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\begin{document}
\newcommand{\Schr}{Schr\"odinger}
\begin{titlepage}
\begin{center}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
{\twelve LANDAU HAMILTONIANS \\
WITH UNBOUNDED RANDOM POTENTIALS}
\vspace{0.3 cm}
\setcounter{footnote}{0}
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{\bf J.-M. Barbaroux \footnote{D\'epartement de Math\'ematiques,
Universit\'e de Toulon et du
Var -
83130 La Garde - France.}, J.M. Combes$^1$ and P.D. Hislop\footnote{
Mathematics Department, University of Kentucky - Lexington KY
40506-0027 - USA.}$^,$\footnote{
Supported in part by NSF grants INT 90-15895 and DMS 93-07438.}
}
\vspace{.5 cm}
{\ten Centre de Physique Th\'eorique\footnote{
Unit\'e Propre de Recherche 7061} - CNRS - Luminy, Case 907}
{\ten F-13288 Marseille Cedex 9 - France }
\vspace{1 cm}
{\bf Abstract}
\end{center}
We prove the almost sure existence of pure point spectrum
for the two-dimensional Landau Hamiltonian with an unbounded
Anderson-like random potential, provided that the magnetic field
is sufficiently large. For these models,
the probability distribution of the coupling
constant is assumed to be absolutely continuous. The
corresponding density
$g$ has support equal to $\R$, and satisfies
$\mbox{sup}_{\lambda \in \R} \{ \lambda^{ 3 + \epsilon } g (
\lambda ) \} < \infty $, for some $\epsilon > 0$.
This includes
the case of
Gaussian distributions.
We show that the almost sure spectrum $\Sigma$ is $\R$, provided
the magnetic field $B \neq 0$.
We prove that for each positive integer $n$, there exists a
field strength $B_n$, such that for all $B \geq B_n $, the almost
sure spectrum $\Sigma$ is pure point at all energies $ E \leq
( 2n + 1 )B - {\cal O} ( B^{-1}) $
and outside of intervals of width
${\cal O} ( B^{-1} )$ about each Landau level $E_m(B) \equiv (2 m
+ 1 ) B $, for $m < n$.
We also prove that the integrated density
of states is Lipschitz continuous away from the Landau energies
$E_n(B)$.
The proof relies on a new Wegner estimate for the finite-area
magnetic
Hamiltonians with random potentials and exponential decay
estimates
for the finite-area Green's functions. As in a previous work,
the proof of the decay
estimates
for the Green's functions uses
fundamental results from two-dimensional bond percolation theory.
\vspace{.2 cm}
\end{titlepage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1
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\section{Introduction and the Main Result}\label{S.1}
Several recent discussions of the integer quantum Hall effect
(IQHE) \cite{[Halp], [Prange-Joynt], [Kunz], [Bellissard1], [Th],
[Prange-Girvin]}
require certain spectral properties of the one-particle \Schr\
operator. As proved in \cite{[Bellissard2]}, the one-particle model
of a free electron moving in two dimensions in the presence of a
constant, transverse magnetic field will not exhibit the IQHE. At
very low temperatures, when dissipative effects are small, it is
commonly believed that random impurities in the sample provide a
necessary mechanism for the IQHE. Typically, the hypotheses
on the spectral properties of one-particle Hamiltonian
include 1) the existence of regions of localized states between
the Landau levels, and 2) the divergence of
the localization length at some energy between these regions.
Here, we prove the first of these hypotheses on the existence of
localized states. In a first paper \cite{[CoHi2]}, we proved the existence of
localized states near the band edges, and to within ${\cal O} (
B^{-1} )$ of the Landau energies, for fixed disorder and large
magnetic field. For these models, there are spectral gaps between
the bands of length ${\cal O} ( B )$. The corresponding eigenfunctions
decay exponentially with respect to distance and the magnetic
field strength $B$. Kunz's proof \cite{[Kunz]}
of the IQHE requires the existence of spectral gaps and localized
states near the band edges with finite localization length.
A consequence of his analysis
is a proof that the localization
length diverges at some energy in each band. It is believed that for
neutral samples, the localization length should diverge precisely
at the Landau energy.
In a recent review article by
Bellissard, van Elst, and Schulz-Baldes \cite{[Bellissard2]}, these authors
point-out that in experiments, the disorder is strong enough
to fill the gap between the Landau levels. In their proof of the
IQHE, they require regions of localized states between the Landau energies
in order that the quantum Hall conductivity exhibits a plateau region.
They do not require a spectral gap.
They prove that if the quantum Hall conductivity jumps by an integer (as a
function of the filling factor), then
the localization length must diverge at some energy between the
localized state regions.
In this paper, we prove localization between Landau energies, and
to within ${\cal O} (B^{-1} )$ of the Landau energies, for Landau
Hamiltonians with unbounded, Anderson-type random potentials and
large magnetic fields. For
these models, there is typically no spectral gap. As in
\cite{[CoHi2]}, we prove that the integrated density of states is
Lipschitz continuous at all energies but the Landau energies. This
indicates that the localization length may diverge there, but we
provide no such estimate here.
We now provide a description of the one-particle Hamiltonian and
state the main results. We consider a one-particle Hamiltonian
which describes an electron in
two-dimensions $(x_1, x_2)$ subject to a constant magnetic field
of
strength $B > 0$ in the perpendicular $x_3$-direction, and a
random
potential $V_{\omega}$. The Hamiltonian $H_{\omega}$ has the form
\beq{1.1}
H_{\omega} = (p - A)^2 + V_{\omega},
\eeq
\noindent on the Hilbert space $L^2 (\R^2)$, where $p \equiv - i
\nabla$, and the vector potential $A$ is
\beq{1.2}
A = {B \over 2} (x_2, -x_1),
\eeq
\noindent so the magnetic field $B = \nabla \times A$ is in the
$x_3$-direction. The random potential $V_{\omega}$ is
Anderson-like
having the form
\beq{1.3}
V_{\omega} (x) = \Sum_{i \in \Z^2} \lambda_i (\omega) u (x - i).
\eeq
\noindent We make the following assumptions on the single-site
potential $u$ and the coupling constants $\{ \lambda_i (\omega)
\}$.
\begin{itemize}
\item[(V1)] $u\ge 0,\ u \in C^2,\ \supp u \subset B (0, {1 \over
\sqrt 2})$, and $\exists C_0 > 0$ and $r_0 > 0$ s.t. $u | B (0,
r_0) >
C_0$.
\item[(V2)] $\{ \lambda_i (\omega) \}$ is an independent,
identically
distributed family of random variables with common distribution
$g \in L^{ \infty } ([- M, M ])$, for some $0 < M \leq \infty$, s.t. $g ( -
\lambda ) = g ( \lambda ), $
$g (\lambda) > 0$ Lebesgue a.e. $\lambda
\in [ -M , M ] $, and for some $\epsilon > 0$,
$\mbox{sup}_{ \lambda \in \R } \{ \lambda^{3 + \epsilon } g (
\lambda ) \} < \infty $.
%\item[(V3)] $\P \{ \lambda \geq \xi > 0 \} = {\cal O} ( \xi^{
%- N} )$, for all $\xi > 0$ large and some $N > 2$.
\end{itemize}
The condition on the decay of $g$ implies that the first two
moments are finite, i.e.\
$\int | \lambda |^k g ( \lambda ) d
\lambda < \infty$ for $ k = 0, 1, 2$.
The above condition does not require any differentiability of the
density $g$. If we require $g$ to be twice differentiable,
we can replace (V2) by another condition.
\begin{itemize}
\item[(V2)'] $\{ \lambda_i (\omega) \}$ is an independent,
identically
distributed family of random variables with common distribution
$g \in C^{ 2} ([- M, M ])$, for some $0 < M \leq \infty$,
s.t. $g ( - \lambda ) = g ( \lambda ), $
$g (\lambda) > 0$ Lebesgue a.e. $\lambda
\in [ -M , M ] $, $ \| g^{ ( p ) } \|_1 < \infty$, for $ p = 0 ,
1, 2$, and $\int_{ - M }^{ M } \lambda^2 g ( \lambda ) < \infty
$.
\end{itemize}
We remark for future use that either condition (V2) or (V2)'
implies that
\beq{1.4}
\P \{ \lambda \geq \xi > 0 \} = {\cal O} ( \xi^{
- N} ), \mbox{for all} \; \xi > 0 \; \mbox{large and some} \; N \geq 2.
\eeq
An important example is the case of Gaussian distributed
coupling-constants for which $ g ( \lambda ) = ( \alpha \pi )^{-
1/2} e^{ - \alpha \lambda^2 }$, for some $\alpha > 0$.
The case of finite $M$ was dealt with in \cite{[CoHi2]},
so here we consider only the case $M = \infty$.
We denote by $H_A \equiv (p - A)^2$, the Landau Hamiltonian. As
is
well-known, the spectrum of $H_A$ consists of an increasing
sequence
$\{ E_n (B) \}$ of eigenvalues, each of infinite multiplicity,
given by
\beq{1.5}
E_n (B) = (2 n + 1) B,\ n = 0,1,2,\dots
\eeq
We will call $E_n (B)$ the \nth Landau level
and denote by $P_n$ the projection onto the corresponding
subspace.
The orthogonal projection is denoted by $ Q_{n} \equiv 1 - P_{n}
$.
The family of random potentials defined in (1.3) is unbounded.
Consequently, we must show that the corresponding family of
\Schr\ operators is self-adjoint with probability one. It is easy
to show, by a Borel-Cantelli type argument,
that the potential is almost surely bounded below by $ - C_0
( 1 + \| x \|^2 )$, for some finite constant $C_0$.
It then follows that the family $H_\omega$ is essentially
self-adjoint on $C_0^{\infty} ( \R^2 )$ with probability one.
We refer to \cite{[BCH]} for the details.
The family of random, self-adjoint \Schr\ operators defined above
on $L^2 ( \R^2 )$ is an ergodic, measurable family. For general
densities $g$,
it follows from Appendix 2 of
\cite{[CoHi2]} that the almost sure spectrum $\Sigma ( B )$ of
the family $H_\omega$ is a closed subset of $\R$
and may have spectral gaps of width at most ${\cal O} ( B^{-1/2} )$.
When $\mbox{supp} g = \R$, we prove below that $ \Sigma ( B ) =
\Sigma =\R $.
Our main theorem concerning localization is the following.
\begin{theorem}\label{T.1.1}
Let $H_{\omega}$ be the family given in~\rf{1.1} with vector
potential
$A$ satisfying~\rf{1.2}, $B > 0$, and the random potential
$V_{\omega}$ as in~\rf{1.3} and satisfying~(V1) and either (V2) or (V2)' with $M =
\infty$.
Let
$ I_n (B) $ denote the unbounded set of energies
$$
I_n (B) \equiv \left( - \infty, B - {\cal O} (B^{-1}) \right] \cup \;
\bigcup_{j = 0}^n \left[ E_{j} (B) + {\cal O} (B^{-1}),
E_{j+1} (B) - {\cal O} (B^{-1}) \right],
$$
where the term $ {\cal O} (B^{-1}) $ depends on $n$.
For each integer $ n > 0$,
there exists $B_n \gg 0$ such that for $B > B_n$,
$$
\Sigma \cap I_n (B)
$$
is pure point and the corresponding eigenfunctions
decay exponentially.
\end{theorem}
We remark that we can also prove localization at energies between the
Landau levels for fixed, nonzero $B > 0$ for the family
$H_{\omega} ( g ) \equiv H_A + g V_{\omega}$, with a coupling
constant $g > 0$, provided that we work in the small coupling regime.
This result does not require percolation theory, but we cannot control the spectrum
to within ${\cal O} ( B^{-1} ) $ of the Landau levels.
We refer the reader to section 6 of \cite{[BCH]} where
results of this type are proved in general situations.
As with previous work, the Wegner estimate for the finite-area
Hamiltonians allows us to control the integrated density of
states.
\begin{theorem}\label{T.2.1}
Let $H_{\omega}$ be the family given in~\rf{1.1} with vector
potential
$A$ satisfying~\rf{1.2}, $B > 0$, and the random potential
$V_{\omega}$ as in~\rf{1.3} and satisfying~(V1) and either (V2) or (V2)'.
Then, the integrated density of states is
Lipschitz
continuous on $\R \backslash \sigma (H_A)$.
\end{theorem}
This note is organized as follows. We first prove that the almost
sure spectrum is the entire real line when $\mbox{supp} \; g = \R$
for any $B \neq 0$.
The proof of Theorem 1.1 is similar to the proof in \cite{[CoHi2]}.
However, since the potential is unbounded,
we need a refined Wegner estimate.
In section 3, we give a new Wegner estimate
for random perturbations of Landau Hamiltonians.
This technique is generalized and applied to band-edge
localization in \cite{[BCH]}.
We sketch the proof of Theorem 1.1 in section 4, providing the
details of the modifications of the estimates in
\cite{[CoHi1],[CoHi2]} required for unbounded potentials.
We mention that the proof of localization at energies below the
first Landau level $B$ and in the center of the gap does
not require percolation estimates, since one is well within the spectral
gap of the unperturbed operator $H_A$. For these energies,
localization may be proved using the Wegner estimate and the
Combes-Thomas method as in \cite{[BCH]}.
In order to prove localization for the energy intervals of Theorem 1.1,
i.e. close to the Landau energies,
the percolation estimates of \cite{[CoHi2]} are needed.
We mention that Wang \cite{[Wang]} has obtained results similar
to those of \cite{[CoHi2]} in the case that the density $g$ has
bounded support. A discussion of the single-band approximation,
including the case of densities with unbounded support,
is given in the papers of Dorlas, Macris, and Pul\'e
\cite{[DMP1], [DMP2]}.
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\section{Location of the Spectrum}\label{S.2}
Let $ \Sigma (B) $ be the almost sure spectrum of the Landau
Hamiltonian as defined in (1.1)-(1.3). For general density
$g$ with compact support, the nature of $ \Sigma (B) $
may be quite complicated (cf.\ \cite{[CoHi2]}). If, however,
$ \mbox{supp} \:g = I \hspace{-.17cm} R $, we can characteristize
$ \Sigma (B) $ quite easily.
\vspace{.2in}
\noindent
{\bf Proposition 2.1}. {\em Assume conditions (V1) and either (V2)
or (V2)', and that $ \mbox{supp} \: g = I \hspace{-.17cm} R $. Then
$ \Sigma (B) = I \hspace{-.17cm} R , ~~ \forall B \neq 0 $}.
\vspace{.2in}
\noindent
{\bf Proof}.
\begin{enumerate}
\item
We will prove that $ I \hspace{-.17cm} R \setminus \sigma
( H_A ) \subset \Sigma $, which proves the result since
$ \sigma ( H_A ) $ is a discrete set. According to the theory
of ergodic Schr\"odinger operators with Anderson-type potentials
(see cf.\ \cite{[Kr]} section 6, Theorem 1),
%if $ \Omega _0 \subset \Omega $ is a base for the probability
%space, then
for any configuration $ \omega \in
\mbox{supp} \: I \hspace{-.17cm} P $, the spectrum $ \sigma
( H_{ \omega } ) \subset \Sigma (B) $. Let $\Omega_1$ be the set
of configurations having measure one for which $H_\omega$ is
essentially self-adjoint
on $ C_0^{ \infty } ( I \hspace{-.17cm} R ^2 ) $.
We define $ \Omega _0 $ to be $ \mbox{supp} \: I \hspace{-.17cm} P $
intersected with this set $\Omega_1$. Then, we have $\P ( \Omega_0 ) = 1 $.
The set $ \Omega _0 $ clearly contains
the set $ ( Z \hspace{-.17cm} Z ^2 )^{\R}_0 $ of all configurations
$ \omega $ which assume a constant value outside some fixed
square $ \tilde{ \Lambda } \subset Z \hspace{-.17cm} Z ^2 $.
Consequently, for any $ E \in I \hspace{-.17cm} R
\setminus \sigma ( H_A ) $, we need to
produce a configuration $ \omega ( E ) \in
( Z \hspace{-.17cm} Z ^2 )^{\R}_0 $ such that $E \in \sigma ( H_\omega )
$.
It then follows from the above comments
that $ E \in \Sigma (B) $. We separately consider
$ E < B $ and $ E > B $ but $ E \neq E_n (B) , ~~ n =
0 , 1 , 2 ,... $.
\item
$ E < B $.
For a fixed $\tilde{ \Lambda} \subset Z \hspace{-.17cm} Z ^2 $
(nonempty and bounded) and any finite $ M > 0 $,
define a configuration $\omega ( E ) \in
( \Z^2 )^{\R}_0 $ by setting $ \lambda _i =
- M , ~~ i \in \tilde{ \Lambda } $, and $ \lambda _j = 0 $,
otherwise. The potential $ V_{ \omega (E) } \leq 0 $ and
has compact support. Let $ \lambda \in I \hspace{-.17cm} R ^+
$
and consider
\beq{2.1}
H_{ \omega (E) } ( \lambda ) = H_A +
\lambda V_{ \omega (E) } = H_A -
\lambda | V_{ \omega (E) } | .
\eeq
Note that because $ \mbox{supp} \: g = I \hspace{-.17cm} R $, the
family
$ H_{ \omega (E) } ( \lambda ) =
H_{ \lambda \omega (E) } $ and $ \lambda \omega (E) \in
\Omega _0 ~~ \forall \lambda \in I \hspace{-.17cm} R ^+ $. Since
$ E
< B $ and the sign of the potential is constant, we can
apply Birman-Schwinger theory. The operator $ H _{ \omega ( E ) }
( \lambda ) $
has an eigenvalue $ E$ if and only if $ K(E) \equiv W^{ 1/2 }
( H_A - E ) ^{-1} W^{1/2} $, where $ W \equiv - V _{ \omega ( E ) } \geq 0
$, has an
eigenvalue $ 1 / \lambda \geq 0 $. The operator $ K(E) $
is a compact, self-adjoint, nonnegative operator bounded above by
$ || W ||_{ \infty } ( B - E ) ^{-1} $. It has an
infinite dimensional kernel but we will show that it is a nonzero
operator. For any $ \phi $ such that
$ W^{ 1/2 } \phi \neq 0 $, we use the spectral representation of
$ H_A = \sum_{ n \geq 0 } E_n (B) P_n $ to write
\beq{2.2}
\langle \phi , K(E) \phi \rangle = \sum_{ n \geq 0 }
( E_n(B) - E )^{-1} \| P_n W^{1/2} \phi \|^2 .
\eeq
Now suppose that $ K(E) = 0 $. Since $( E_n(B) - E ) > 0 $, for
all $n \geq 0$,
it follows from (2.2) that $ \| P_n W^{1/2} \phi \| = 0 $, for
all $n \geq 0$. By the completeness of the projectors $P_n$,
we conclude that $ W^{ 1 / 2 } \phi = 0 $, a contradiction.
Note that any $ \phi \neq 0 $ with
$ \mbox{supp} \: \phi \subset \mbox{supp} \: u $ satisfies
$ W \phi \neq 0 $. Hence, the operator $ K(E) $ has at least
one
strictly positive eigenvalue $ \mu _0 (E) > 0 $. We adjust
$ \lambda > 0 $ so that $ { \lambda }^{-1} = \mu _0 (E) $.
The configuration $ \tilde{\omega} ( E ) \equiv
{\mu_0 (E)}^{-1} \omega ( E ) \in ( \Z^2 )^{\R}_0 $.
This shows that $E \in \sigma ( H_{ \tilde{\omega} ( E ) } )
\in \Sigma ( B ) $.
\vspace{.1in}
\item
$ E > B $ and $ E \not \in \sigma ( H_A ) $. We proceed
as above and arrive at the statement: $ H _{ \omega ( E ) } (
\lambda ) $
has an eigenvalue $E$ if and only if $ K(E) \equiv
W^{ 1/2 } ( H_A - E ) ^{-1} W^{ 1/2 } $ has an eigenvalue $ 1 /
\lambda > 0 $.
Now, however, the compact self-adjoint operator $ K(E) $ is no
longer
nonnegative. It still has an infinite dimensional kernel. We must
show that $ K(E) $ is nonzero.
We first note that by
the argument in (2.2), the operator $W^{ 1/2 } ( H_A - E )^{ -2 }
W^{ 1/2
} $
is nonzero.
The operator $ K_{E}
( \epsilon ) \equiv W^{ 1/2 } ( H_A - E - \epsilon ) ^{-1} W^{
1/2
} $ is
a real analytic function of $ \epsilon $ in a small interval
about $ E $. We will show that for all $ \epsilon $ small,
$ K_{E} ( \epsilon ) $ is nonzero. By the second resolvent
formula,
\[ \begin{array}{rl}
K_{E} ( \epsilon ) & = K(E) + \epsilon W^{ 1/2 }
( H_A - E - \epsilon ) ^{-1} ( H_A - E ) ^{-1} W^{ 1/2 } \\ & \\
& = K(E) + \epsilon S_{E} ( \epsilon ) \end{array} \]
Now, we know that
$$
\displaystyle{ n- \lim_{ \epsilon \rightarrow 0 }}
S_E ( \epsilon ) = W^{ 1/2 }
( H_A - E ) ^{-2} W^{ 1/2 } \neq 0 ,
$$
so for all $ \epsilon $ small,
$ \epsilon S_E ( \epsilon ) \neq 0 $.
If $ K(E) = 0 $, then we can find an open interval $ I(E) \setminus
E $
of nearby energies $ E + \epsilon $ for which
$ K_E ( \epsilon ) = K ( E + \epsilon ) \neq 0 $.
This operator $ K ( E + \epsilon ) $
has a positive eigenvalue $
\mu _0 ( E + \epsilon ) $.
By adjusting $ \lambda > 0 $, we find
$ K ( E + \epsilon ) $ has an eigenvalue $ 1 / \lambda $
and hence $ E + \epsilon \in \Sigma (B) $. This implies
$\overline{ I (E) } \subset \Sigma (B) $. Finally, since
$ \Sigma (B) $ is closed, $ \Sigma (B) = I \hspace{-.17cm} R $. \eop
\end{enumerate}
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\section{The Wegner Estimate}\label{S.3}
We give a new proof of the Wegner estimate which holds for any
$ E \not \in \sigma ( H_A ) $, for any field strength
$ B > 0 $ and for any density $g$ satisfying either (V2) or (V2)'.
This result improves Theorem 3.1 of \cite{[CoHi2]} in two ways.
First, it applies to the case when $ \mbox{supp} \: g =
I \hspace{-.17cm} R $. Secondly, it gives a better dependence
of the upper bound on the field strength $B$. Let $ H_\Lambda $ be a
local Hamiltonian (see below). In \cite{[CoHi2]}, the bound depends on
$B$
through $ [ \mbox{dist} \: ( \sigma ( H_A ) , E ) ] ^{-2} B $,
so if $ | E - B | = {\cal O} ( B ^{-1} ) $, the estimate is
$ {\cal O} ( B^3 ) $. Here, the dependence is given as
$ [ \mbox{dist} \: ( \sigma ( H_A ) ,
E ) ] ^{-1} B^{-1} $, so when $ | E - B |
= {\cal O} ( B ^{-1} ) $, the bound is $ {\cal O} (1) $.
We define local Hamiltonians as relatively compact perturbations of
the Landau Hamiltonian $H_A = (p - A)^2$.
Let $\Lambda
\subset \R^2$ denote an open connected region in $ \R^2 $.
We let $\Lambda_{\ell} (x)$ denote a square of side $\ell$ centered at $x
\in \R^2$,
$$
\Lambda_{\ell} (x) \equiv \left\{ y \in \R^2 |\ | x_i - y_i | <
\ell,\ i = 1,2 \right\}.
$$
Given $\Lambda \subset \R^2$, the local potential
$V_{\Lambda}$ is defined as follows.
Let us denote the lattice points in $\Lambda$ by $ \tilde{ \Lambda }
\equiv \Z^2 \cap \Lambda $.
We define the local potential by $V_{\Lambda} ( x ) \equiv \sum_{ i \in
\tilde{ \Lambda } } \lambda_i (\omega) u ( x - i ) $, and the
local Hamiltonians by $H_{\Lambda}
\equiv H_A + V_{\Lambda}$ on $L^2 (\R^2)$. We denote by $\P_\Lambda$
and $\E_\Lambda$ the probability and expectation with respect
to the random variables in $\Lambda$. Note that
$\sigma_{\ess}(H_{\Lambda}) = \sigma_{\ess}
(H_A)$, since $V_{\Lambda}$ is relatively compact.
We denote the resolvent of $H_A$ by $R_A ( z ) \equiv
( H_A - z )^{-1}$, whenever it exists.
For two general regions $\Lambda_i, i = 1 , 2$, the local
Hamiltonians $H_{\Lambda_i}$ are
not independent since $\mbox{supp} u$ is not contained in a unit cube.
This lack of
independence is characteristic of models on $\R^d$. However,
since $\mbox{supp} u $ is compact, this does not pose
any problem as explained in \cite{[CoHi2]}. Henceforth, we
will assume that the potentials $V_{\Lambda_{1}}$ and
$V_{\Lambda_{2}}$ are independent if $\Lambda_{1} \cap \Lambda_{2}
= \emptyset$.
\begin{theorem}\label{T.3.1}
Assume (V1) and either (V2) or (V2)'.
For any $E_0 \not\in \sigma ( H_A )$, and
for any $\eta > 0$ such that $\eta < 1/2
\mbox{ dist} ( E_0 , \sigma ( H_A ) ) $,
and for all $B > 0$,
\beq{3.1}
\P_\Lambda \{ \dist (\sigma
(H_{\Lambda}),\ E_0) < \eta \} \le \frac{\eta}{ \pi B } C_g
| \Lambda | C ( B , E_0 , \eta ),
\eeq
where, if (V2) holds,
$C_g$ depends upon $\mbox{ supp } \{ \lambda^{ 3 + \epsilon } g (
\lambda ) \}$, or, if (V2)' holds, $C_g$
depends upon $ \| g^{ ( p )} \|_1 $, for $ p = 0 , 1 , 2 $,
and the second moment
\beq{3.2}
\E ( \lambda^2 ) \equiv \int_{ \R } g ( \lambda ) \lambda^2 d \lambda .
\eeq
Assuming, without loss of generality, that $E_N(B) < E_0
< E_{ N + 1 }( B)$, the constant in (3.1) is given by
\beq{3.3}
C ( B , E_0 , \eta ) \equiv \left[ \frac{2}{ E_0 - \eta - ( 2N + 1)B }
- \frac{1}{ E_0 - \eta - B } + \frac{1}{ E_0 - \eta - ( 2N + 3)B }
\right] .
\eeq
\end{theorem}
\begin{prf}
Let $I_\eta \equiv [ E_0 - \eta , E_0 + \eta ]$, so that by the hypotheses,
$I_\eta \cap \sigma ( H_A ) = \emptyset$. We write $R_0$ for the
resolvent $ ( H_A - E_0 )^{ -1 } $. By the definition of $\eta$,
it follows that $ \eta \| R_0 \| \leq 1/2 $.
Chebyshev's inequality
shows that it is sufficient to estimate $ \E_\Lambda ( tr E_\Lambda ( I_\eta
) )$, where $tr$ is the trace on $L^2 ( \R^2 )$.
Let $E \in I_\eta $ be an eigenvalue of $ H_\Lambda $ and let
$P_E$ be the corresponding finite-rank projector. We define an
operator $K_0 \equiv V_\Lambda R_0 $. From the
eigenvalue equation, we easily derive
\beq{3.4}
P_E = - K_0 P_E + ( E - E_0 ) R_0 P_E ,
\eeq
and the related, adjoint equation,
\beq{3.5}
P_E = - P_E K_0^* + ( E - E_0 )P_E R_0 .
\eeq
Since the spectrum of $H_\Lambda$ in $I_\eta$ is discrete,
we find from (3.4) that
\bea
tr E_\Lambda ( I_\eta )& = & \Sum_{ E \in I_\eta \cap \sigma ( H_\Lambda ) }
tr P_E \nonumber \\
& \leq & | E - E_0 | \; | tr ( R_0
E_\Lambda ( I_\eta ) ) | + | tr ( K_0 E_\Lambda ( I_\eta ) ) | \nonumber
\\
& \leq & \eta \| R_0 \| \; \| E_\Lambda (
I_\eta ) \|_1 + \| K_0 E_\Lambda ( I_\eta ) \|_1 .
\eea
By the positivity of the projector $E_\Lambda ( I_\eta ) $, we
have
$$
tr E_\Lambda ( I_\eta ) = \| E_\Lambda ( I_\eta ) \|_1 .
$$
By the fact that $ \eta \| R_0 \| \leq 1/2 $, it follows from
(3.6) that
\beq{3.7}
tr E_\Lambda ( I_\eta ) \leq 2 \| K_0 E_\Lambda ( I_\eta ) \|_1 .
\eeq
We next use equation (3.5) to find an expression for $ K_0
E_\Lambda ( I_\eta ) $,
\beq{3.8}
K_0 P_E = - K_0 P_E K_0^* + ( E - E_0 ) K_0 P_E R_0 .
\eeq
Upon taking the trace norm of (3.8) and using the same argument
as above,
we find that
\beq{3.9}
\| K_0 E_\Lambda ( I_\eta ) \|_1 \leq 2 \| K_0 E_\Lambda ( I_\eta
) K_0^* \|_1 .
\eeq
It follows from equations (3.7) and (3.9) that
\beq{3.10}
tr E_\Lambda ( I_\eta ) \leq 4 tr ( K_0 E_\Lambda ( I_\eta )
K_0^*) .
\eeq
Hence, it remains to evaluate the trace on the right side (3.10).
For any $E \not\in
\sigma ( H_A )$ and any pair $ i , j \in \tilde{ \Lambda }$,
define the operator $K_{i j } ( E )$ by
\beq{3.11}
K_{i j } ( E ) \equiv u_i^{1/2} R_A ( E )^2 u_j^{1/2}.
\eeq
We will prove below that this operator, which is analytic in $E$
on $\C \backslash \sigma ( H_A )$, is trace class.
Let $ \{ \phi_k^{ i j , E} \}$ and
$ \{ \psi_k^{ i j , E} \}$ be two complete orthonormal sets of functions
occurring in the canonical representation of $K_{i j } ( E )$,
and let $ \{ \mu_k^{ i j , E} \}$ be the singular values of $K_{i j } ( E )$.
Then the canonical representation of $K_{i j } ( E )$ is
\beq{3.12}
K_{i j } ( E ) = \Sum_{k \in \N} \mu_k^{ i j , E} | \phi_k^{ i j , E}
\rangle \langle \psi_k^{ i j , E} |.
\eeq
We write the potential as $V_\Lambda = \sum_{ i \in \tilde{
\Lambda } }
\lambda_i u_i $. By the cyclic property of the trace, the right
side
of (3.10) can be written as
\beq{3.13}
\E_\Lambda
\left\{ \Sum_{ i , j \in \tilde{ \Lambda } }
\lambda_i \lambda_j ~tr [ K_{i j } ( E_0 )
( u_j^{1/2} E_\Lambda ( I_\eta ) u_i^{1/2} ) ] \right\}.
\eeq
Inserting the canonical representation (3.12)
for $K_{i j } ( E_0 )$,
into
(3.13), and evaluating the trace in the orthonormal basis $ \{
\phi_k^{ i j , E_0} \} $ yields
\beq{3.14}
\E_\Lambda
\left\{ \Sum_{ i , j \in \tilde{ \Lambda } }
\lambda_i \lambda_j \; \Sum_{k} \mu_k^{ i j , E_0} \langle
\phi_k^{ i j , E_0} , \;
( u_j^{1/2} E_\Lambda ( I_\eta ) u_i^{1/2} ) \psi_k^{ i j
, E_0}
\rangle \right\} .
\eeq
Since $E_\Lambda ( I_\eta ) $ is a projector,
the $k$-sum can be
bounded above by
\bea
\frac{1}{2} \Sum_{k} \mu_k^{ i j , E_0} \E_\Lambda \{
\lambda_j^2
\langle \phi_k^{ i j , E_0} , \;
( u_j^{1/2} E_\Lambda ( I_\eta ) u_j^{1/2} )
\phi_k^{ i j , E_0}
\rangle \nonumber \\
+ \lambda_i^2 \langle \psi_k^{ i j , E_0} , \;
( u_i^{1/2} E_\Lambda ( I_\eta ) u_i^{1/2} )
\psi_k^{ i j , E_0}
\rangle \} \nonumber \\
\leq \frac{1}{2} \parallel K_{i j } ( E_0 ) \parallel_1
\displaystyle{
\sup_{k} }
\E_\Lambda \{ \lambda_j^2 \langle \phi_k^{ i j , E_0} , \;
( u_j^{1/2} E_\Lambda ( I_\eta ) u_j^{1/2} )
\phi_k^{ i j , E_0}
\rangle \nonumber \\
+ \lambda_i^2 \langle \psi_k^{ i j , E_0} , \;
( u_i^{1/2} E_\Lambda ( I_\eta ) u_i^{1/2} )
\psi_k^{ i j , E_0}
\rangle \} .
\eea
The spectral averaging theorem of \cite{[CHM]} can be
applied
to the matrix elements in (3.15). Recalling the definition of the
constant $C_g$ given in the theorem, we obtain, for example,
\beq{3.16}
\E_\Lambda \left( \lambda_j^2 \langle \phi_k^{ i j , E_0} ,
\; ( u_j^{1/2} E_\Lambda ( I_\eta ) u_j^{1/2} )
\phi_k^{ i j , E_0}
\rangle \right) \leq C_g .
\eeq
Consequently, the right side of (3.10) is bounded above
by
\beq{3.17}
\left( \Sum_{ i , j \in \tilde{ \Lambda } }
\parallel K_{ij} ( E_0 ) \parallel_1 \right) C_g .
\eeq
The trace norm in (3.17)
is evaluated in the same way as in \cite{[CoHi2]}
using the exponential decay of the localized resolvent
$R_A ( E )$, for any $E \not\in \sigma ( H _A ) $. This
exponential decay can be obtained by the Combes-Thomas method. Let
$\chi_x$ be a bounded function with compact support
localized near $x \in \R^2$.
Let $ \mbox{dist} ( \sigma ( H_A ) , E ) \equiv d ( E )$.
Then there exist finite, positive constants $C_1 , C_2 > 0$,
depending only on $E$,
such that for $x , y \in \R^2$ with $\| x - y \|$ large enough,
such that
\beq{3.18}
\| \chi_x R_A ( E ) \chi_y \| \leq C_1 d ( E )^{-1} e^{ - C_2 d ( E )
\parallel x - y \parallel }.
\eeq
Following \cite{[CoHi2]}, we first consider $i , j \in \tilde{ \Lambda }$
such that $\parallel i - j \parallel < 2$. Let $\chi_{ij}$ be the
characteristic function on the $\mbox{supp} ( u_i + u_j )$. We can then
write
\beq{3.19}
\Sum_{ \parallel i - j \parallel < 2 } \parallel K_{i j } ( E ) \parallel_1
\leq
\Sum_{ \parallel i - j \parallel < 2 } \parallel \chi_{ij} R_A ( E )^2
\chi_{ij} \parallel_1 .
%\right)
\eeq
Since the operator on the right in (3.19) is positive, we can evaluate the
trace using the integral kernel
\bea%{3.20}
\parallel \chi_{ij} R_A ( E )^2 \chi_{ij} \parallel_1 & = & \Sum_{n}
\int \chi_{ij}^2 ( x ) P_n ( x , x ) ( E_n(B) - E )^{-2} dx \nonumber \\
& \leq & | \mbox{supp} \chi_{ij} | C ( E , B ),
\eea
where $ C ( E , B ) \equiv \sum_n ( E_n(B) - E )^{-2} $.
Consequently, this gives
\beq{3.21}
\Sum_{ \parallel i - j \parallel < 2 } \parallel K_{i j } ( E ) \parallel_1
\leq C ( E , B ) .
\eeq
As for the complementary set of indices, $i , j \in \tilde{ \Lambda }$ such
that $\parallel i - j \parallel > 2$, let $ \chi_{ij}^+ ( x ) $ be
the characteristic function on the set $ \{ x \in \tilde{ \Lambda } \;
| \; \parallel x - i \parallel ~<~ \parallel x - j \parallel \} $, and define
$ \chi_{ij}^- \equiv 1 - \chi_{ij}^+ $. Using the identity,
\beq{3.22}
\parallel A B \parallel_1 ~\leq ~\parallel A \parallel_{HS} ~\parallel B
\parallel_{HS},
\eeq
where $HS$ denotes the Hilbert-Schmidt norm, we write
\bea%{3.23}
\parallel K_{i j } ( E ) \parallel_1 & = & \parallel u_i^{1/2} R_A ( E )
( ( \chi_{ij}^+ )^2 + ( \chi_{ij}^- )^2 ) R_A ( E ) u_j^{1/2} \parallel_1 \\
& \leq & \parallel u_i^{1/2} R_A ( E )
\chi_{ij}^+ \parallel_{HS} ~ \parallel \chi_{ij}^+ R_A ( E ) u_j^{1/2}
\parallel_{HS} \nonumber \\
& & + \parallel u_i^{1/2} R_A ( E )
\chi_{ij}^- \parallel_{HS} ~ \parallel \chi_{ij}^- R_A ( E ) u_j^{1/2}
\parallel_{HS} . \nonumber
\eea
As in \cite{[CoHi2]}, the condition on $\mbox{supp} u$ implies that there
exists a constant $ 0 < a < 2 $ such that
\beq{3.24}
\mbox{dist} ( \mbox{supp} \chi_{ij}^+ , u_j ) \geq a \parallel i -
j \parallel .
\eeq
It follows from this and the Combes-Thomas estimate (3.18) that there
exists a positive constant $C_E$, depending only on $E$, such that
\beq{3.25}
\parallel \chi_{ij}^+ R_A ( E ) u_j^{1/2} \parallel_{HS} \leq d ( E )^{-1}
\parallel u \parallel_1^{1/2} C_E e^{ - a d ( E ) \parallel i - j \parallel
/ 2 } ,
\eeq
and
\beq{3.26}
\parallel u_i^{1/2} R_A ( E ) \chi_{ij}^+ \parallel_{HS} \leq d ( E )^{-1}
\parallel u \parallel_1^{1/2} C_E ,
\eeq
and similarly for the other two terms in (2.18). So for the second sum we
obtain
\beq{3.27}
\Sum_{ \parallel i - j \parallel > 2 } \parallel K_{i j } ( E ) \parallel_1
\leq d ( E )^{-2} C ( a , E ) | \mbox{supp} u | ~ | \Lambda | .
\eeq
This estimate and (3.21) give the desired upper
bound on (3.17). The
constant $C ( E , B )$, defined above,
can be approximated by an integral giving (3.3).
This proves the theorem. \end{prf}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\setcounter{chapter}{4}
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\section{Sketch of the Proof of Localization}\label{S.4}
The proof of localization for the model (1.1)-(1.3)
follows the basic ideas of \cite{[CoHi1]}. Once
we control the decay of the localized resolvents of the
infinite-volume Hamiltonian, the absence of continuous spectrum
is proved using the Simon-Wolff \cite{[SiWo]} type
argument in \cite{[CoHi1]}. Hence, in this section, we
will comment on the modifications of sections 4 and 5 of
\cite{[CoHi2]} and of the multiscale analysis of
\cite{[CoHi1]} necessary when the potential is unbounded
(see also section 6 of \cite{[BCH]}).
In addition to the Wegner estimate, Theorem 3.1,
we need to verify an initial decay hypothesis on the
localized resolvent of the finite-area Hamiltonian $ H_{ \Lambda }
$.
We fix the band index $ n $. For each $ E \in
I_n (B) = ( - \infty ,
B - {\cal O} ( B ^{-1} ) ] \bigcup \{
{\displaystyle \cup_{ j = 0 }^n }
\: [ E_j (B) + {\cal O} ( B ^{-1} ) , E_{j + 1} (B) -
{\cal O} ( B ^{-1} ) ] \} $, we show that the following condition
holds. For $ \delta > 0 $ small, let
$ \chi_{ l , \delta } $ be a smoothed characteristic
function so that $\mbox{supp} \; \chi_{ l , \delta }
\subset \Lambda_l $, $ \chi_{ l , \delta } |
\{ x \in \Lambda _l | \mbox{dist} \: ( x , \partial
\Lambda _l ) \geq \delta \} $,
and $\mbox{supp} \; \nabla \chi_{ l ,
\delta } \subset \{ x \in \Lambda _l | \mbox{dist} \: ( x , \partial
\Lambda _l ) \leq
\delta \} $. Let $ W ( \chi_{ l , \delta } ) \equiv [ -
\Delta ,
\chi_{ l , \delta } ] $ be a first-order differential
operator localized near the boundary of $\chi_l$.
\vspace{.1in}
\begin{verse}
[H1]$( \gamma _0 , l_0 ) $ For some $ \gamma _0 > 0 $,
$ l_0 > 1 $ such that \ $ l_0 \gamma _0 > 1 , ~~
\exists ~ B_n > 0 $ such that $ \forall ~ B > B_n, ~~
\exists ~ \xi > 4 $ such that
\end{verse}
\beq{4.1}
I \hspace{-.17cm} P \left \{ \begin{array}[t]{c}
\mbox{sup} \\ \vspace{-.1in} \mbox{\small $ \epsilon > 0 $}
\end{array} \: || W
( \chi_{ l_0 , \delta } ) R_{ \Lambda _{ l_0 } }
( E + i \epsilon ) \chi _{ l_0 / 3 } || < e^{ - \gamma _0 l_0 }
\right \} \geq
1 - l_0^{- \xi }
\eeq
\vspace{.1in}
\noindent
Note that the size of the magnetic field $ B_n $ depends on the
number $n$ of bands treated (but not on the energy $E$).
This estimate (4.1) is established following the lines
in \cite{[CoHi2]}. We use percolation theory to prove the existence
of effective magnetic barriers with large probability. Geometric
perturbation
theory is then used to prove the decay of the localized resolvent
across these barriers. The main new feature
in the case of densities with unbounded support is that
the potential $V_{\omega}$ is no longer bounded. However,
since we work only with finite-area Hamiltonians, we can compute
the
probability that $ V_{ \Lambda _l } $ is bounded by a
constant increasing with $ l $.
This probability approaches one as $l \rightarrow
\infty$,
and can be absorbed into the probability on the
right side of
[H1]$ ( \gamma _0 , l_0 ) $.
As a consequence of this $l$-dependent bound on
$ V_{ \Lambda _l } $ ,
the constants at each step in the multiscale analysis depend
polynomially on $l$. We will indicate below
how this is easily absorbed into the
exponent. The second main difference is that we will prove
localization at all energies below $ B - {\cal O} ( B ^{-1} ) $.
We first turn to the percolation and decay estimates of section 4
in
\cite{[CoHi2]}. We recall that we associate the lattice
$ \Gamma \equiv e^{ i \pi / 4 } \sqrt{2} ~ Z \hspace{-.17cm} Z ^2 $
with
$ Z \hspace{-.17cm} Z ^2 $. To each site $ j \in Z \hspace{-.17cm}
Z ^2 $
we associated a bond $ b_j \in \Gamma $. For simplicity
of exposition we will work with $ E \in I_0 (B) $. The
case of any $n$ is analogous.
We first consider $ E < B $. We define a bond $ b_j
\in \Gamma $ {\it occupied} if $ \lambda _j ( \omega )
> ( E - B ) / 2 $. Note that the probability $p$
that $ b_j $ is occupied is $ p =
\int_{ \frac{ E - B }{ 2 }}^{ \infty } \:
g ( \lambda ) \: d \lambda > \frac{1}{2} = p_c $, the critical
probability. We define a ribbon ${\cal R}$ associated with
a closed circuit $ {\cal C} $ of occupied bonds in $ \Gamma $
as in \cite{[CoHi2]}. Given that such a closed circuit
${\cal C}$ exists, we have for all $ x \in {\cal R} $,
\beq{4.2}
V(x) + B - E > ( B - E ) / 2 \equiv a > 0 ,
\eeq
with a good probability. For $ E > B $, we consider the bond
$ b_j $ to be {\it occupied} if $ \lambda _j ( \omega ) <
( E - B ) / 2 $. The probability that $ b_j $
is occupied is $ p = \int_{ - \infty }^{ ( E
- B ) / 2 } \: g ( \lambda ) \: d \lambda > \frac{1}{2}
= p_c $. On a closed ribbon of occupied bonds we have
\beq{4.3}
V(x) + B - E < ( B - E ) / 2 = - a < 0 .
\eeq
Percolation theory (cf. \cite{[Gr]})
states that if $ p > p_c $ then the
probability that there exists a closed circuit ${\cal C}$
of occupied bonds lying in the annular region $a_l$ between a square
of side $l$ and a square of side $ 3l $ is $ \geq 1
- C l e^{ - m( 1 - p ) l } $,
where $ m (p) $ is a strictly positive, monotone function satisfying
$ m (p) \downarrow 0 $ as $ p \uparrow p_c $. Consequently,
the following proposition (Proposition 4.1 of
\cite{[CoHi2]}) is valid with the probability $p$ given
above. For an open region $\Omega \subset \R^2$, we define $inrad \Omega \equiv
\mbox{sup} \; \{ R > 0 \; | \; B_R ( x ) \subset \Omega, \forall x \in \Omega \} $.
\begin{proposition}\label{P.4.1}
Assume hypotheses (V1)--(V2) (or (V2)') and suppose that $\mbox{supp}
\; u
\subset B_{ r_u } ( 0 ) $ for some $ r_u < 1 / \sqrt{ 2 }
$.
Let $ l > \sqrt{2}, ~E \in ( - \infty , 3B ) \backslash \{
B \} $,
and $a > 0$. Then for $m ( p ) $ and the constant $C$
introduced above,
there exists a ribbon ${ \cal R } $ satisfying
\beq{4.4}
\mbox{inrad} \; {\cal R } \geq 2 \left( \frac{1}{ \sqrt{
2} } -
r_u \right) ;
\eeq
\beq{4.5}
\mbox{dist} \; ( { \cal R }, \partial r_{ 3 l } ) , ~
\mbox{dist} \; ( { \cal R }, \partial r_l ) \geq \frac{1}{
\sqrt{ 2} } +
r_u ;
\eeq
\beq{4.6}
{ \cal R } \subset a_l ,
\eeq
and such that (4.2)--(4.3) hold with a probability larger
than
\beq{4.7}
1 - C l e^{ - m ( 1 - p ) l } ,
\eeq
where
\beq{4.8}
p \equiv \int_{- \infty }^a g ( \lambda ) d \lambda =
\int_{- a }^{ \infty } g ( \lambda ) d \lambda .
\eeq
\end{proposition}
As for the decay estimates of section 4.2 of
\cite{[CoHi2]}, we recall that
the potential $V$ of that section is supposed to be bounded, to have
compact support, and satisfy either condition (4.2)
or (4.3) $ \forall x \in I \hspace{-.17cm} R ^2 $. When we apply
this to the
finite-area Hamiltonians $ H_{ \Lambda }$, the potential
$ V $ of section 4.2 is the local
potential
$ V_{ \Lambda } $. As we have seen above, this potential
satisfies (4.2) or
(4.3), depending on the energy, with a probability given
in
(4.8). As for the boundedness of $V_\Lambda$,
the probability $P_M$ that
$ | V_{ \Lambda_l } | < M $ is given by
\beq{4.9}
P_M \equiv I \hspace{-.17cm} P \{ | V_{ \Lambda _l } (x) |
< M ~~ \forall x \in \Lambda_l \}
\geq \left [ 1 -
2 \int_M^{ \infty } \: g ( \lambda ) \: d \lambda
\right ]^{ | \Lambda_l | } .
\eeq
We are interested in the asymptotics as $ l \uparrow \infty $
and $ M \uparrow \infty $.
Using the decay on the probability given in (1.4),
we find
\beq{4.10}
P_M \geq 1 - C_0 l^2 M^{ - N } ,
\eeq
for some $N \geq 2$.
We want this to be larger than $ 1 - l^{ - \xi } $, for some
$ \xi > 4 $. Since $ N \geq 2 $, we take $ M = {\cal O}
( l^{ 3 + \epsilon } ) $, for some $ \epsilon > 0 $.
As a consequence, Lemma 4.1 of \cite{[CoHi2]}
is modified as follows.
Each local Hamiltonian $ H_{\Lambda_l} = H_A + V_{\Lambda_l} $ admits
an analytic
continuation onto a strip $ S = \{ \alpha \in \C
| ~ | \mbox{Im} \: \alpha |
< \eta_p B ^{ \frac{1}{2} } \} $, whose width is independent of $l$,
with a probability $ \geq 1 - l^{ - \xi } $, for $ \xi > 4 $ as
above. The decay estimates on the analytic family $ H_{ \Lambda_l
} ( \alpha ) $, as given in Theorem 4.3, hold with probability
$ \geq 1 - l^{ - \xi } $. The constant $ C_2 \leq \eta_p $,
and hence $ \gamma $, are independent of $l$. With regard
to the a priori estimates on the projections given in
Appendix 1, we have, for example, $ \forall ~ \alpha $ with
$ | \mbox{Im} \: \alpha | < \gamma $,
\beq{4.11}
|| P ( \alpha ) V_{ \Lambda_l } Q ( \alpha ) || < C_l B^{ -
\frac{1}{2} },
\eeq
with probability $ \geq 1 - l^{ - \xi } $, some $ \xi > 4 $,
and $ C_l = {\cal O} ( l^{ 3 + \epsilon } ) $. The constant
in Lemma A.2 is independent of $l$. We obtain the following
version of Corollary 4.1.
\vspace{.1in}
\noindent
{\bf Proposition 4.2}. {\em Let $ \chi_{ l/3 } $ be the characteristic
function on $ \Lambda _{ l/3 } $. For $ \delta > 0 $ small, let
$ \chi_{ l , \delta } $ be the characteristic function for
$ \{ x \in \Lambda _l | \mbox{dist} \: ( x , \partial \Lambda _l )
< \delta \} $. There exists constants $ C_0 > 0 $
and $ C_1 > 0 $ such that for any $ l \gg 1 , ~~ \exists ~ B_0 (l)
= {\cal O} ( l^{ 3 + \epsilon } ) $, such that $ \forall ~
B > B_0 (l) $ and $ \forall ~ E \in ( - \infty ,
B - {\cal O} ( B ^{-1} ) ] \cup [ B + {\cal O} ( B ^{-1} ) , ~~
3 B - {\cal O} ( B ^{-1} ) ] $, we have
\[ \begin{array}[t]{c}
\mbox{sup} \\ \vspace{-.1in} \mbox{\small $ \epsilon \neq 0 $}
\end{array} \| \chi _{ l , \delta } ( H_A +
V_{ \Lambda_l } - E - i \epsilon ) ^{-1} \chi _{ l / 3 } \|
\leq C_0 l^{ 3 + \epsilon }
\mbox{\rm max} \: \{ a ^{-1} , B ^{-1} \}
e^{ - \gamma l } , \]
with probability $ \geq 1 - l^{ - \xi } , ~ \xi > 4 $,
where $ \gamma \equiv C_1 \: \mbox{\rm min} \:
\{ B ^{ \frac{1}{2} } , a B \} $ and $ a = ( B - E ) / 2 $
if $ E < B $ and $ a = ( E - B ) / 2 $ if $ E > B $}.
This proposition is the main component in verifying
[H1]$( \gamma _0 , l_0 ) $.
Note that there is a gradient term in
[H1]$( \gamma _0 , l_0 ) $
coming from the operator $W (
\chi_{ l , \delta } )$.
This is now controlled as in Lemma
5.1 of \cite{[CoHi2]} with a constant
$ {\cal O} ( l^{ 3 + \epsilon } ) $ and with a probability
$ \geq 1 - l^{ - \xi } , ~~ \xi > 4 $. We have the following
analog of Proposition 5.1 of \cite{[CoHi2]}.
\vspace{.2in}
\noindent
{\bf Proposition 4.3}. {\em For any $ l_0 \gg \sqrt{2} ~~
\exists ~ B_0 \equiv B_0 ( l_0 ) $ such that for any
$ E \in I_0 (B) $, and for $ a \equiv ( B - E ) / 2 $,
if $ E < B $, and $ a \equiv ( E - B ) / 2 $, if $ E > B $,
and $ a = {\cal O} ( B^{ -1 + \sigma } ) $,
for any $ \sigma > 0 $, hypothesis
{\rm [H1]}$( \gamma _0 , l_0 ) $ holds for some $ \xi > 4 $
and for some $ \gamma _0 > C_0 \gamma $, where
$ \gamma = C_1 \: \mbox{min} \: \{ B^{ \frac{1}{2} } , B^{ \sigma }
\} $}.
As for the multiscale analysis of \cite{[CoHi1]}, at each
step of the iteration, we use (4.4)-(4.5) so we can work with a
bounded local potential $ V_{ \Lambda _{ l_n }} $. As a
consequence,
the coefficient of the estimate on the resolvent of the
finite-volume
Hamiltonian is $ {\cal O} ( l_n^{ 3 + \epsilon } ) $. This,
however, contributes a vanishing small amount $ {\cal O}
( l_{ n + 1 } ^{-1} \: \log \: l_n ) $ to the new decay
constant $ \gamma _{ n + 1 } $, and hence the iteration
is unchanged. The probability estimate is also
modified by an amount $ \geq l_n^{ -
\xi } , ~~ \xi > 4 $, which
guarantees the boundedness of the local potential.
This, however, can be easily absorbed into the probability
at the $(n+1)^{st}$
step.
Let us remark that for energies near the center of the unperturbed
gaps, in $ \sigma ( H_A ) $, the hypothesis
[H1]$( \gamma _0 , l_0 ) $ can be obtained using a Combes-Thomas
type argument without percolation theory. We refer the
reader to \cite{[BCH]} for discussion of this method.
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\end{document}