$ $\phin||^2_{L^2}$ on boxes of size $4\delta$
and centered on $4\delta k$, $k\in\Z^\nu$, and apply Theorem IX.26
of \cite{rs}.
It therefore remains to state and prove the analog of
Lemma~\ref{lem124}. Following \cite{124} let's denote by
$R(\gamma,L,x,y)$ the set $$
R(\gamma,L,x,y)\equiv\{\omega\in\Omega |\,\forall\, E\in I,\,
\Lambda_L(x)\, or\, \Lambda_L(y)\, is \,(\gamma,E)-regular\}. $$
As in the discrete case, for all $\alpha\in]1,2[$ and $L_0>1$ we
define the sequence $(L_k)_{k\in\N}$ by $L_{k+1}=L_k^\alpha$.
\begin{pro} \label{propcont} For any $\gamma\in]0,\gamma_0[$,
$p>\nu$ and $\alpha\in]1,2-2\nu/(p+2\nu)[$ there exist
$L_*=L_*(\gamma,I,\alpha)$ such that if [H1]($\gamma_0,I,p,L_0$)
and [H2](I,$\tilde{L}$) hold for $L_0>L_*$, $L_0>\tilde{L}$, then
for all $k\geq 0$: $$ |x-y|>L_k+2R\; \Longrightarrow \;
\P(R(\gamma,L_k,x,y))>1-\frac{1}{L_k^{2p}}. $$ \end{pro}
The proof of Proposition~\ref{propcont} follows upon adapting the
arguments of the proof of Theorem 2.2 in \cite{124} to the
continuum. Various authors \cite{combhis,klopp1} have written up
versions of the multi-scale analysis for continuous Schr\"odinger
operators, but, to our knowledge, the version we need is not
available in the literature. Nethertheless, nobody seems to doubt
that any such argument can be carried over from the discrete to
the continuous case. Since multi-scale arguments are in addition
to this painful, we have chosen to put the proof of
Proposition~\ref{propcont} in the appendix, while making an effort
to give a clear, complete and relatively simple argument.
\vskip5mm
%%%%%%%%%% Application / Hypothesis Validity %%%%%%%%%%%%%
\section{Applications} \label{appl}
\setcounter{equation}{0}
We briefly indicate two applications of Theorem \ref{thcont} to an
Anderson model on $\R^\nu$ \cite{combhis} and to random
Schr\"odinger operators with a magnetic field.
\vskip 3mm \noindent {\bf The Anderson tight-binding model}
\vskip2mm Here the free Hamiltonian $H_0$ of equation (\ref{eq:H})
is $-\Delta$. We suppose, following \cite{combhis}, that $$
u(x)>\chi_{3/2}(x), $$ where $\chi_{3/2}$ is the characteristic
function of the cube $\Lambda_{3/2}(0)$. Putting together
Proposition 4.5 and Theorem 5.1 of \cite{combhis}, one has
immediately from Theorem~\ref{thcont}: \begin{th} Suppose that
$\mu$ has a $L^\infty$ density $g(\lambda)$ with support
$[0,\lambda_{max}]$ and disorder $\delta_0=||g||_\infty^{-1}>0$
then, for energy $E_A>0$ fixed and disorder $\delta_0$ high
enough, or for disorder $\delta_0>0$ fixed and energy $E_A$ low
enough, the conclusions of Theorem \ref{thcont} hold on $[0,E_A]$.
\end{th}
\noindent {\bf The Landau Hamiltonian} \vskip2mm
Here the Hamiltonian has the general form described in equation
(\ref{eq:H}), {\em i.e.} $A\neq 0$ and $V_{per}=0$. Although the
result is still valid in arbitrary dimension under further
assumptions (see \cite{babar}) we prefer to state the application
in the well-known two dimensional version \cite{bacombhis1}
\cite{combhisL} \cite{wang}. In that case, the vector potential
$A$ is given by $$ A=\frac{\textstyle B}{\textstyle 2} (x_2,-x_1),
$$ with $B>0$. Recall that the spectrum of the free Landau
Hamiltonian $H_0$ consists of a sequence of eigenvalues $$
E_n(B)=(2n+1)B,\;n\in\N. $$ We suppose that $u>0$, $supp\:u\subset
B(0,1/\sqrt{2})$, and that there exist $C_0$ and $r_0>0$ such that
$u_{|B(0,r_0)}>C_0$. We suppose that the common measure $\mu$ (of
the $\lambda_i(\omega)$) has a bounded density function $g\in
C^2_0(\R)$, $g$ being even and positive for almost every
$\lambda\in\;supp\,g$.
Under those assumptions it is well-known that
$M_0=\sup\{|V_\omega(x)|,\,x,\omega\}<+\infty$. Let's define the
following bands: \[ \begin{array}{c}
I_0(B)=[-M_0,B-\epsilon_0(B)], \\
I_{n+1}(B)=[E_n(B)+\epsilon_n(B)),E_{n+1}(B)-\epsilon_n(B)]
\end{array} \] with some $\epsilon_n(B)>0$. It follows from
\cite{combhisL} and Theorem~\ref{thcont}:
\begin{th} Let $V$ be as described above. Then for $B$ high
enough, there exist some $\epsilon_n(B)=O(B^{-1})$ such that the
conclusions of Theorem~\ref{thcont} hold on each interval
$I_n(B)$, $n\in\N$. \end{th}
A similar result holds for the model treated by Wang \cite{wang},
where $u$ can be negative and its support is included in $B(0,r)$,
$00$. Two boxes $\Lambda_1$ and
$\Lambda_2$ will be called $r$ non-overlap\-ping iff
d($\Lambda_1,\Lambda_2)>2r$. \end{de} Note that, if $\Lambda_1$
and $\Lambda_2$ are $R\; non-overlapping$, then, since
$\mbox{supp\ }u\subset [-R,R]^\nu$, two events depending
respectively on the $\lambda_i$ with $i$ in $\Lambda_1$ and in
$\Lambda_2$ are necessarily independent.
%%%%%%%%%%%%%% definition of a non resonant box %%%%
\begin{de}
\label{defnr} Let $\beta\in]0,1[$ be given. A box $\Lambda_L$ will
be called non resonant at energy $E$ (we'll write $E-NR$) if $$
||R_{\Lambda_L}(E)||\leq 2e^{L^\beta} $$ This means, in other
words, that $d(E,\sigma(H_{\Lambda_L}))>(1/2)e^{-L^\beta}$.
\end{de}
Remark that the commutator $W_L$ does not appear in Definition
\ref{defnr} as it did in Definition \ref{defregcont}. In fact one
can replace $W_L$ with
the characteristic function $\tilde{\chi}_{L,x}$ defined above
(see \cite{combhis} for the Anderson case and \cite{combhisL}
Lemma 5.1 and lines (5.27 - 5.29) if $A\neq 0$) as follows. There
exists a constant $C(\delta,I)$ with: \begin{equation}
||\chi_{4\delta,x}R_{\Lambda_L(x)}(E)W_{L,x}|| \leq C(\delta,I)
||\chi_{4\delta,x}R_{\Lambda_L(x)}(E)\tilde{\chi}_{L,x}||.
\label{eq:majW} \end{equation}
\noindent Remark then that this last bound tells us that
\begin{equation} \Lambda_L\,is\,E-NR\,
\Longrightarrow\,||\chi_{4\delta}R_{\Lambda_L}(E)W_L|| \leq
2C(\delta,I)e^{L^\beta} \label{eq:nonres} \end{equation}
\noindent An essential ingredient of the proof of Proposition
\ref{propcont} is the following deterministic lemma.
%%%%%%%%%%%%%%%%%%%%%%%%%% lemma %%%%%%
\begin{lem}
\label{lemcont} Let $L=l^\alpha$ with $\alpha\in]1,2[$ and
$x\in\R^\nu, l>12R$. Denote by $s_\nu$ the number of faces of a
cube in dimension $\nu$. Assume that for some $$
\gamma>\left(27l^\beta+4(\nu-1)ln(l/\delta))+4ln(2s_\nu)\right)/l,
$$
with $\delta$ and $\beta$ defined previously, and for some energy
$E$
\noindent i) $\Lambda_L(x)$ is $E-NR$;
\noindent ii) Each box of size 4j(l+R), j=1,2,3, centered in
$x+l\Z^\nu$ and contained in $\Lambda_L(x)$ is E-NR;
\noindent iii) Among all the $(\gamma,E)-singular$ boxes of size
$l$ contained in $\Lambda_L(x)$, there are no more than three that
are two by two R non-overlapping.
\noindent Then $\Lambda_L(x)$ is $(\gamma',E)-regular$ with
\begin{equation}
\gamma'=\gamma\left(1-\frac{27}{l^{\alpha-1}}\right)
-\frac{2}{l^{\alpha(1-\beta)})}
- \frac{ln\left(2s_\nu
C(l/\delta)^{\nu-1}\right)}{l}, \label{eq:defgamma} \end{equation}
with $C=C(\delta,I)$ defined in (\ref{eq:majW}). \end{lem}
In \cite{combhis}, Combes and Hislop have proved a simpler
version of this result. In fact, they have adapted to the
continuous case a simplified version of \cite{124} which is
contained in chapter IX of \cite{carlac} (see also \cite{vdk2}).
But, as in the discrete case, this simplified version does not
seem to suffice to obtain the results of this paper, since we
need to obtain regular boxes at any size with good probability,
uniformly in a compact interval of energy, and no longer at some
fixed energy $E$. So we turn to \cite{124} and adapt it to the
continuous case.
%%%%%%%% Proof of the deterministic Lemma %%%%%%%%%
\vskip2mm \noindent {\bf Proof of Lemma
\ref{lemcont}:} The aim is to bound
$||\chi_{4\delta,x}R_{\Lambda_L(x)}(E) \mbox{ } W_{L,x}||$. Using
first inequality (\ref{eq:majW}), we are reduced to control
$||\chi_{4\delta,x}R_{\Lambda_L(x)}(E)\tilde{\chi}_{L,x}||$. This
is achieved in (\ref{eq:control}) below. We recall that $\delta>1$
has been chosen small, so, without loose of generality one can
suppose $l>3\delta$.
In order to achieve our goal, we will recursively construct inside
$\Lambda_L(x)$ a chain of $n$ boxes $\Lambda_{l}(v_k)$,
$k=0,...,n-1$, being most of the time $(\gamma,E)-regular$, and
starting at $v_0\equiv x$. At each step of this process, we will
use the geometric resolvent equation as follows.
Let $l'>3\delta$ and consider any box
$\Lambda_{l'}(z)\subset\Lambda_L(x)$ with
$d(\Lambda_{l'}(z),\tilde{\Lambda}_L(x))>0$. For
$E\not\in\sigma(H_{\Lambda_{l'}(z)}) \cup
\sigma(H_{\Lambda_L(x)})$, the resolvent identity
(\ref{eq:resolvant}) gives: $$ \chi_{4\delta,z}
R_{\Lambda_L(x)}(E)\tilde{\chi}_{L,x} = \chi_{4\delta,z}
R_{\Lambda_{l'}(z)}(E)W_{l',z}R_{\Lambda_L(x)}(E)\tilde{\chi}_{L,x}.
$$ The support of $W_{l',z}$ can be covered by a family of boxes
$\Lambda_{4\delta}(v)\subset\Lambda_L(x)$, indexed by points $v$
that satisfy $|v-z|=l'/2$ and so that the sum over all the
corresponding characteristic functions $\chi_{4\delta,v}$ is equal
to $1$ on $SuppW_{l',z}$ (Note that $s_\nu
(1+l'/3\delta)^{\nu-1}0$, whereas one of these
conditions fails for $\Lambda_l(v_{k^*})$. As a result, if
$k^*>0$, we have \begin{equation}
||\chi_{4\delta,x}R_{\Lambda_L(x)}(E)\tilde{\chi}_{L,x}|| \leq
(s_\nu(l'/\delta)^{\nu-1})^{k^*}
e^{-\gamma k^*l/2}
||\chi_{4\delta,v_{k^*}}R_{\Lambda_L(x)}(E)\tilde{\chi}_{L,x}||.
\label{eq:good} \end{equation} If $k^*=0$, this equation holds
trivially. The important point here is that we gained a factor
$e^{-\gamma k^*l/2}$: if there were no $(\gamma,E)-singular$
boxes $\Lambda_l$ inside $\Lambda_L(x)$, this would end the proof.
Indeed, in that case, the process could only end when
$\Lambda_l(v_{k^*})$ gets too close to
the boundary of $\Lambda_L(x)$, implying $k^*\geq(L/l)$, so that
(\ref{eq:good}) immediately yields the result upon using
hypothesis (i).
Of course, there may be $(\gamma, E) - singular$ boxes in
$\Lambda_L(x)$ and we now use hypothesis (iii) of the lemma to
control the case in which the above process stops because
$\Lambda_l(v_{k^*})$ is $(\gamma, E)-singular$ and $v_{k^*}$ is at
a distance greater than $12(l+R)$ from the boundary of
$\Lambda_L(x)$. Using hypothesis (iii) and drawing a few pictures
one easily convinces oneself that one can pack all the singular
boxes of size $l$ in $t\leq 3$ slightly bigger and disjoint boxes
$\Lambda_{l_i}\subset\Lambda_L(x)$, centered in $x+l\Z^\nu$, and
so that each box $\Lambda_l(z)$, where $z$ belongs to the edge of
one of those $\Lambda_{l_i}$, is $(\gamma,E)-regular$. More
precisely, the $l_i$ are taking on one of the values $4j(l+R)$,
$1\leq j\leq 3$, they satisfy $\sum_{i=1}^t l_i \leq 12(l+R)\equiv
l_0$, and the two following facts are simultaneously true:
\begin{eqnarray}
(1) & \left( \begin{array}{c}
d(z,\partial\Lambda_L(x))\geq l/2 \\
z\in\Lambda_L(x)\backslash \bd \bigcup_{i=1}^r \ed
\Lambda_{l_i}
\end{array}
\right) \Longrightarrow \left( \Lambda_l(z)\, is\,
(\gamma,E)-regular \right).\label{eq:zz}\\
(2) & \mbox{\ if\ }z \mbox{\ belongs to the edge of one of the\ }
\Lambda_{l_i},\mbox{\ then\ }
z\not\in\bigcup_{i=1}^r \Lambda_{l_i}.\label{eq:zzz}
\end{eqnarray} So, if $\Lambda_{l}(v_{k^*})$ is
$(\gamma,E)-singular$, there exists $i\in\{1,..,t\}$ so that
$\Lambda_l(v_{k^*})\subset\Lambda_{l_i}$. Define a new family of
points $v$ on the boundary of $\Lambda_{l_i}$, such that the boxes
$\Lambda_{4\delta}(v)$ cover $supp \,W_{l_i}$. Then use the
equivalent of (\ref{eq:inegresolv}) with $\Lambda_{l'}(z) =
\Lambda_{l_i}$ and $z=v_{k^*}$. This produces some $v_{k^*+1}$
that belongs to the edge of $\Lambda_{l_i}$, and consequently
(\ref{eq:zz})-(\ref{eq:zzz}) implies that $\Lambda_l(v_{k^*+1})$
is $(\gamma,E)-regular$, provided
$d(v_{k^*+1},\partial\Lambda_L(x))>l/2$. By checking how
$v_{k^*+1}$ is positioned with respect to $v_{k^*}$ one sees that
the latter condition is satisfied because $v_{k^*}$ is at least at
a distance $12(l+R)$ from $\partial\Lambda_L(x))$. Use now
Hypothesis (ii) and apply once again (\ref{eq:inegresolv}) to
$\Lambda_{l}(v_{k^*+1})$, to obtain that for some $v_{k^*+2}$ on
the edge of $\Lambda_{l}(v)$,
with $|v_{k^*}-v_{k^*+2}|\leq l_i$: \begin{equation}
||\chi_{4\delta,v_{k^*}}R_{\Lambda_L(x)}(E)\tilde{\chi}_{L,x}||
\leq 2s_\nu^2\left(\frac{l_0l}{\delta^2}\right)^{\nu-1}
e^{-\gamma l/2+l_i^\beta}
||\chi_{4\delta,v_{k^*+2}}R_{\Lambda_L(x)}(E)\tilde{\chi}_{L,x}||.
\end{equation} Using the preliminary condition on $\gamma$, this
leads to \begin{equation}
||\chi_{4\delta,v_{k^*}}R_{\Lambda_L(x)}(E)\tilde{\chi}_{L,x}||
\leq
||\chi_{4\delta,v_{k^*+2}}R_{\Lambda_L(x)}(E)\tilde{\chi}_{L,x}||.
\label{eq:case2} \end{equation} This is the way in which we get
past a singular box $\Lambda_{l}(v_{k^*})$ far from the edge of
$\Lambda_L(x)$. We have now completely described the recursive
construction of the $v_k$ and it is clear that the process grinds
to a standstill only when, for some $n$, $\Lambda_l(v_n)$ is too
close to $\partial\Lambda_L(x)$. From (\ref{eq:case2}) one sees
that, when meeting a singular box, we do not gain a factor
$\exp^{-\gamma l/2}$, so that we have to assure ourselves this
does not happen to often before the process ends. We therefore
need to count how many of the boxes $\Lambda_l(v_k),\ 0\leq k\leq
n$ are regular. Since $|v_{k+1}-v_k|=l/2$ in that case and
$|v_{k+2}-v_k|\leq l_i$ if not, it is not hard to see that the
process cannot stop before $n=n^*=n_1^*+2t$ where \[ n_1^* =
\left[ \frac{L/2 - \sum_{i=1}^t l_i}{l/2}\right], \] or
\begin{equation} [L/l]-27\leq n_1^*\leq L/l. \label{eq:n1*}
\end{equation} Hence, using (i) of the lemma, one has
\begin{equation}
||\chi_{4\delta,x}R_{\Lambda_L(x)}(E)\tilde{\chi}_{L,x}||
\leq \left( s_\nu(l/\delta)^{\nu-1} e^{-\gamma l/2}
\right)^{n_1^*} 2e^{L^\beta}. \label{eq:control}\end{equation}
Putting together relations (\ref{eq:majW}) and (\ref{eq:control})
and using (\ref{eq:n1*}) as well as the definition of $\gamma'$
stated in the lemma leads to the desired result. \fin
%%%%%%%%% Proof of the proposition %%%%%%%%%%%%%
\vskip3mm \noindent {\bf Proof of Proposition \ref{propcont}:} Take
$\alpha\in]1,2-2\nu/(p+2\nu)[$ and $\gamma\in]0,\gamma_0[$. Let
$L\equiv L_{k+1}=L_k^\alpha$, and use equation~(\ref{eq:defgamma})
to produce a sequence of exponents $\gamma_k\in]0,\gamma_0]$. It
will be enough to show that
a) $\forall k\geq0$, $\gamma\leq\gamma_k\leq\gamma_0$;
b) $\forall k\geq 0:\; |x-y|>L_{k+1}+2R \Longrightarrow
\P\left(R(\gamma_k, L_{k+1},x,y)\right)>1-1/L_{k+1}^{2p}$.
\vskip2mm To prove (a), choose $L_0>0$: the sequence
($\gamma_k)_{k\geq 0}$ produced by repeatedly using
(\ref{eq:defgamma}) decreases, so for all $k\geq 0$,
$\gamma_{k+1} \leq \gamma_k \leq \gamma_0$. Then, using equation
(\ref{eq:defgamma}), it is clear there exists
$L_*=L_*(\gamma,\gamma_0,\beta,\alpha,\nu)$ so that, if $L_0>L_*$,
the sequence ($\gamma_k)_{k\geq 0}$ satisfies \begin{eqnarray*}
0<\sum_{k=0}^\infty (\gamma_k-\gamma_{k+1})
& \leq & 15\gamma_0 \sum_{k=0}^\infty L_k^{1-\alpha}
+ \sum_{k=0}^\infty
L_k^{-\frac{1}{2}min(1,\alpha(1-\beta))} \\
& \leq & \gamma_0-\gamma \end{eqnarray*} Hence (a) follows,
and we turn to (b), which clearly follows from: \begin{eqnarray}
\lefteqn{|x-y|>L_{k+1}+2R \Longrightarrow} \nonumber \\ &
\hspace{-.2in} \P \left(\! \begin{array}{ll}
\forall\,E\in I,\, the\, hypotheses\, of\, Lemma\,
\ref{lemcont}\, with \,\gamma=\gamma_k \\
and\,L=L_{k+1}\,are\, satisfied\,for\, either\, point\, x\,
or\, y
\end{array}
\!\right)\! >\! 1-1/L_{k+1}^{2p}. \label{eq:bbis}
\end{eqnarray} Firstly, since for all $k\geq 0$,
$\gamma_k\geq\gamma$, provided $L_0$ is large enough, one has: $$
\forall k\geq 0,\: \gamma_k>\frac{\textstyle 1}{\textstyle L_k}
\left(27L_k^\beta+4(\nu-1)ln(L_k/\delta))+4ln(2s_\nu)\right) $$
Let's now define $I_l=\cap \{E\in\R,\;d(E,I) \leq
e^{-l^\beta}/2\}$ and $\sigma'(H_{\Lambda_l}) =
\sigma(H_{\Lambda_l}) \cap I_l$. It is easy to estimate the
probability that the distance between the respective spectrum of
two R non-overlapping boxes $\Lambda_{l_1}$ and $\Lambda_{l_2}$,
$(l_1, l_2 > \tilde{L}$, is greater than $\eta$, $0<\eta<1$. Using
first (\ref{eq:wegner}) and then Hypothesis [H2], one has, with
some abuse of notations: \begin{eqnarray}
\P(d(\sigma'(H_{\Lambda_{l_1}}),\sigma'(H_{\Lambda_{l_2}}))<\eta)
& \leq & \!\int\! \sum_{E\in I\cap \sigma'(H_{\Lambda_{l_2}})}\!
\P_{\Lambda_{l_1}}
\!\left(d(\sigma'(H_{\Lambda_{l_1}}),E)<\eta\right)
d\omega_2 \nonumber \\
& \leq & C_W |\Lambda_{l_1}| \eta
\,\E(Tr(E_{\Lambda_{l_2}}(I_l))) \nonumber \\
& \leq & C_W^2(|I|+1)|\Lambda_{l_1}|\:|\Lambda_{l_2}|\eta
\nonumber \\
& = & C_{W,I}|\Lambda_{l_1}|\:|\Lambda_{l_2}|\eta.
\label{eq:distsigma} \end{eqnarray}
Hence, for all $k$, and writing for convenience $L\equiv L_{k+1}$
and $l=L_k$: if $|x-y|>L+2R$, it follows from (\ref{eq:distsigma})
that \begin{eqnarray} \lefteqn{ \P(\exists\;
u\in(x+l\Z^\nu)\cap\Lambda_L(x),\;
v\in(y+l\Z^\nu)\cap\Lambda_L(y)\;
and\; l_1,\; l_2 =\;L\; or}
\nonumber \\
& & 4j(l+R)l,\; j=1,..,3\; with \;\Lambda_{l_1}(u)
\subset\Lambda_L(x)\; and\;
\Lambda_{l_2}(v)\subset\Lambda_L(y), \nonumber \\
& & with\, d(\sigma'(H_{\Lambda_{l_1}}),
\sigma'(H_{\Lambda_{l_2}}))<\eta) \leq C_{W,I}
(L/l)^{2\nu}|\Lambda_L|^2\eta. \label{eq:prob} \end{eqnarray} But
consider this elementary exercise in logic: let $A_i$ and $B_j$,
$i$ and $j=1,..,J$ be $2J$ intervals, then \begin{eqnarray*}
\lefteqn{\left(\forall\, i,j=1,...,J,\; d(A_i,B_j)>\eta\right)} \\
& \Longleftrightarrow & \left(\forall\,E\in\R\; \forall\,
i,j=1,...,J,\;
(d(E,A_i)>\eta/2\, or\, d(E,B_j)>\eta/2)
\right) \\
& \Longleftrightarrow &
\left(\begin{array}{c}
\forall\,E\in\R,\; either\,
(\forall\,i=1,...,J,\;d(E,A_i)>\eta/2) \\
or\, (\forall\,j=1,...,J,\;
d(E,B_j)>\eta/2)
\end{array}
\right). \end{eqnarray*}
\noindent This, combined with inequality (\ref{eq:prob}) and
$\eta=e^{-L_k^\beta}$, gives for all $k\geq 0$, and if
$|x-y|>L_{k+1}+2R$ that \begin{eqnarray} \P \left(\!
\begin{array}{ll}
\forall\,E\in I,\, (i)\, and\; (ii)\, of\, Lemma\,
\ref{lemcont}\, with \,
\gamma=\gamma_k \\
and\,L=L_{k+1}\,are\,
satisfied\,for\, either\, point\, x\, or\, y
\end{array}
\!\right)\! >\! 1-1/L_{k+1}^{2p+1} \label{eq:i)ii)}.
\end{eqnarray}
Let's finish the proof: for $L_0$ large enough, Hypothesis
[H1]($\gamma,I,p,L_0$) gives (\ref{eq:bbis}) at rank 0. Suppose it
is true at rank $k$: points (i) and (ii) of Lemma \ref{lemcont},
with $\gamma = \gamma_k$ and $L=L_{k+1}$, are satisfied for
either points $x$ or $y$, with probability evaluated line
(\ref{eq:i)ii)}). Now, \begin{eqnarray} \lefteqn{\P(for\; any\;
E\in I, (iii)\; of\; Lemma\;\ref{lemcont}\; holds)} \hspace{.3in}
\nonumber \\ & = &
1-\P(\exists\,E\in I\, s.t.\; there\; are\; at\; least\; 4\; R\,
non-overlapping
\nonumber \\ & &
\hspace{.4in} (\gamma_k,E)-singular\; boxes\; \Lambda_{L_k}
contained\, in\,
\Lambda_{L_{k+1}}(x)) \nonumber \\ &\geq
&1-\P(\exists\,E\in I\, s.t.\; there\; are\; at\; least\; 2\; R\,
non-overlapping
\nonumber \\ & &
\hspace{.4in} (\gamma_k,E)-singular\; boxes\; \Lambda_{L_k}
contained\, in\,
\Lambda_{L_{k+1}}(x))^2 \nonumber \\ &\geq
&1-\left(\frac{(L_{k+1}/L_k+1)^{2\nu}}{L_k^{2p}}\right)^2,
\label{eq:(iii)} \end{eqnarray} where we obtained the last
inequality using the recurrence hypothesis. Hence, since
$\alpha<2-2\nu/(p+2\nu)$, combining (\ref{eq:i)ii)}),
(\ref{eq:(iii)}), there exists a constant
$L_*=L_*(p,\gamma,\gamma_0,\nu)$ such that if $L_0>L_*$ and
$|x-y|>L_{k+1}+2R$: $$
\P(R(\gamma_k,L_{k+1},x,y))>1-\frac{1}{L_{k+1}^{2p}}. $$ Use now
that $\gamma_k>\gamma$, and Proposition~\ref{propcont} is proved.
\fin
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\end{document}