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Anderson localization, localization center
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\begin{document}
%%%%%%% DOUBLE SPACED %%%%%%%%
\setlength{\baselineskip}{15pt}
%
\title{
The repulsion between localization centers 
in  the Anderson model
}
\author{Fumihiko Nakano
\thanks{Faculty of Science, 
Department of Mathematics and Information Science,
Kochi University,
2-5-1, Akebonomachi, Kochi, 780-8520, Japan.
%
e-mail : 
nakano@math.kochi-u.ac.jp}}
\date{}
\maketitle
%%%%%%% ABSTRACT %%%%%%%%%%%%%
\begin{abstract}
In this note
we show that, a simple combination of deep results 
in the theory of random Schr\"odinger operators yields 
a quantitative estimate of the fact that the localization centers 
become far apart, as corresponding energies are close together. 
\end{abstract}

Mathematics Subject Classification (2000): 82B44, 81Q10

%%%%% INTRODUCTION %%%%% 
\section{Introduction}
%
In this paper, 
we consider a simple random system and its spectral region where the Anderson localization holds (i.e., we have dense point spectrum with exponentially decaying eigenfunctions). 
%
We study 
the ``center" of these localized eigenfunctions and prove a repulsive property on the distribution of those in relation to their corresponding energies, that is, 
%
``as eigenvalues get closer, the corresponding localization centers become far apart"
$\cdots (*)$.
%
A naive explanation 
which supports the observation $(*)$ is : 

(1) 
Suppose we have 
two eigenvalues 
$E_1, E_2$
with their corresponding localization centers 
$x_1, x_2$
satisfy 
$| x_1 - x_2 | \sim L$. 
%
Because 
eigenfunctions are exponentially localized, 
we can find a finite box 
$\Lambda$ 
of size in the order of
$L$ 
surrounding
$x_1, x_2$, 
so that 
$H_{\Lambda}$
has two eigenvalues close to 
$E_1, E_2$. 
%
If the density of state is finite, 
eigenvalues of 
$H_{\Lambda}$ 
would arrange in the order of 
$| \Lambda |^{-1}$
so that we could have
$| E_1 - E_2 | \ge | \Lambda |^{-1} \sim L^{-d}$.
%

(2)
(Carmona-Lacroix \cite[p.338]{CL})
Because locally we have no repulsion between eigenvalues(Molchanov, Minami \cite{Molchanov, Minami}), 
the overlap between the eigenfunctions should be small as the corresponding energies get closer. 

The phenomenon $(*)$
has been observed in numerical calculations
\cite[p.338]{CL}.
%
\cite{Molchanov}
proves $(*)$ in a special model on one space dimension, with a complicated statement. 
%
On the other hand, the observation 
$(*)$
was used by Mott in the study of the ac-conductivity of random systems whose mathematical study is done recently
(Kirsch-Lenoble-Pastur, Klein-Lenoble-M\"uller \cite{KLP, KLM}).
%
The purpose of this paper 
is to obtain a quantitative statement of $(*)$ which holds for almost surely. 
%
To study the same property 
for the averaged quantity would require more elaborate analysis as is done in \cite{KLM}. 

%
Our model 
is the standard tight binding Hamiltonian with random potential on 
${\bf Z}^d$
as was treated by 
\cite{Minami}.
%
\[
(H \varphi)(x)
:=
\sum_{|y-x| =1} \varphi (y)
+
\lambda V_{\omega} \varphi (x), 
\quad
\varphi \in l^2 ({\bf Z}^d)
\]
%
where
$\lambda > 0$
is the coupling constant and 
$\{ V_{\omega} (x) \}_{x \in {\bf Z}^d}$
is independent, identically distributed 
real-valued random variables on a probability space
$(\Omega, {\cal F}, {\bf P})$
whose common distribution is assumed to have a bounded  density 
$\rho$. 
%
Under this assumption, the following facts are well-known.
%
(1)
$\sigma (H) = \Sigma := [-2d, 2d] + \mbox{supp } \rho$, a.s.
(Kunz-Souillard \cite{Kunz-Souillard}),
%
(2)
We can find a bounded interval
$I (\subset \Sigma)$
such that the spectrum of 
$H$ 
in 
$I$
is almost surely pure point with exponentially decaying eigenfunctions
(Anderson localization). 
%
$I$
can be taken, for instance
(Fr\"olich-Spencer, von Dreifus-Klein, Aizenman \cite{FS, vonDK, Aizenman}), 
(i)
(high disorder) 
$I = \Sigma$ 
if 
$\lambda \gg 1$, 
(ii)
(extreme energy) away from the origin, 
(iii)
(weak disorder) 
away from the spectrum of the free Laplacian if 
$| \lambda | \ll 1$, and 
(iv) band edges. 
%
Before stating our results, 
we define some notations. 
\\

%
\noindent
{\bf Definition}\\
%
{\it 
(1)
$\Lambda_L (x) = 
\{ y \in {\bf Z}^d : | y_j - x_j | \le \frac L2, 
j=1, 2, \cdots, d \}$
is the finite box in 
${\bf Z}^d$
of length 
$L > 0$
with its center 
$x = (x_1, \cdots, x_d) \in {\bf Z}^d$.
%
For simplicity, 
$\Lambda_L := \Lambda_L (0)$. 
%
$\partial \Lambda := 
\{ y \in \Lambda : \exists z \notin \Lambda, 
|y - z | = 1\}$
be the boundary of the box 
$\Lambda$. 
%
$H_{\Lambda}$
is the restriction of 
$H$
on 
$\Lambda$
with Dirichlet boundary condition. 
%
$G_{\Lambda}(E; x,y) :=
\langle 
\delta_x, (H_{\Lambda}- E)^{-1} \delta_y
\rangle_{l^2({\bf Z}^d)}$
is the matrix element of the resolvent 
$(H_{\Lambda}- E)^{-1}$
%
where
$\delta_x(z) = 1 (z=x), 0 (z\ne x)$
and 
$\langle \cdot, \cdot \rangle_{l^2({\bf Z}^d)}$
is the inner-product in 
$l^2({\bf Z}^d)$.
%
$|\Lambda| = \sharp \Lambda$
is the volume of a box 
$\Lambda (\subset {\bf Z}^d)$
and 
$| I | = b - a$
is the width of an interval 
$I = (a,b) (\subset {\bf R})$. 
%
$\chi_{\Lambda}$
is the characteristic function of a box 
$\Lambda$. 
\\
%
(2)
We say the box
$\Lambda_L (x)$
is
$(\gamma, E)$-regular
iff 
$E \notin \sigma (H_{\Lambda_L (x)})$
and for any
$y \in \partial \Lambda_L (x)$, 
%
\[
| G_{\Lambda_L (x)} (E; x, y) | 
\le
e^{- \gamma L/2}.
\]
%
(3)
For 
$\phi \in l^2 ({\bf Z}^d)$, 
let 
$X({\phi})$
be the set of its localization centers given by
%
\[
X({\phi}) := 
\left\{
x \in {\bf Z}^d : 
| \phi (x) | = \max_{y \in {\bf Z}^d} | \phi (y) |
\right\}.
\]
%
This notion 
is due to Germinet-De Bi\`evre \cite{BG}.
%
Since
$\phi \in l^2 ({\bf Z}^d)$, 
$X({\phi})$
is a finite set. 
%
Moreover, 
for the set 
$\{ E_j (\omega) \}_{j \ge 1}$
of eigenvalues of 
$H$
counting multiplicity, we take 
the  corresponding eigenfunctions
$\{ \phi_j (\omega) \}_{j \ge 1}$
and let 
$X(E_j(\omega)) := X(\phi_j  (\omega))$.
\\
%
(4)
For 
a finite box
$\Lambda$
and 
$\phi \in l^2({\bf Z}^d)$, 
we say 
$\phi$
is localized in 
$\Lambda$
iff
$X({\phi}) \cap \Lambda \ne \emptyset$.
%
For an eigenvalue 
$E_j (\omega)$
of 
$H$, 
we say 
$E_j (\omega)$
is localized in 
$\Lambda$
iff
$X(E_j(\omega)) \cap \Lambda \ne \emptyset$}. 
\\

Let 
$I (\subset \Sigma)$
be the bounded interval where the initial length scale estimate of the multiscale analysis holds : \\
%

%%%%%
\noindent
{\bf Assumption}
\\
%
{\it 
We have an interval 
$I (\subset \Sigma)$
with 
%
\[
{\bf P}\left(
\mbox{for $\forall E \in I$, 
$\Lambda_{L_0}$
is
$(\gamma, E)$-regular}
\right)
\ge
1 - L_0^{-p}
\]
%
for some 
$\gamma > 0$, $p > 2d$
and some 
$L_0 > 0$
sufficiently large.
}\\
%

\noindent
$I$
can be taken in regions mentioned in the paragraph preceding  Definition in this section. 
%
This condition, together with Wegner's estimate, guarantee to apply the multiscale analysis, and from which the following fact is deduced
\cite{vonDK} : 
%
we can find 
$\alpha = \alpha(p,d)$, 
$1 < \alpha < 2$
such that, putting
%
\[
\Lambda_k (x) = \Lambda_{L_k} (x), 
\quad
L_k = L_{k-1}^{\alpha},
\quad
1 < \alpha < 2, 
\quad
x \in  {\bf Z}^d, 
\]
%
we have 
%
\begin{equation}
{\bf P}
\left\{
\mbox{For all 
$E \in I$
either 
$\Lambda_k(x)$
or
$\Lambda_k(y)$
is
$(\gamma, E)$-regular}
\right\}
\ge 
1 - L_k^{-2 p}
\label{MSA}
\end{equation}
%
for any 
$x, y$
with
$|x - y | > L_k$.
%
The first result
implies the distribution of localization centers are ``thin" in 
${\bf Z}^d$. 
%
\begin{theorem}
{\bf (localization centers are thin)}\\
%
Let
$d_k = | \Lambda_k |^{-1} k^{-2}$, 
$E \in I$, 
$J_k = (E - \frac {d_k}{2}, E + \frac {d_k}{2})
(\subset {\bf R})$, 
$k=1, 2, \cdots$.
%
Then for
a.e. 
$\omega$, 
we can find 
$k_0 = k_0 (\omega)$
such that, if
$k \ge k_0$, 
there are no eigenvalues of 
$H$
in 
$J_k$
localized in 
$\Lambda_k$.
%
\label{localized centers are thin}
\end{theorem}
%
\begin{remark}
Theorem 
\ref{localized centers are thin}
states that, for any 
$E \in I$, 
localization center run away from the origin, as the corresponding eigenvalue approaches 
$E$. 
%
It implies 
the distribution of the localization centers is thin,
while the eigenvalues are dense in 
$I$. 
%
A naive explanation of this fact is : 
we have infinite number of eigenvalues near 
$E$
while the number of states is proportional to the volume, 
by the finiteness of the density of states. 
\end{remark}
%
\begin{remark}
As the density of states obeys 
the Lifschitz tail asymptotics on  the bottom of the spectrum, another estimate is obtained there : 
%
$d_k$
can be replaced by 
$d_L = (a, a + \frac {1}{| \Lambda_L |^{2/d}})$, 
$a = \inf \sigma (H)$
(Simon \cite{Simon-Lifschitztail}).
\end{remark}
%
\begin{remark}
Theorem \ref{localized centers are thin}
holds true for random Hamiltonians 
where the multiscale analysis is applicable and Wegner's estimate holds. 
%
\end{remark}
%
\begin{theorem}
{\bf (localization centers are repulsive)}\\
%
Let 
$d_k = | \Lambda_k |^{-2}k^{-2}$, 
$k =1, 2, \cdots$.
%
For 
a.e. 
$\omega$, 
the following event occurs. 
%
For any 
$x \in {\bf Z}^d$, 
there exists 
$k_0 = k_0(\omega, x)$
such that for 
$k \ge k_0(\omega, x)$
and any interval 
$J \subset I$
with 
$|J| \le d_k$, 
there is at most one eigenvalue of 
$H$
in 
$J$
localized in 
$\Lambda_k (x)$. 
%
\label{localized centers are repulsive}
\end{theorem}
%
\begin{remark}
%
Theorem \ref{localized centers are repulsive}
implies : 
%
for a.e. $\omega$ and for any eigenvalue 
$E = E_j (\omega) \in I$
of 
$H$, 
%
we can find
$k_1 = k_1 (\omega,E_j(\omega))$
such that, for any 
$k \ge k_1$, 
we have no eigenvalues of 
$H$
in 
$J_k = (E_j(\omega) - \frac {d_k}{2}, 
E_j(\omega) + \frac {d_k}{2})$
localized in 
$\Lambda_k(x)$
for any 
$x \in X(E_j(\omega))$
except 
$E_j(\omega)$ 
itself.
%

Hence
this theorem roughly states that,  for a localization center 
$x$
with energy 
$E$, 
any other localization center must be away from 
$x$
at least in the distance of 
$L_k/2$, 
if the corresponding eigenvalues are within the distance of 
$| \Lambda_k |^{-2}$
from 
$E$. 
%
And 
this happens simultaneously for all eigenvalues in 
$I$
almost surely. 
\end{remark}
%
\begin{remark}
Theorem \ref{localized centers are repulsive}
also proves that eigenvalues in 
$I$
are simple almost surely. 
%
Indeed, this is done by Klein-Molchanov \cite{KM}
by the argument similar to ours, but without relying on the multiscale analysis. 
\end{remark}
%
\begin{remark}
One of the essential ingredients of the proof of 
Theorem \ref{localized centers are repulsive}
is the estimate of the probability to have more than two eigenvalues on a given interval obtained by Minami 
\cite[Lemma 2]{Minami}, 
which is also essential to prove the absence of repulsion of eigenvalues, 
and known to hold in the Anderson model only so far. 
%
Hence 
something different could be expected for the acoustic type operator, in which the level repulsion is known to occur
(Grenkova-Molchanov-Sudarev \cite{Grenkova}). 
\end{remark}
%
\begin{remark}
This result concerns 
the distribution of the localization centers which holds for almost surely. 
%
If one is interested in the fluctuation of those, 
by proceeding along the ideas in \cite{Minami}, 
one could expect the Poisson-type behavior also for localization centers, under a suitable scaling.
%
To verify this observation 
would be an interesting problem\footnote{The author would like to thank Rowan Killip for pointing this out. }
\cite{Nakano}. 
\end{remark}
%

In the following section, 
we prove
Theorem \ref{localized centers are thin}, 
\ref{localized centers are repulsive}, 
along the naive argument given at the beginning of this section, which is done by making use of the machinery developed by 
Germinet-De Bi\`evre,
 Damanik-Stollmann, 
 Klein-Molchanov and  
 Minami \cite{BG, DS, KM, Minami} : 
%
(i)
If we have 
an (resp. at least two) eigenvalue in an interval 
$J (\subset I)$, 
the corresponding eigenfunction is exponentially small outside a box 
$\Lambda$
surrounding its localization center
\cite{BG, DS}.
%
Then 
$H_{\Lambda}$
has an (resp. at least two) eigenvalue in 
$J$
\cite{KM}.
%
(ii)
We estimate the event that 
$H_{\Lambda}$
has an (resp. at least two) eigenvalue in 
$J$
by Wegner's (resp. Minami's) estimate.
%
Then 
the usual Borel-Cantelli argument gives the assertion.
%
In Appendix, 
we collect lemmas used in these proofs borrowed from 
\cite{BG, DS, KM, Minami}.
%
\section{Proof of Theorems}
%
We first set 
%
\begin{eqnarray*}
E_j &=& \Bigl\{
\omega \in \Omega : 
\mbox{
For some 
$E \in I$
and some
$x, y \in \Lambda_{3 L_{j+1}}$
with 
$\Lambda_j (x) \cap \Lambda_j (y) = \emptyset$, }
\\
&&\qquad\qquad
\mbox{
$\Lambda_j (x)$, $\Lambda_j(y)$
are both 
$(\gamma, E)$-singular}
\Bigr\}
\\
%
\Omega_k &=& \bigcap_{j \ge k} E_j^c.
\end{eqnarray*}
%
Then by (\ref{MSA}), 
%
\begin{equation}
{\bf P} (\Omega_k)
\ge
1 - C_1(\alpha, d, p) L_k^{2d \alpha - 2 p}
\label{Omega}
\end{equation}
%
for some 
$C_1= C_1(\alpha, d, p)$.
%
In what follows, 
we take and fix any 
$0 < \gamma' < \gamma$, 
and 
$k_0 (\alpha, d, \gamma)$, 
$k_1(\alpha, d, \gamma, \gamma')$, 
$L_0(\gamma')$, 
and 
$C_2$
are positive constants given  in Appendix.
%
\begin{lemma}
{\bf \mbox{}}\\
%
Let 
$J=(a,b)( \subset I)$
and let 
$\epsilon_{L_k} = C_2 e^{-\gamma' L_k/2}$. 
%
If 
$k \ge k_0(\alpha, d, \gamma) \vee k_1(\alpha, d, \gamma, \gamma')$, $L_k \ge L_0(\gamma')$, 
the following estimates hold.
%
\label{whole Hamiltonian}
%
\begin{eqnarray*}
(1) \quad &&
{\bf P}
\left(
\mbox{we have an eigenvalue of 
$H$ 
in 
$J$ localized in $\Lambda_k$}
\right)
\\
&& \qquad
\le
\| \rho \|_{\infty}
( | J |+ 2\epsilon_{L_k} )   | \Lambda_{3k} |
+
C_1(\alpha, d, p)
L_k^{2d \alpha -2 p}.
\\
%
(2) \quad &&
{\bf P}
\left(
\mbox{we have at least 2 eigenvalues of 
$H$ 
in 
$J$ 
localized in 
$\Lambda_k$}
\right)
\\
%
&& \qquad \le 
\pi^2 \| \rho \|^2_{\infty}
| \Lambda_{3 L_k} |^2
( | J | + 2\epsilon_{L_k} )^2
+
C_1(\alpha, d, p)
L_k^{2d \alpha -2 p}.
\end{eqnarray*}
%
\end{lemma}
%
\begin{proof}
(1)
Let 
%
\[
A_k := \left\{
\omega \in \Omega : 
\mbox{we have an eigenvalue of 
$H$ 
in 
$J$ localized in $\Lambda_k$}
\right\}.
\]
%
Let 
$\omega \in A_k \cap \Omega_k$.
%
Then we have an eigenvalue 
$E \in J$
localized in 
$\Lambda_k$
and since
$j \ge k_0 \vee k_1$, $L_k \ge L_0$, 
the corresponding eigenfunction 
$\phi$
satisfies
%
\[
\| ( 1- \chi_{3 L_k} ) \phi \|
\le
e^{- \gamma' L_k/2}
\]
%
by Lemma \ref{decay estimate2}.
%
Then 
the argument of the proof of 
Lemma \ref{approximation}
shows that 
$H_{\Lambda_{3 L_k}}$
has an eigenvalue in 
$(a - \epsilon_{L_k}, b + \epsilon_{L_k})$.
%
By 
Wegner's estimate:
\footnote{$\{ E_j (\Lambda) \}_{j=1}^{| \Lambda |}$
is the set of eigenvalues of 
$H_{\Lambda}$}
%
\[
{\bf P}(\sharp \{ E_j (\Lambda) \in J \} \ge 1)
\le
\| \rho \|_{\infty} | \Lambda | \cdot | J |,
\]
%
we have
%
$
{\bf P} (A_k \cap \Omega_k)
\le
\| \rho \|_{\infty}  | \Lambda_{ 3 L_k } | ( | J |+ 2\epsilon_{L_k} ).
$
\\
%
(2)
Let 
%
\[
B_k := \left\{ \omega \in \Omega : 
\mbox{we have at least 2 eigenvalues of 
$H$ 
in 
$J$ 
localized in 
$\Lambda_k$}
\right\}
\]
%
By the same argument 
in the proof of Lemma \ref{whole Hamiltonian}(1), 
if
$\omega \in B_k \cap \Omega_k$, 
$H_{\Lambda_{3 L_k}}$
has at least two eigenvalues in 
$(a - \epsilon_{L_k}, b + \epsilon_{L_k})$.
%
By Minami's estimate
\cite[Lemma 2]{Minami}, 
\cite[Appendix]{KM}:
%
\[
{\bf P}\left(
\{ \sharp \{ E_j (\Lambda) \in J \} \ge 2 \}
\right)
\le
\pi^2 \| \rho \|^2_{\infty} | \Lambda |^2 | J |^2, 
\]
%
we have
%
$
{\bf P} (B_k \cap \Omega_k)
\le
\pi^2 \| \rho \|^2_{\infty}
\| \Lambda_{3 L_k} |^2 ( | J |+ 2\epsilon_{L_k} )^2.
$
%
\QED
\end{proof}
%




%
\noindent
{\it Proof of Theorem \ref{localized centers are thin}}
\\
%
We consider the following event. 
%
\[
A_k = \left\{
\omega \in \Omega : 
\mbox{ 
We have an eigenvalue of $H$ in $J_k$
localized in $\Lambda_k$
} 
\right\}.
\]
%
By Lemma \ref{whole Hamiltonian}, 
if 
$k \ge k_0(\alpha, d, \gamma) \vee k_1(\alpha, d, \gamma, \gamma')$
and
$L_k \ge L_0(\gamma')$, 
we have
%
\[
{\bf P}(A_k)
\le
\| \rho \|_{\infty} | \Lambda_{3 L_k} | 
 ( d_k + 2\epsilon_{ L_k} )
+
C_1 (\alpha, d, p) L_k^{2d(\alpha-1) - 2 p}
\]
%
%
Since
$d_k +2 \epsilon_{L_k} \le 2 d_k$
for sufficiently large 
$k$, 
$\sum_k {\bf P} (A_k) < \infty$. 
%
The
Borel-Cantelli
argument then proves the assertion of 
Theorem \ref{localized centers are thin}. 
\QED\\
%%%%%

%%%%%
\noindent
{\it Proof of Theorem \ref{localized centers are repulsive}}
\\
%
We consider the following events. 
%
\begin{eqnarray*}
B_k (x, J)
&=&
\left\{
\omega\in \Omega : 
\mbox{at least two eigenvalues of $H$ in $J$
 localized in $\Lambda_k(x)$}
\right\}, 
\\
%
B_k (x)
&=&
\Bigl\{
\omega\in \Omega : 
\mbox{at least two eigenvalues of $H$ in $J$}
\\
%
&&\qquad\qquad
\mbox{ 
for some 
$J(\subset I)$ with 
$|J| \le d_k$ in $\Lambda_k(x)$}
\Bigr\}.
\end{eqnarray*}
%
Suppose
$| J | \le 2 d_k$, $J \subset I$. 
%
Then
by the argument in the proof of 
Lemma \ref{whole Hamiltonian}(2), 
%
\[
{\bf P}(B_k(x, J) \cap \Omega_k)
\le
\pi^2 \| \rho \|_{\infty}^2 ( 3 d_k )^2
\cdot
| \Lambda_{3 L_k} |^2
\]
%
for 
$k$
sufficiently large. 
%
Here we use the argument in 
\cite[Lemma 2]{KM}
and cover the interval 
$I$
by those 
$J(i, d_k)$, $i = 1, 2, \cdots, N_k$
of width 
$2 d_k$
such that the left end of 
$J(i+1, d_k)$
coincides with the mid point of 
$J(i, d_k)$. 
%
Then 
$N_k \le \frac {| I |}{2 d_k} \cdot 2 = \frac {| I |}{d_k}$
and any interval (in $I$) of width less than 
$d_k$
is contained by some 
$J(i, d_k)$.
%
Hence
%
\[
{\bf P} (B_k (x) \cap \Omega_k)
\le
\pi^2 \| \rho \|_{\infty}^2 
( 3 d_k )^2 \cdot \frac {|I|}{d_k}
\cdot
| \Lambda_{3 L_k} |^2
\]
%
for large 
$k$
and therefore 
$\sum_k {\bf P}(B_k(x)) < \infty$
by (\ref{Omega}).
%
By the Borel-Cantelli lemma, 
$\Omega(x) = \liminf_{k \to \infty} B_k^c(x)$
satisfies
${\bf P}(\Omega(x)) = 1$
and for 
$\omega \in \Omega(x)$
we can find
$k_0 = k_0(\omega,x)$
such that for any
$k \ge k_0(\omega, x)$
and for any interval 
$J (\subset I)$ 
with 
$| J | \le d_k$, 
we have at most one eigenvalue of 
$H$
in 
$J$
localized in 
$\Lambda_k (x)$. 
%
For 
$\omega \in \Omega' = \bigcap_{x\in {\bf Z}^d} \Omega(x)$, 
the event described in the statement of 
Theorem \ref{localized centers are repulsive} occurs. 
%
\QED
%%%%%

%%%%%
\section{Appendix}
%
In this section, 
we state Lemmas used in section 2, which are borrowed from 
\cite{BG, DS, KM, Minami}. 
%
The following lemma is 
\cite[Lemma 3.5]{BG}.
%
\begin{lemma}
{\bf (\cite[Lemma 3.5]{BG})}\\
%
We can find a constant 
$k_0 = k_0 (\alpha, d, \gamma)$
such that, if 
$k \ge k_0$
and 
$\phi \in l^2 ({\bf Z}^d)$
satisfies
$H \phi = E \phi$, 
then 
$\Lambda_{L_k} (x_{\phi})$
is
$(\gamma, E)$-singular.
%
\label{localized centers live in bad box}
\end{lemma}
%
In what follows, we take and fix any 
$\gamma'$ 
with 
$0 < \gamma' < \gamma$.
%
\begin{lemma}
{\bf \mbox{}}\\
%
We can find a constant 
$k_1 = k_1 (\alpha, d, \gamma, \gamma')$ 
such that, if 
$k\ge k_0(\alpha, d, \gamma) \vee k_1 (\alpha, d, \gamma, \gamma')$, 
$\omega \in \Omega_k$
and if 
$\phi \in l^2({\bf Z}^d)$
satisfies 
$H \phi = E \phi$, $\| \phi \| = 1$
and localized in 
$\Lambda_k$, 
%
\label{decay estimate2}
%
\[
\| ( 1 - \chi_{\Lambda_{3 L_k}}) \phi \|
\le
e^{- \gamma' L_k/2}, \quad \gamma' < \gamma.
\]
%
\end{lemma}
%
Lemma \ref{decay estimate2}
is proved along the argument in 
\cite[Step 3, Theorem 3.1]{DS}.\\
%

\noindent
{\it Sketch of proof }
%
We divide
$\Lambda_{3 L_k}^c$
into annulus : 
%
$
\Lambda_{3 L_k}^c 
=
\bigcup_{i \ge k} M_i, 
\;
M_i = \Lambda_{3 L_{i+1}} \setminus \Lambda_{3 L_i}, 
\;
i \ge k.
$
%
Then we have
%
$
\| (1 - \chi_{3 L_k}) \phi \|^2
\le 
\sum_{i = k}^{\infty}
\| \chi_{M_i} \phi \|^2
%
\le 
\sum_{ i = k }^{\infty} \sum_{x \in M_i}
| \phi (x) |^2.
$
%
Since 
$\Lambda_{L_i} (x_{\phi}) \cap \Lambda_{L_i}(x) = \emptyset$
for any 
$x \in M_i$, 
$\Lambda_{L_i}(x)$
is 
$(\gamma, E)$-regular
by Lemma \ref{localized centers live in bad box}.
%
\QED
%
%
\begin{lemma}
{\bf \mbox{}}\\
%
We can find positive constants
$C_2$, 
$L_0 = L_0 (\gamma')$
with the following property. 
%
If 
$\varphi_1, \varphi_2$
satisfy
%
\begin{eqnarray*}
&&
H \varphi_j = E_j \varphi_j, 
\quad
\| \varphi_j \| = 1, 
\quad
E_j \in J = (a,b), 
\quad
j=1,2,
\\
%
&&
\| ( 1 - \chi_{\Lambda_L}) \varphi_j \|
\le
e^{ - \gamma' L/2}, 
\quad
j = 1,2, 
\quad
L \ge L_0, 
\end{eqnarray*}
%
then 
$H_{\Lambda_L}$
has at least two eigenvalues in 
$(a - \epsilon_L, b + \epsilon_L)$
where
$\epsilon_L = C_2 e^{-\gamma' L/2}$.
%
\label{approximation}
\end{lemma}
%
The proof 
is found in \cite{KM} : 
we orthonormalize
$\varphi_j^{\Lambda} = \chi_{\Lambda} \varphi_j$, 
$j=1,2$
and estimate from below the trace of the spectral projection of 
$H$
corresponding to the interval 
$(a- \epsilon_L, b+\epsilon_L)$.\\
%

\noindent {\bf Acknowledgement }
The author would like to thank Professors 
Abel Klein, Rowan Killip, Nariyuki Minami and a referee 
for their discussions and comments. 

%%%%% REFERENCES %%%%%%%%%%%%%%%%%%%%%
%
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%
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\end{document}

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