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random operators, localization, density of states, Wegner estimate, point interactions, wavelets
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%preliminary version 25.06.01, 30.11.01, 01.12.01, 12.02.02
% 06.03.02, 11.04.02, 07.06.02, 19.07.02 of Hislop-Kirsch-Krishna %
%12.08.02

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\begin{document}


\begin{titlepage}

\begin{center}

{\bf SPECTRAL AND DYNAMICAL PROPERTIES OF RANDOM MODELS \\
WITH NONLOCAL AND SINGULAR INTERACTIONS}

\vspace{0.3 cm}

\setcounter{footnote}{0}
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{\bf Peter D.\ Hislop \footnote{Supported in part by NSF grants 
DMS-0202656 and INT-9810322.}} 

\vspace{0.2 cm}

{\ten Department of Mathematics \\
University of Kentucky \\
Lexington, KY 40506--0027 USA}

\vspace{0.3 cm}

{\bf Werner Kirsch \footnote{Supported in part by SFB 237.}}

\vspace{0.2 cm}

{\ten Mathematics Department \\  
Ruhr Universit\"at Bochum \\
Bochum, GERMANY }

\vspace{0.3 cm }

{\bf M.\ Krishna \footnote{Supported in part by NSF grant
INT-9810322 and the DST grant DST/INT/US(NSF-RP014)/98.}}

\vspace{0.2 cm }

{\ten Institute of Mathematical Sciences \\
CIT Campus \\  
Taramani 600 113  \\   
Chennai, INDIA }

\end{center}

\vspace{0.1 cm}

\begin{center}
{\bf Abstract}
\end{center}

\noindent
We give a spectral and dynamical description of certain models of
random Schr\"odinger operators on $L^2 ( \R^d)$
for which a modified version of the small moment method of Aizenman and
Molchanov \cite{[AizenmanMolchanov]} can be applied.
One family of models includes includes \Schr\ operators
with random, nonlocal interactions constructed from a wavelet basis. The second family
includes \Schr\ operators
with random singular interactions randomly located on sublattices of 
$\Z^d$, for $d = 1 , 2, 3$.
We prove that these 
models are amenable to Aizenman-Molchanov-type
analysis of the Green's function, thereby eliminating the
use of multiscale analysis.
The basic technical result is an estimate on the expectation of small moments 
of the Green's function. Among our results,
we prove a good Wegner estimate and the H\"older continuity of
the integrated density of states, and spectral and dynamical localization
at negative energies. 

\vspace{0.1 cm}

\noindent
\today

\end{titlepage}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\section{Introduction and Main Results}\label{S.1}

In this paper, we treat two new interesting models in the theory of random
\Schr\ operators. One model is a nonlocal, random interaction model, and
the second is a random, singular interaction model. It turns out that, with
an appropriate reduction for the second model, 
both models can be treated with the same
techniques. The theoretical basis of our paper is the analysis of the following
abstract model.
We consider
a family of random \Schr\ operators on a separable Hilbert space ${\cal
H}$ having the form
\beq
H_\omega = H_0 + \Sum_{j \in I} \lambda_j ( \omega ) P_j .
\eeq
The index set $I$ is countable and
the family $\{ P_j \; | \; j  \in I \}$ is a complete family of rank-one
orthogonal projectors on ${\cal H}$. The family of random variables $\{
\lambda_j ( \omega ) \; | \; j \in I, ~\mbox{and} ~\omega \in \Omega \}$
is a family of independent random variables with distributions $\mu_j$ with
bounded first moments. We will assume that
we have some {\it a priori} information
about the self-adjoint and, in general, unbounded operator $H_0$.
We show that the 
method of Aizenman-Molchanov \cite{[AizenmanMolchanov]}, and of Aizenman
\cite{[Aizenman]}, can be extended to models (1) of random operators on $\R^d$.
This extension allows us to by-pass the multiscale analysis for
obtaining estimates on the resolvents that is currently used for
models on $\R^d$. This provides a rather direct proof of localization for
models of the type (1) that cannot be treated by the methods of multiscale
analysis (MSA).
It also allows a more direct treatment of dynamical localization for continuous
models.

Our main results on this family of operators (1) are stated precisely in
section 2. We obtain {\it a priori} and {\it uniform} exponential  
bounds on the expectation of the small
moments of the Green's functions in the small and large disorder regimes.
These estimates are valid for an interval of energy $I$ on which
the unperturbed operator $H_0$ satisfies certain estimates.
These bounds lead to a proof of exponential localization
in the energy interval $I$. 
The main hypotheses on the resolvent of the unperturbed operator $H_0$
are as follows. 

\begin{enumerate}
\item{[Weak Disorder]}. For any $0 < s < 1$, suppose there is a
finite constant $K_s > 0$ such that
\beq
\int \frac{ |x|^s }{ | x-\alpha |^s } ~ d \mu_n (x) \: \leq \:
K_s \: \int \frac{1}{ |x - \alpha |^s }
~ d \mu_n ( x ) ,
\eeq
for all $\alpha \in \C$.
Suppose further that $I = [a , b ] \subset \sigma ( H_\omega )$, almost
surely, and
there exists $\gamma = \gamma (E) > 0$ such that for every $E \in I$, the
condition
\beq
K_s \left\{  \Sup_n \left[ \Sum_m \: \| P_n ( H_0 - E )^{-1} P_m \|^s
e^{ s \gamma \| n - m\| } \right] \right\} \; < \; 1 ,
\eeq
is satisfied.

\item{[Strong Disorder or High Energy]}. For any $0 < s < 1$, suppose there
is a finite constant $C_s > 0$ defined by
\beq
\int \frac{ |x - \beta |^s }{ | x - \alpha|^s } ~d \mu_n (x) \; \geq \: C_s
\:
\int \frac{1}{ | x - \alpha |^s } ~d \mu_n (x) ,
\eeq
for all $\alpha , \beta \in \C$.
Suppose that $I = [a,b] \subset \sigma ( H_\omega ) $, almost surely,
and there exists a $\gamma = \gamma (I) > 0$ so that 
the condition
\beq
C_s \left\{ \Sup_n \: \Sum_m \left[ \: \| P_n H_0 P_m \|^s
e^{ s \gamma \| n - m\| } \right] \right\}^{-1}
\; > \; 1 ,
\eeq
is satisfied.
\end{enumerate}
\noindent
We prove that, under either condition,
there exists a finite constant
$D_s = D_s (\gamma, I) > 0$, so that for every $E \in [a , b]$,
\beq
\Sup_{\epsilon \rightarrow 0}
\E \{ \; \| P_n ( H_\omega - E - i \epsilon )^{-1} P_m \|^s \; \}
\leq D_s e^{- s \gamma \| n - m \| } .
\eeq
We will show below how to use result (6) to prove localization for
one of the family of models, and to prove exponential  and dynamical
localization for another family of models, both of which we 
describe below.
Result (6) is a generalization of the weak disorder result of Aizenman
\cite{[Aizenman]} and the strong disorder or high energy result of Aizenman
and Molchanov \cite{[AizenmanMolchanov]}, 
who studied lattice models. In their papers,
$P_n$ is the rank-one orthogonal projector on the site $j \in \Z^d$, and
$H_0$ is a bounded, self-adjoint operator like the lattice Laplacian.

Our first application of this result is to a specific
{\it random nonlocal model} on ${\cal H} = L^2 (\R^d)$. The unperturbed
operator is $H_0 = - \Delta$, and the $P_n$ are
orthogonal projections onto a basis of $L^2 ( \R^d)$ given
by an orthonormal basis of wavelets. For this explicit model,
we can verify hypothesis (3) with $\gamma = 0$,
and prove localization at sufficiently negative energies.
It appears that this model cannot be treated by the methods of MSA.

We call the second family of models {\it random singular interaction
models}. These are formally defined by the family of \Schr\ operators 
\beq
H_\omega =  H_0 + \Sum_{ j \in \Gamma ( \omega ')} \lambda_j ( \omega )
\delta ( x - j ) ,
\eeq
on $L^2 ( \R^d)$, for $d = 1 , 2 , 3$.
The Dirac delta interactions are located on a subset $\Gamma ( \omega
') \subset \Z^d$ that is also random. 
Although this Hamiltonian does not have the form (1), we will show that
all of the interesting questions for this model can be obtained from a related
$\Z^d$-lattice model with an energy dependent Hamiltonian of the form
\beq
h_\omega (z) = t(z) + v , ~\mbox{on} ~\ell^2 ( \Gamma ( \omega ')) ,
\eeq
where $t(z)$ is zero on the diagonal and has exponentially-decaying
off-diagonal matrix elements, and $v$ is a random multiplication operator.
This reduction appears in the related article by A.\ Boutet de
Monvel and Grinshpun \cite{[BDMG]}, 
who proved exponential localization at negative energies for this models
with $\Gamma ( \omega ' ) = \Z^d$. 
However, they used the multiscale analysis to 
analyze $h_\omega ( z)$, whereas we use 
the method of fractional
moments presented above. Not only is this method simpler,
but we are able to prove dynamical localization for this model
using with this technique.

\subsection{The Main Results}

With these preparations, we can now state the main localization
results for these two families of models.
Specific statements are given in Theorem 3.5, Theorems 6.2 and 6.4,
and Theorems 8.2 and 8.5.  

\vspace{.1in} 
\noindent
{\bf A.\ Random nonlocal model.}

\vspace{.1in}
\noindent
{\bf Theorem 1.1.} {\it Let $H_\omega = - \Delta + \Sum_n \lambda_n (\omega)
P_n$ be a random family of \Schr\ operators as given in (1) with wavelet
interactions described in (41)--(44) of section 3. 
Under Hypotheses 3.1 and 3.2, there exists an energy $E_1(\mu) > 0$
so that $\sigma (H_\omega) \cap (-\infty , -E_1 (\mu) ] \neq \emptyset$, with
probability one, and the model has only pure point spectrum almost surely in $( - \infty,
-E_0 ]$.} 

\vspace{.1in}
\noindent
{\bf B.\ Random singular interaction model.}

\vspace{.1in}
\noindent
{\bf Theorem 1.2.} {\it Let $H_{\omega , \omega'}$ be the family
of random \Schr\ operators with singular interactions
formally given in (77), and defined using the resolvent formula
(81). We assume that the random variables satisfy Hypotheses 4.1 and 4.2.
There exists a constant $E_1(a) > 0$ so that $\Sigma \cap ( - \infty ,
-E_1 (a) ]$ is nonempty and pure point almost surely, with
exponentially decaying eigenfunctions. 
The family of operators exhibits dynamical localization at all orders $q
\in \N$ on $I \equiv \Sigma \cap ( - \infty ,-E_1 (a) ]$. That is, 
with probability one,
\beq
\sup_{t>0 } \| ~\|x\|^{q/2} e^{- i H_\omega t } E_\omega ( I )
\phi \|_{HS} < \infty,
\eeq 
for any $\phi \in L_0^2 ( \R^d)$, for $d=1,2,3$.
The integrated density of states $N(E)$ is Lipschitz continuous for $E < 0$.
}

\vspace{.1in}
\noindent
We also consider the case when the impurities are located on random subsets of
the lattice $\Z^{d-j}$ of codimension $j \geq 1$, for $d = 2 , 3$.
In this case, the methods of scattering theory are used
to prove that there is absolutely
continuous spectrum at positive energies and localized states in an interval
of negative energies with probability one.

\vspace{.1in}
\noindent
{\bf Theorem 1.3.} {\it Suppose that the random set $\Gamma ( \omega')$ is
a subset of a sublattice $\Z^{d-j}$ of codimension $j \geq 1$.
Let $H_{\omega, \omega'}$ be the corresponding random \Schr\ operator
with singular interactions formally given in (77), and 
defined using the resolvent formula (81). 
We assume that the random variables satisfy
Hypotheses 4.1 and 4.2. Let $\Sigma$ be the almost sure spectrum,
and let $\Sigma_{ac}$ be the almost sure absolutely continuous spectrum.
Then, we have that $[0, \infty ) \subset \Sigma_{ac}$.
Furthermore, there exists a constant $E_1 (a) > 0$
so that $\Sigma \cap ( - \infty , -E_1 (a) ]$ is nonempty
and pure point almost surely, with exponentially decaying eigenfunctions.
The model exhibits exponential localization in this energy interval
as in Theorem 1.2.}

\subsection{Contents}

The contents of this paper are as follows. In section 2,
we prove the main exponential estimates on the expectation
of the small powers of $\| P_n (H_\omega - E - i \epsilon )^{-1} P_m \|$,
$m,n \in I$, for the abstract model (1).
We prove a spectral averaging result and Kotani's trick for this
family of operators. We then prove the localization results. In
section 3, we apply these results to the random nonlocal model.
We introduce the multidimensional 
wavelet basis and locate the negative part of the 
essential spectrum
of this family with probability one. We prove the key
estimate on the matrix elements of the resolvent of the Laplacian in this basis. 
The random singular interaction
models are defined in section 4. The almost sure spectrum of the
model is described in detail. Basic resolvent estimates are proved. 
Spectral averaging and the perturbation of 
singular spectrum results are proved in
section 5. Exponential and dynamical localization are proved in section
6. The Wegner estimate, implying the Lipschitz continuity of the integrated
density of states, is proved in section 7. 
In section 8, we discuss the case of scattering theory for 
random singular interactions
with impurities
randomly located on subsets of $\Z^{d-j}$ of codimension $j \geq 1$.
Using the methods of scattering theory,
we prove the persistence of
absolutely continuous spectrum at positive energies,
and the existence of an interval of pure point spectrum at negative energies,
almost surely, for these models.

\subsection{Related Results}

There are a few results directly related to this work. The idea
of estimating the expectation
of small moments of the Green's functions for lattice
models is due to Aizenman and Molchanov \cite{[AizenmanMolchanov]}, 
and its application
to weak disorder is due to Aizenman \cite{[Aizenman]}.  
These papers apply to lattice models only. Quite recently, Aizenman,
Elgart, Naboko, Schenker, and Stolz \cite{[AENSS]} announced an extension
of this method to a restricted family of random Anderson models
in the continuum. It is not clear that these methods apply to
the models treated in this paper. The methods of Aizenman and Molchanov
have been used to study random Landau models on $L^2 ( \R^2)$ 
by Dorlas, Macris, and Pul\'e \cite{[Pule]}. 
The results for the lattice case suffice for 
their analysis since they reduce the model to a lattice one using the
eigenfunctions of the unperturbed Landau model.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Exponential Localization for an Abstract Nonlocal Model}\label{S.2}

We begin with an abstract theorem concerning the exponential
decay of the expectation of small moments of the Green's function
for operators of the form $H_\omega = H_0 + V_\omega$, 
on a separable Hilbert space
${\cal H}$. We assume that we have control on the Green's function 
for the self-adjoint operator $H_0$.
The potential $V_\omega$ is an Anderson-type potential of the form
\beq
V_\omega = \Sum_{j \in I } \lambda_j ( \omega ) P_j ,
\eeq
where $I$ is a countable set, and
$\{ P_k \; | \; k \in I \}$ is a complete family of one-dimensional,
orthogonal projectors in ${\cal H}$.
That is, the projectors satisfy $P_l P_k = \delta_{lk} P_k$, and
$\sum_k P_k = 1$. We assume that the index set $I$ has a metric $d: I \times I
\rightarrow R^+ \cup \{ 0 \}$.
We treat both the weak disorder and the strong disorder
regimes, as in the articles of Aizenman \cite{[Aizenman]}, 
and Aizenman and Molchanov \cite{[AizenmanMolchanov]}, respectively.
In the next section, we construct specific models with $H_0 = - \Delta$,
and the projections $P_n$ constructed from various wavelet bases of ${\cal H}
= L^2 ( \R^d)$. We will then prove estimate (13) for these models.

\subsection{Basic Resolvent Estimates}

We prove the
exponential decay of the Green's function given {\it a priori} estimates on
the unperturbed operator $H_0$. 

\vspace{.1in}
\noindent
{\bf Theorem 2.1.}{ \it Suppose that $H_0$ is a self-adjoint
operator on a separable Hilbert
space $\cal H$, and that $\{ P_n \; | \; n \in I \}$ is a countable family of
one-dimensional orthogonal projections such that 
$\sum_n P_n = 1$. Let $H_\omega$
be a family of random \Schr\ operators given by
\beq
H_\omega = H_0 + \Sum_{n \in I } \lambda_n ( \omega ) P_n .
\eeq
The random variables $\{ \lambda_n ( \omega ) \}$ are independent,
with distributions $\mu_n$ satisfying $B_n
\equiv \int |x| d \mu_n (x) < \infty$. 
Consider the following two hypotheses:
\begin{enumerate}
\item{{\bf [Weak Disorder].}} For any $0 < s < 1$, suppose there is a
finite constant $K_s > 0$ such that 
\beq
\label{eq:decouple1}
\int \frac{ |x|^s }{ | x-\alpha |^s } ~ d \mu_n (x) \: \leq \:
K_s \: \int \frac{1}{ |x - \alpha |^s }
~ d \mu_n ( x ) ,
\eeq
for all $\alpha \in \C$.
Suppose that the interval $I = [ a , b ] \subset \R$, with $b <
\infty$, satisfies $[a , b ] \subset \sigma ( H_\omega )$, a.\ s., and
there exists $\gamma = \gamma (E) > 0$ such that for every $E \in [a , b]$, the condition
\beq
K_s \left\{  \Sup_n \left[ \Sum_m \: \| P_n ( H_0 - E )^{-1} P_m \|^s 
e^{ s \gamma d( n , m ) } \right] \right\} \; < \; 1 ,
\eeq
is satisfied. 

\item{{\bf [Strong Disorder or High Energy].}} 
For any $0 < s < 1$, suppose there
is a finite constant $C_s > 0$ defined by
\beq
\label{eq:decouple2}
\int \frac{ |x - \beta |^s }{ | x - \alpha|^s } ~d \mu_n (x) \; \geq \: C_s \: 
\int \frac{1}{ | x - \alpha |^s } ~d \mu_n (x) ,
\eeq
for all $\alpha , \beta \in \C$.
Suppose that the interval $I = [ a , b ] \subset \R$, with $b < \infty$,
satisfies $[a,b] \subset \sigma ( H_\omega ) $, a.\ s., and
there exists a $\gamma = \gamma (I) > 0$ so that 
the condition
\beq
C_s \left\{ \Sup_n \: \Sum_m \left[ \: \| P_n H_0 P_m \|^s 
e^{s \gamma d( n , m ) } \right] \right\}^{-1}
\; > \; 1 ,
\eeq
is satisfied. 
\end{enumerate}

\noindent
Then, under either condition,
there exists a finite constant
$D_s = D_s ( \gamma, I) > 0$, so that for every $E \in I = [a , b]$,
\beq
\label{eq:decay}
\Sup_{\epsilon \rightarrow 0} 
\E \{ \; \| P_n ( H_\omega - E - i \epsilon )^{-1} P_m \|^s \; \}
\leq D_s e^{- s \gamma d( n , m ) } .
\eeq  
}

\vspace{.1in}
\noindent
{\bf Remarks.}   
\begin{enumerate}  
\item To understand the notion of weak or strong disorder, we replace
the potential $V_\omega$ by $\lambda V_\omega$. Carrying out the
same calculations, we arrive at (12) and (14) with $K_s$ and $C_s$ replaced
by $\lambda^s K_s$ and $\lambda^s C_s$, respectively. Then, we see that small
$\lambda$ improves (13), and large $\lambda$ improves (15). 
\item The constants $K_s$ and $C_s$ depend on the distributions 
$\mu_n$. We implicitly assume that the constants in (12) and (14) are
uniform in $n$. An explicit expression for $K_s$
is given in Proposition 6.1, cf.\ \cite{[KrishnaSinha]}.
\item For localization only, we may take $\gamma = 0$ in (13) and (15). We
obtain the bound (16) with $\gamma = 0$. This suffices to prove
localization.
\end{enumerate} 

\vspace{.1in}
\noindent
{\bf Proof.} \\
\noindent
1. {\bf Weak Disorder.} Let $E \in I$.
We begin by writing the resolvent equation
as
\bea
P_n ( H_\omega - E - i \epsilon )^{-1} P_m & = & P_n 
 ( H_0 - E - i \epsilon )^{-1} P_m 
 \nonumber \\
 & & - \Sum_k P_n ( H_0 - E - i \epsilon )^{-1} \lambda_k (
\omega) P_k  ( H_\omega - E - i \epsilon )^{-1} P_m .
 \nonumber \\
 & &
\eea
Taking the norm of both sides, the $s^{th}$-power, and then
the expectation, we obtain
\bea
\label{eq:exp}
\lefteqn{ \E \{ \| P_n ( H_\omega - E - i \epsilon )^{-1} P_m \|^s \} 
 } \nonumber \\
& \leq & \| P_n ( H_0 - E - i \epsilon )^{-1} P_m \|^s \nonumber \\
 & & + \Sum_k \| P_n ( H_0 - E - i \epsilon )^{-1} P_k \|^s
    \nonumber \\
  &  & \times \E \{ | \lambda_k ( \omega ) |^s \; \| P_k ( H_\omega - E - i
    \epsilon )^{-1} P_m \|^s \} . \nonumber \\
  & &  
\eea 
We now use a decoupling inequality to estimate the expectation of
the product of the random variables in (\ref{eq:exp}). 
Let $P_n = | \phi_n \rangle \langle \phi_n |$, where $\phi_n$ is
a normalized vector in the one-dimensional range of $P_n$. Let
us define an operator with no interaction at site $k$ by
$H_{\omega, k} \equiv H_\omega - \lambda_k ( \omega ) P_k $. 
Comparing this operator with $H_\omega$ via the resolvent
identity, we easily find
\bea
\| P_k ( H_\omega - z )^{-1} P_m \|^s & = & | \langle \phi_k , (H_\omega - z 
 )^{-1} \phi_m \rangle |^s  \nonumber \\
 & = &  \frac{ | \langle \phi_k , (H_{\omega , k } - z)^{-1} \phi_m \rangle 
        |^s }{ | 1 + \lambda_k ( \omega ) \langle \phi_k , ( H_{\omega , k } -z 
         )^{-1} \phi_k \rangle |^s } ,   
\eea
where we write $z = E + i \epsilon$.
The decoupling inequality (\ref{eq:decouple1}) now gives
\beq
\label{eq:exp2}
\E \{ | \lambda_k (\omega )|^s \; \| P_k ( H_\omega - z)^{-1} P_m \|^s \}
\; \leq \; K_s \; \E \{ \| P_k ( H_\omega - z )^{-1} P_m \|^s \} .
\eeq
Inserting this estimate back into (\ref{eq:exp}) gives
\bea
\lefteqn{ \E \{ \| P_n ( H_\omega - z)^{-1} P_m \|^s \} } \nonumber \\
 & \leq & \| P_n ( H_0 - z )^{-1} P_m \|^s  \nonumber \\
 & & + K_s \; \Sum_k \| P_n ( H_0 - z )^{-1} P_k \|^s \; \E \{ \| P_k
      ( H_\omega - z )^{-1} P_m \|^s \} .
\eea
For $\gamma (I) > 0$ as in (13),  
we define two functions.
For the unperturbed operator $H_0$, 
we define $M_0 ( s,E, \gamma)$ by
\beq
M_0 ( s, E , \gamma ) \equiv \sup_i \left\{ 
\; \Sum_j \| P_i ( H_0 - z )^{-1} P_j \|^s
e^{s \gamma d( i , j ) } \right\} , 
\eeq
and, for the random operator $H_\omega$,
we define
\beq
M ( s , E , \gamma ) \equiv \sup_i \left\{
 \; \Sum_j \E \{ \| P_i ( H_\omega - z )^{-1}
P_j \|^s \} e^{ s \gamma d( i , j ) } \right\} .
\eeq
With these definitions, we can rewrite inequality (\ref{eq:exp2}) as
\beq
M( s , E , \gamma ) \leq M_0 ( s , E , \gamma ) + K_s M_0 ( s , E , \gamma ) 
\: M ( s , E , \gamma  ) .
\eeq
Hence, if the condition (13), 
\beq
K_s M_0 ( s , E , \gamma ) \; < \; 1,
\eeq
is satisfied, we can solve form $M ( s , E , \gamma )$, 
showing that it is bounded independently of $\epsilon$. This
proves the exponential decay result (16) .

\vspace{.1in}
\noindent
2. {\bf Strong Disorder or High Energy.}  We begin with the basic operator 
equation satisfied by the resolvent
\beq
( H_\omega - z ) R_\omega ( z ) = 1 ,
\eeq
where we write $z = E + i \epsilon$. As above, we write $P_k = | \phi_k \rangle
\langle \phi_k |$. Taking the matrix element of this equation between
$\phi_k$ on the left, and $\phi_m$ on the right, we obtain
\beq
\Sum_l \langle \phi_k, H_0 \phi_l \rangle \langle \phi_l , R_\omega (z) \phi_m
\rangle  + ( \lambda_k ( \omega ) - z) 
\langle \phi_k , R_\omega (z) \phi_m \rangle
= \delta_{km} .
\eeq
We take the $s^{th}$-power of this equation and rearrange it.
We then take the expectation, use formula (19)
and the lower decoupling result (\ref{eq:decouple2}), to obtain,
\bea
\lefteqn{ C_s \E \{ \; | \langle \phi_k , R_\omega (z) \phi_m \rangle |^s \; \}
} \nonumber \\
 & \leq & \E \{ \; | \lambda_k ( \omega ) - z |^s 
\; | \langle \phi_k , R_\omega (z)
\phi_m \rangle |^s \; \} \nonumber \\
 & \leq  & \delta_{km} + \Sum_l | \langle \phi_k , H_0 \phi_l \rangle |^s \;
  \: \E \{ \; | \langle \phi_l , R_\omega (z) \phi_m \rangle |^s \; \} .
\eea
For $\gamma > 0$ as in (15),
we now define a function $M_0 ( s , \gamma)$ by
\beq
M_0 ( s , \gamma ) \equiv \Sup_k \left\{ 
\Sum_l \; | \langle \phi_k , H_0 \phi_l 
\rangle |^s e^{s \gamma d( k,l ) } \right\} ,
\eeq
for the unperturbed operator, and for $H_\omega$,
we define
\beq
M ( s , E , \gamma ) \equiv \Sup_k \left[ 
\Sum_l \; \E \{ \; | \langle \phi_k ,
 R_\omega ( z ) \phi_l \rangle |^s \} e^{s\gamma d( k,l ) } \right] .
\eeq
In terms of these functions, inequality (28) becomes
\beq
C_s M ( s , E , \gamma ) \leq 1 + M_0 ( s , \gamma ) M(s , E , \gamma ) .  
\eeq
Solving for $M ( s , E , \gamma )$ under condition (15),
we obtain
\beq
M(s , E , \gamma ) \; \leq \; \frac{1}{ M_0 ( s , \gamma ) } \; \left\{ 
\frac{ 1 }{C_s / M_0 (s, \gamma )  - 1 } \right\}  ,
\eeq
proving the second result. $\Box$ 

\vspace{.1in} 

The resolvent estimate (\ref{eq:decay}) is the key ingredient in the proof of
exponential localization as presented in \cite{[CH1]}.
Since the projectors $P_n$ are rank-one, a modification of the Simon-Wolff
\cite{[SW]} argument will allow us to prove exponential localization on the
energy interval $I = [a , b]$ of Theorem 2.1.
The modification is necessary since the unperturbed
operator $H_0$ is, in general,
an arbitrary, unbounded, self-adjoint operator satisfying, of course,
either condition (13) or (15) on $I = [a , b]$.

\subsection{Spectral Averaging for the Nonlocal Family}

We restate the spectral averaging theorem of \cite{[CHM]}
in its simplest setting that suffices to study the family $H_\omega = H_0 +
V_\omega$, with the potential $V_\omega$ given
in (10). 

\vspace{.1in}
\noindent
{\bf Theorem 2.2.} {\it 
Let $H_\lambda = H_0 + \lambda P $, $\lambda \in \R$,
be a one-parameter family of
self-adjoint operators on ${\cal H}$, with $P$ a rank-one orthogonal
projector. For each real, nonnegative function $g \in C_0^1
(\R)$, there exists a finite constant $C > 0$, depending on $\|g^{(j)}
\|_1$, for $j = 0 , 1$, such that for all $E \in \R$, and
for all $\phi \in {\cal H}$, we have
\beq
\label{eq:spave}
\Sup_{\epsilon > 0} \left| \int_{\R} g( \lambda )  ~ \langle \phi , P (
H_\lambda - E - i \epsilon )^{-1}  P \phi \rangle \right | 
\; \leq \; C_g \| \phi \|^2 .
\eeq 
}

\vspace{.1in}
We note that (\ref{eq:spave}) holds for each $P_n$ with a
constant uniform in $n$.
There are two 
immediate consequences of Theorem 2.2.
The first states that for any real, nonnegative function $g \in C_0^1 (
\R)$, and any Borel subset $J \subset \R$, we have
\beq
\left\| \int ~d \lambda g ( \lambda ) P_j E_\lambda (J) P_j  \right\| \; = \;
 \int ~d \lambda g ( \lambda ) \langle \phi_j , E_\lambda (J) \phi_j \rangle
\;  \leq \; C | J| ,
\eeq
for some finite constant $C > 0$, depending on $g$ and its
derivative, but independent of $j \in I$. 
Note that this implies that if $J \subset \R$ has Lebesgue measure zero,
then the left side vanishes for almost every $\lambda$. 
Kotani's trick states that for any
Borel subset $J \subset \R$ with $| J | = 0$, we have $E_\lambda ( J ) =
0$, for Lebesgue almost every $\lambda \in \R$. 
This follows from (33)--(34). For each $j \in I$, we have a set $\Gamma_j$
of full measure, so that,
\beq
\langle \phi_j , E_\lambda (J) \phi_j \rangle = 0 , ~\mbox{for}
~\lambda \in \Gamma_j .
\eeq
Let $\Gamma_\infty = \Cap_{j \in I} \Gamma_j$, so that $\Gamma_\infty$ has
full measure. Then, since
\bea 
| \langle \phi_j , E_\lambda (J) \phi_k \rangle | & \leq & \|  E_\lambda (J)
\phi_j \| \; \| E_\lambda (J) \phi_k \| \nonumber \\
 & \leq & \frac{1}{2} \{ \langle \phi_j , E_\lambda
  (J) \phi_j \rangle + \langle \phi_k , E_\lambda (J) \phi_k \rangle \},
\eea
it follows that for any $\phi$ in a dense set, we have
\beq
\langle \phi , E_\lambda (J) \phi \rangle = 0 , ~\mbox{for} ~\lambda \in
\Gamma_\infty.
\eeq
By a standard argument, this extends to any vector in ${\cal H}$, so
that the result follows from this.

\subsection{Perturbation of Singular Spectra for the Nonlocal Family}

The classical Simon-Wolff criterion \cite{[SW]}
for the Anderson model on $\Z^d$ is the following. If for Lebesgue almost
every $E \in (a,b)$, there is a set $\Omega_E \subset \Omega$ 
of full measure so that
\beq
\lim_{\epsilon \rightarrow 0}
\Sum_{n \in \Z^d } | G_\omega ( n , 0 ; E+ i \epsilon )|^2 < \infty,
\eeq
then $H_\omega$ has only pure point spectrum in $(a, b)$ almost surely.
If, in addition, the Green's function decays exponentially almost surely,
then the corresponding eigenfunctions also decay exponentially almost
surely.

For our model (11), the estimate analogous to (38) is the following.
The completeness of the projectors $P_n$ implies the identity
\beq
\| R_\omega ( E + i \epsilon ) P_n \|^2 = \Sum_m \| P_m R_\omega ( E + i
\epsilon ) P_n \|^2 .
\eeq
Consequently, the exponential decay result (16) implies the following
bound. 
For Lebesgue almost every energy
$E \in I$, there exists a finite constant $C(E) > 0$, and a set $\Omega_E$
with $| \Omega_E | = 1$, so that
\beq
\Sup_{\epsilon > 0} \| R_\omega ( E + i \epsilon ) P_n \| \; \leq \; C(E).
\eeq

In order to use this result to prove the almost sure pure point spectrum in
$I$, we first consider one-parameter perturbations obtained by varying one
coupling constant.
Consider variations $\omega \rightarrow \omega'$ for which $\lambda_j (
\omega ) = \lambda_j ( \omega')$, for $j \neq k$, and $\lambda_k ( \omega')
- \lambda_k ( \omega ) \equiv \lambda$. We then have $H_{\omega '} =
  H_\omega + \lambda P_k$. We consider this Hamiltonian
as the Hamiltonian $H_\lambda = H_0 + \lambda
P$ treated in the previous subsection.
 
\vspace{.1in}
\noindent
{\bf Theorem 2.3.} {\it  
Assume that $H_0$ satisfies the estimate (40) for $E \in I_0$, for some $I_0
\subset I$ with $| I_0 | = |I|$. Then, we have that for almost every
$\lambda \in \R$,  
$\sigma_{ac} (H_\lambda) \cap I = \emptyset = \sigma_{sc} (H_\lambda) \cap
I$, and the spectrum of $H_\lambda$ in
$I$ is pure point with finitely degenerate eigenvalues.}

\vspace{.1in}

The argument that Theorem 2.3 implies that the spectrum of $H_\omega$ in
$I$ is pure point almost surely follows as in \cite{[CH1],[SW]}. The
exponential decay of the corresponding eigenfunctions requires a separate
argument, cf.\ \cite{[CH1]}. We note that the model (10) need not be
ergodic.
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Random Nonlocal Model 
with Wavelet Interactions}\label{S.3}

We now consider a specific the random nonlocal model 
on $L^2 ( \R^d)$
for which the results of the previous section
can be directly applied. For these models, the unperturbed operator $H_0 =
- \Delta$ and the random potential $V_\omega$, as in (10), 
is diagonal in a wavelet basis.  In order to construct the projections
$P_j$, we first recall the definition of wavelets in higher dimensions.

A {\it wavelet} in one dimension is a function $\psi$ with the
property that the collection of translated and diadically dilated functions
$\{ \psi_{j,k}(x) = 2^{j/2} \psi(2^{j}x - k) \; | \;  j, k \in \ZZ\}$, 
forms an orthonormal basis for $L^2(\RR)$.
Associated with the wavelet $\psi$ is the {\it scaling function} $\phi$.
The scaling function $\phi$ is used to construct the wavelet $\psi$
through a procedure called {\it multiresolution analysis} (cf.
\cite{[D],[Wo]}).

To define a wavelet in higher dimensions (as in \cite{[Wo]}, Proposition
5.2), we first start with
a collection $\{\phi_1, \dots, \phi_d, \psi_1, \dots, \psi_d\}$
of $2d$ functions on $\R$ of which the $\phi_j$ are scaling functions and
the $\psi_j$ are the associated wavelets constructed from the $\phi_j$.
We note that we may take all the $\phi_j \equiv \phi, ~~ \psi_j \equiv
\psi, ~~ j=1, \dots, d.$, although this is not necessary. 
Let us define an index set
$F = \{ c = (c_1 , \ldots , c_d ) \in \{0,1\}^d \setminus (0,0,\dots, 0) \}$.
For each $c \in F$, we define a function on $\R^d$ by   
\begin{equation}
\label{eqn100}
\Psi_c (x) = \prod_{j=1}^d (\delta_{c_j, 0} \phi_j + \delta_{c_j, 1}
\psi_j)
(x_j), ~~ c \in F.
\end{equation}
Here, the $\delta_{c_j , k}$, for $c = ( c_1 , \ldots , c_d ) \in F$
and $k = 0 , 1$, is the Kronecker delta. In the product, the
function $\phi_j (x_j)$ is present if the index $c_j$ is zero, 
and $\psi_j (x_j)$ is present otherwise.
Note that there is at least one factor $\psi_j$ in $\Psi_c$ for any $c \in
F$. We consider the set of diadic dilations and 
$\Z^d$-lattice translations of these functions.
We denote by $I$ the countable index
set $I = F \times \Z \times \Z^d$. An element
$\n \in I$ is a triple $\n = ( c(\n) , n_1 , n_2 )$.
The collection of dilated and translated functions
\beq 
\Phi_\n ( x) = 2^{n_1 d/2} \Psi_{c(\n)}(2^{n_1}  x - n_2 ), ~~ c \in F,
~~ n_1 \in \ZZ ~~ n_2 \in \ZZ^d,
\eeq  
is called a {\it multivariable wavelet}
if the collection forms an orthonormal basis for $L^2(\RR^d)$.

In the following we
shall, notationally, always refer to the collection of functions $\{\Psi_c
\; | \; c \in F\}$ simply as $\Psi$, and any property stated for $\Psi$
is by definition to be take to be valid for each member of this collection.
Thus a statement that the property P is valid for $\widehat{\Psi}$ means
that P is valid for each of the Fourier
transforms $\widehat{\Psi_c}$, for each $c \in F$, and so on.

We assume the following conditions on the 
multivariable wavelet  and the distribution
of the random variables $\{ \lambda_{\n}  ( \omega ) \; | \; \n \in I \}$.
\begin{hyp}
\label{hyp1.1}
Let $\Psi$ be a multivariable wavelet formed out of the scaling functions
$\phi_i, i = 1, \dots, d$ and the wavelets $\psi_i, i=1, \dots, d$ such
that
\begin{enumerate}
\item the functions $\widehat{\phi_j}, \widehat{\psi_j} \in \cc^4(\RR), ~~ j = 1, \dots, d$;
\item the functions $\widehat{\psi_j}$ have compact support in $\R \backslash 0 , ~~j = 1, \dots, d $;
\item the functions $\widehat{\phi}_j^{(\alpha)}$, 
for $|\alpha | \leq 4 $, decay rapidly;
\item the functions are normalized, $\int | \Psi |^2 ~dx = 1$.
\end{enumerate}
\end{hyp}

\begin{hyp}
\label{hyp1.2}
Let $I = F \times \ZZ \times \ZZ^d$, and
let $\{ \lambda_\n (\omega) \; | \;  \n \in I\}$ be independent and 
identically distributed random variables with their common 
probability distribution $\mu$ being
absolutely continuous and of compact support in $\RR$.
\end{hyp}

{\noindent \bf Remarks:} 
\begin{enumerate}
\item Any one-dimensional Lemari\'e-Meyer wavelet
$\psi$, and its related scaling function $\phi$, satisfy Hypothesis
\ref{hyp1.1}. Typically, a Meyer wavelet can be constructed to be in the
Schwartz class, $\psi \in {\cal S} ( \R)$, and its
Fourier transform $\hat{ \psi}$ is compactly
supported in the set $[ - 8 \pi / 3 , - 2 \pi / 3] \cup 
[ 2 \pi / 3 , 8 \pi / 3 ]$. The corresponding scaling
function can also be chosen to satisfy
$\phi \in {\cal S} ( \R)$, and so that $\hat{ \phi}$ has compact
support in $[- 4 \pi / 3 , 4 \pi / 3]$, cf.\ \cite{[LM],[Wo]}.
A large number of additional examples are 
constructed in the paper of Auscher, Weiss, and Wickerhauser \cite{[AWW]}.
\item As discussed in \cite{[Wo]}, there is another method of constructing
multidimensional wavelets using tensor products. We could also use this
construction below. Wavelets constructed by tensoring one-dimensional
wavelets have some undesirable support properties.
\end{enumerate}

\vspace{.1in}

As in section 1, we consider the operator $H_0 = -\Delta$, and
define the random family of nonlocal interaction Hamiltonians 
on $L^2(\RR^d)$ by  
\begin{equation}
\label{eqn1.1}
H_\omega = -\Delta + V_\omega, ~~  V_\omega = \sum_{\n \in I} 
\lambda_\n (\omega) P_\n ,
\eeq
where $P_\n$ is the rank-one projection defined by
\beq
P_\n \equiv  |\Phi_\n \rangle \langle \Phi_\n|, ~~\mbox{with} ~\n \in I.
\end{equation}

Since $\mu$ has compact support, each $\lambda_\n ( \omega) $ 
takes values in a
compact set, so the operator $V_\omega$ is bounded for each $\omega$.
Then, the operator $H_\omega$ is self-adjoint on the same domain as
$H_0$.
Although the operator $H_\omega$ is not ergodic, we can determine the essential spectrum
using the action of the subgroup of dilations. 

\vspace{.1in}

\begin{prop}
\label{prop:spectrum}
Suppose that the support of $\mu$ is $[- M_0 , 0]$. Then, the essential spectrum of
$H_\omega$ is $[ - M_0 , \infty )$ almost surely.
\end{prop}

\vspace{.1in}
\noindent
{\bf Proof.} We concentrate on the negative part of the essential spectrum.
For any $E \in supp(\mu)$, the Borel-Cantelli theorem
implies that for any $\epsilon > 0$, the event
$$
\Omega(\epsilon, E) = \{ \omega \; | \; |\lambda_{(c, k, 0)} ( \omega) - E| 
< \epsilon, \\
\text{for infinitely-many {\bf negative} integers k }, 
~~ c \in F \},
$$
has measure one. Consequently, the set
\beq     
\Omega_0 = \bigcap_{l \in \ZZ^+} \; \bigcap_{r \in \QQ \cap supp(\mu)}
\Omega( 1/l , r) ,
\eeq      
also has measure one. 
For each rational $r \in supp(\mu)$,
and for each $l \in \ZZ^+$, there is a sequence
$s_l$ of {\bf negative} integers with $|\lambda_{(c, s_l, 0)}(\omega) - r| <
1 / l$, for all $\omega \in \Omega_0$. For each $l \in \ZZ^+$,
we choose one element of $s_l$, so that $\{ s_l \}$ is monotone decreasing,
and define a sequence of vectors
$f_l = \Phi_{(c, s_l, 0)}$, with $\|f_l\| = 1$. In the lemma below, we
prove a bound on $\| \Delta f_l \|$. Using these vectors $f_l$,
we have,
\begin{equation}
\begin{split}
\|(H_\omega - r)f_l\| &\leq \|\Delta \Phi_{(c, s_l, 0)}\| + \|(\lambda_{(c,
s_l, 0)}(\omega) - r)f_l\| \\
& \leq  2^{4 s_l}C + \frac{1}{l},
\end{split}
\end{equation}
which obviously goes to zero as $l$ goes to infinity, since $s_l$ are {\bf
negative} by choice.  Thus, we produced a sequence $f_l$ for which
$\|(H_\omega - r)f_l\|$ goes to zero as $l$ goes to $\infty$, for all
$\omega \in \Omega_0$. Since the sequence $\{ f_l \}$ is orthonormal,
we have $r \in \sigma_{ess} (H_\omega), ~~ \omega \in \Omega_0$.
Since this happens for all rationals in the support of $\mu$, we see that
$supp(\mu) \subset \sigma(H_\omega), ~~ \omega \in \Omega$. $\Box$ 

\vspace{.1in}

\begin{lem}
For any $\n \in I$, we have
$$
\|\Delta \Phi_\n\| \leq  2^{2 n_1}C, 
$$
where $C$ is a constant that is independent of $\n$.
\end{lem}

\vspace{.1in}
\noindent
{\bf Proof.} 
We recall equations (41) and (42) and the Hypothesis
3.1 on the functions $\phi_j$ and $\psi_j$ that are used to define the
functions $\Psi_\n$ and the orthonormal basis $\{\Phi_\n, \n \in I = F\times \ZZ \times \ZZ^d\}$.
We also recall that the functions $\psi_j$ have compact support in
$\RR\setminus \{0\}$,
from which it follows that the following integral is finite for any
polynomial $P$.
In the following, the differential operator $P(\nabla)$ is defined using the
(polynomial) symbol $P(\xi), \xi \in \RR^d$ and the "hat" denotes Fourier
transform.
\begin{equation}
\label{se}
\begin{split}
\langle \Phi_\m, P(\nabla) \Phi_\n\rangle &=
2^{- (m_1-n_1)d/2} \int_{\RR^d}
e^{i ( 2^{-(m_1 - n_1)} m_2 - n_2 ) \cdot \xi} \\
& ~~~~~ \times \overline{\widehat{\Psi_{c(\m)}}} ( 2^{-(m_1 - n_1)} \xi)
~\widehat{\Psi_{c(\n)}}(\xi)
P(2^{n_1}\xi) ~ d\xi .
\end{split}
\end{equation}
Using the equation (\ref{se}), where we take $\m = \n$ and the
polynomial
$P(\xi) = (\sum_j \xi_j^2)^2)$, we have
\begin{equation}
\label{te}
\begin{split}
\|\Delta \Phi_\n\|^2  &=
2^{- (n_1-n_1)d/2} \int_{\RR^d}
e^{i ( 2^{-(n_1 - n_1)} m_2 - n_2 ) \cdot \xi}
\overline{\widehat{\Psi_{c(\n)}}} ( 2^{-(n_1 - n_1)} \xi) \\
& ~~~~~~~ \times
~\widehat{\Psi_{c(\n)}}(\xi)
(\sum_j (2^{n_1}\xi_j)^2)^2 ~ d\xi \\
&\leq
2^{4n_1} ~~ \int_{\RR^d}
|\widehat{\Psi_{c(\n)}} (\xi)|
~ |\widehat{\Psi_{c(\n)}}(\xi)|
(\sum_j (\xi_j)^2)^2 ~ d\xi \\
& \leq  C 2^{4n_1},
\end{split}
\end{equation}
where we take
$$
C = \sup_{c \in F} \int_{\RR^d} |\widehat{\Psi_{c}} (\xi)|
~ |\widehat{\Psi_{c}}(\xi)|
(\sum_j (\xi_j)^2)^2 ~ d\xi,
$$
which gives the lemma. $\Box$

\vspace{.1in}

\noindent
The main result is the following theorem.

\vspace{.1in}

\begin{thm}
\label{wavethm1}
Let $H_\omega$ be a family of random nonlocal Schr\"odinger operators as in
equation (\ref{eqn1.1}),
with the multivariable wavelet $\Psi$ 
satisfying the Hypothesis \ref{hyp1.1}, and
the random variables $\lambda_\n (\omega), \n \in I$ satisfying Hypothesis
\ref{hyp1.2}.  Then, there is a number $E_1 (\mu) > 0$ such that
the spectrum of $H_\omega$ in ($-\infty$, -$E_1 (\mu)]$ is pure point
for almost every $\omega$.
\end{thm}

The proof of Theorem \ref{wavethm1} follows the argument of section 2.
It is necessary to prove either hypothesis (13) on the
resolvent of $H_0$ or hypothesis (15) on $H_0$.
We give a proof that hypothesis (13) is satisfied.

\vspace{.1in}

{\noindent \bf Proof of Theorem \ref{wavethm1}.} We have to verify the
condition (13) of Theorem 2.1 when $H_0 = -\Delta$. Let us begin by
estimating the matrix elements
\begin{equation}
\label{eqn1}
T(\m,\n, E) = \langle \Phi_{\m}, (-\Delta - E)^{-1} \Phi_{\n} \rangle, 
~~ \n, \m \in I , ~~E \in (-\infty, 0).
\end{equation}
Since $-\Delta \geq 0$, the resolvent $( -\Delta - E)^{-1}$, for $E < 0$,
is  bounded. 
This implies that $T(\m, \n, E)$ is finite for all $\n, \m \in I , ~  E
< 0$. Our aim is to show that for suitable $1/4 < s < 1/2$, we actually have a
$E(\mu) > 0$ such that
\begin{equation}
\label{eqn3}
K_s \left( \sup_{m \in I} \sum_{n \in I} |T(\m,\n, E)|^s \right)
< 1 ~~ {\text for }
~~  E \in (-\infty, - E(\mu)),
\end{equation}
where $K_s$ is a constant given in the inequality (\ref{eq:decouple1}). Let
$S_d = \{ (2l+1 ) / 2 \; | \;  l=0,1, \dots, 4d-1\}$, for $d \geq 1$.
Given the estimate
in the inequality (\ref{eqn2}), in Lemma 3.6 below, we have
\begin{equation}
\label{eqn4}
K_s \left( \sup_{m \in I} \sum_{n \in I} |T(\m,\n, E)|^s \right) \leq K_s
\left[ \frac{C(\Psi, s, d)}{\sqrt{|E|}} \max_{j \in S_d}  \frac{1}{|E|^j}
\right].
\end{equation}
This inequality makes it clear that for either $K_s$ small enough, or for
$|E|$ large, there is a critical $E(\mu) > 0$ such that for all
$E < -E(\mu)$, the right hand side of (51)
is smaller than 1. In order to see that the set of $E < 0$ for which
these conditions hold, is nonempty,
we need to examine the dependence of the constant
$K_s$ on the probability distribution $\mu$. For this, we use estimates
(154)--(155) of Proposition 6.1. For example, if $d \mu (\lambda) =
\chi_{[-M_0 , 0]} ( \lambda ) d \lambda$, it follows from (154)--(155)
that $K_s \leq C_0 M_0^\tau$, 
for any $0 < \tau \leq 1$, where the
constant $C_0$ is an absolute number.
With upper bound (51), the inequality (13) is satisfied
provided the energy $E < 0$ satisfies: 
\beq
\label{eq:bound}
C_1 M_0^\tau | E |^{-1} \; < \; 1,
\eeq
with $C_1$ depending on $C_0$ and $C( \Psi, s ,
d)$. Certainly, if $0 < \tau < 1$, and
$| E | = {\cal O} ( M_0 )$, then for $M_0 > 0$ large enough,
the inequality (\ref{eq:bound}) is satisfied for a nonempty interval of energy.
As Proposition 3.3 indicates that $\mbox{inf} ~\sigma ( H_\omega ) \geq -
M_0$, we see that there is a nonempty region of overlap with the spectrum.
The remainder of the proof of the theorem follows from sections 2.2 and 2.3. 
$\Box$ 

\begin{lem}
\label{lem1}
For any $E \in (-\infty, 0)$, there is a constant $C = C( \Psi, s, d )$,
independent of $E$, such that
\begin{equation}
\label{eqn2}
\sup_{\m \in I} \sum_{\n \in I} |T(\m,\n, E)|^s \leq  
\frac{C(\Psi, s, d)}{\sqrt{|E|}} \max_{j \in S_d}  \frac{1}{|E|^j},
\end{equation}
where $S_d = \{ (2l+1)/2 \; | \; l=0,1, \ldots, 4d-1 \}$.
\end{lem}

{\noindent \bf Proof:}  \\
\noindent
1. Recall that by definition
$\Phi_{\n}(x) = 2^{n_1 d/2} \Psi_{c(\n)}(2^{n_1} x - n_2), ~~
(c(\n), n_1 , n_2) \in I = F \times  \Z \times \Z^d$,
and similarly for $\m = (c(\m), m_1, m_ 2) \in I$.
By Parseval's identity, we write the matrix element
(\ref{eqn1}) in terms of the Fourier transform of the wavelets
as 
\begin{equation}
\label{eqn41}
T(\m,\n, E) = \int_{\RR^d} \overline{ \widehat{\Phi_{\m}}} (\xi)
\widehat{ \Phi_{\n}}(\xi) \frac{1}{(\xi^2 - E)} ~ d\xi ,
\end{equation}
where $\xi^2 = \sum_{j=1}^d \xi_j^2$.
Using the definition of the Fourier transform and the definition of
$\Phi_{\n}$, we see that
\begin{equation}
\label{eqn5}
\begin{split}
\widehat{\Phi_{\n}}(\xi) & =  \frac{1}{(2\pi)^{d/2}} \int_{\RR}
e^{-i \xi\cdot x} \Phi_\n ( x) ~d x \\
&= \frac{2^{d n_1/2}}{(2\pi)^{d/2}} \int_{\RR} e^{-i\xi\cdot x}
\Psi_{c(\n)}(2^{n_1} x - n_2) ~dx \\
&= 2^{-d n_1/2} e^{-i2^{-n_1} n_2\cdot \xi} ~~
\widehat{\Psi_{c(\n)}}(2^{-n_1} \xi).
\end{split}
\end{equation}
Then, the matrix element (\ref{eqn1}) becomes
\begin{equation}
\label{eqn6}
\begin{split}
T(\m,\n, E) &= 2^{-(n_1+m_1)d/2} \int_{\RR^d} e^{i(2^{-m_1} m_2 -
2^{-n_1} n_2) \cdot \xi} ~\overline{\widehat{\Psi_{c(\m)}}} (2^{-m_1}\xi) ~
\widehat{\Psi_{c(\n)}}(2^{-n_1}\xi) \\
& ~~~  \frac{1}{(\xi^2 - E)} ~ d\xi .
\end{split}
\end{equation}
Changing variables $\xi \rightarrow 2^{-n_1}\xi$ we find that the above
equation is
\begin{equation}
\label{eqn7}
\begin{split}
T(\m,\n, E) &= 2^{- (m_1-n_1)d/2} \int_{\RR^d} 
e^{i ( 2^{-(m_1 - n_1)} m_2 - n_2 ) \cdot \xi} 
\overline{\widehat{\Psi_{c(\m)}}} ( 2^{-(m_1 - n_1)} \xi)
~\widehat{\Psi_{c(\n)}}(\xi) \\
&~~~ \frac{1}{(2^{2n_1} \xi^2 - E)} ~ d\xi .
\end{split}
\end{equation}

\noindent
2. Since $\widehat{\Psi} \in C^4 ( \R^d )$ and is rapidly decaying in all of
its variables, we integrate 
by parts four times. The boundary terms are zero due to the
rapid decay Hypothesis \ref{hyp1.1}. Let us assume that there exist
finite $a , b > 0$ so that all the $\psi_j$ have compact support in
$[-b , -a ] \times [a,b]$. 
To simplify some notation, we introduce vector-valued phases
$\omega ( \m , \n ) \in \R^d$,
with components 
\beq
\omega_j ( \m , \n ) \equiv 2^{- ( m_1 - n_1 )} m_{2j} - n_{2j} ,
\eeq
for $j = 1 , \ldots , d$.
Our calculations will depend upon whether this vanishes or not.
We write $\alpha_j$ for the function
\beq
\alpha_j = \left\{  \begin{array}{ll}  
                  0  & \mbox{if} ~~\omega_j ( \m , \n ) = 0 \\
                  1  & \mbox{if} ~~\omega_j ( \m , \n ) \neq 0  
                      \end{array}  
                  \right.  
\eeq
With this notation, we have
\begin{equation} 
\label{eqn8}
\begin{split}
T(\m,\n, E) & = 2^{- (m_1-n_1)d/2} \int_{\RR^d} ~ d\xi ~e^{i 
\omega ( \m , \n ) \cdot \xi} \\
& \left\{ \prod_{j=1}^d \left( \alpha_j \frac{1}{\omega_j ( \m, \n )^4} \cdot 
\frac{\partial^4}{\partial \xi_j^4 }  
+ (1-\alpha_j) \right) \right\} \\ 
& \left\{ \overline{\widehat{\Psi_{c(\m)}}} ( 2^{-(m_1- n_1)} \xi) ~
 \widehat{\Psi_{c(\n)}}( \xi) 
 \frac{1}{((2^{n_1}\xi)^2 - E)} \right\}.
\end{split}
\end{equation}   
Since $c(\n) \in F = \{0,1\}^d\setminus (0,\dots,
0)$, at least one of $c(\n)_1, \dots, c(\n)_d$ equals 1.
This means that in the
expansion of $\Psi_{c(\n)}$ as a product using equation (\ref{eqn100}),
at least one $\widehat{\psi_j}$
occurs, for any $c(\n)$. 
By Hypotheses \ref{hyp1.1}, $\widehat{\psi_j}$ has compact
support away from zero in $[-b , -a ] \cup [a, b], ~a > 0$, for
any $j=1,\dots , d$.
Then, for that index $j$ for which $\psi_j$ occurs in the product,
we see that unless $m_1 - n_1$ satisfies
the condition
$$
a \leq | 2^{n_1 - m_1}\xi_j | \leq b, ~~
a \leq | \xi_j | \leq b  ~~ \implies ~~
0 \leq |n_1 - m_1| \leq  \log_2(b/a) ,
$$
we have that $T(\m, \n, E)$ is zero.  
This restricts the values of $n_1$ to a bounded
set for any fixed $m_1$, so T is summable in $n_1$ for any fixed $m_1$.
From the equation (\ref{eqn8}), it is clear that
there is enough decay in $n_2 \in \Z^d$ to obtain summability in that variable.
We will show that the required condition is indeed valid.

\noindent 
3. We begin by expanding (\ref{eqn8}) 
using the binomial expansion into terms for which $\omega_j ( \m , \n )
\neq 0$, and those for which $\omega_j ( \m , \n ) = 0$. This gives us
%\begin{equation}
%\label{eqn81}
%\begin{split}
%T(\m,\n, E) & = 2^{(n_1-m_1)d/2} \int_{\RR^d} ~~ d\xi ~~ e^{i(\n_3 -
%2^{n_1-m_1} \m_3)\cdot \xi} \\
%&\left\{\sum_{k=0}^d(\substack{d\\k})
%\prod_{j=1}^k\frac{\partial^4}{(i({\n_3}_j
% - 2^{n_1-m_1}{\m_3}_j))^4\partial \xi_j^4}\right\} \\
%&\left\{\widehat{\overline{\Psi_{\n_2}}} (\xi) ~~
%\widehat{\Psi_{\m_2}}(2^{n_1- m_1}\xi) \frac{1}{((2^{n_1}\xi)^2 -
%E)}\right\}. 
%\end{split}
%\end{equation}
%We write this equation as
\begin{equation}
\label{eqn9}
\begin{split}
T(\m,\n, E) & = 2^{-(m_1-n_1)d/2} \left( \sum_{k=1}^d \left( \substack{d\\k}
\right) \prod_{j=1}^k \frac{1}{\omega_j ( \m , \n )^4}
(\prod_{j=k+1}^d (1-\alpha_j)) G_k(\m,\n,E) \right. \\
&~~~ \left. + \prod_{j=1}^d (1 - \alpha_j) G_0(\m,\n, E) \right) ,
\end{split}
\end{equation}
where
\begin{equation}
\label{eqn91}
\begin{split}
G_0(\m,\n, E) & =
\int_{\RR^d} ~~ d\xi ~~ e^{i \omega ( \m , \n )\cdot
\xi} \overline{\widehat{\Psi_{c(\m )}}} (\xi) ~~
\widehat{\Psi_{c(\n) }}( 2^{n_1-m_1}\xi) \frac{1}{((2^{n_1}\xi)^2 - E)}. \\
G_k(\m,\n, E) & = \int_{\RR^d} ~~ d\xi ~~ e^{i \omega ( \m , \n ) \cdot
\xi} \prod_{j=1}^k \frac{\partial^4}{\partial \xi_j^4} 
\left\{\overline{\widehat{\Psi_{c(\m)}}} (\xi) ~~
\widehat{\Psi_{c(\n)}}(2^{n_1- m_1}\xi) \frac{1}{((2^{n_1}\xi)^2 -
E)}\right\}.
\end{split}
\end{equation}

\noindent
4. Now note that once we have the above expression we get the bound
\bea 
\label{eqn10}
\lefteqn{ 
\sup_{\m \in I}\sum_{\n \in I} |T(\m,\n, E)|^s } \nonumber \\
& \leq \Sup_{\m \in I} \Sum_{\n \in I}
  2^{(n_1-m_1)d/2} \left[ \sum_{k=1}^d(\substack{d\\k})^s  
  \prod_{j=1}^k \frac{1}{| \omega_j ( \m , \n ) |^4s} \left( \prod_{j=k+1}^d
  (1-\alpha_j) \right) |G_k(\m,\n,E)|^s \right. \nonumber \\
& \left. 
  + \Prod_{j=1}^d (1 - \alpha_j) |G_0(\m,\n, E)|^s \right]. 
\eea  
where the expressions $\omega_j ( \m , \n )$
occurring in the denominator are non-zero. 
We now bound the right side of (63).
Recall that the set of $n_1 \in \Z$, for which $T( \m , \n , E)$ is
nonzero, is bounded for each fixed $m_1$.
Let $N > 0$ be the least integer greater than or equal to $ \log_2 ( b/a) 
$. Then, we have the constraint that $|n_1 - m_1| \leq N$.
We can then express the upper bound in (63) as 
\bea
\label{eqn101}
\lefteqn{ \sup_{\m \in I}\sum_{\n \in I} |T(\m,\n, E)|^s } \nonumber \\
&\leq 2^{Nds/2} K(d,s) \Sup_{\m  \in I} \left[ \Sum_{n_2 \in \Z^d} \Sum_{k=1}^d
\prod_{j=1}^k \frac{1}{|\omega_j ( \m, \n )|^{4s}}
\prod_{j=k+1}^d (1-\alpha_j) |G_k(\m,\n,E)|^s \right.  \nonumber \\
& ~~~~ \left. + \Sum_{n_2 \in \Z^d}
(\prod_{j=1}^d (1-\alpha_j))|G_0(\m,\n,E)|^s \right] . 
\eea
Consider the sum over $n_2 \in \Z^d$ in the second term of (64).
For these terms, we have that $\omega_j ( \m , \n ) \neq 0$, for all $j$,
so that for fixed $(m_1, m_2) \in \Z \times \Z^d$, there is a bounded set
of $(n_1 , n_2)$ for which $| n_1 - m_1 | \leq N$ and $n_2 = 2^{n_1 - m_1}
m_2$. Consequently, for each $m_2$, there are at most $2^N$ nonzero
terms in the second sum.   
%We simplify the inequality by changing variables ${m_3}_j \rightarrow
%{m_3}_j - [2^{n_1-m_1}{n_3}_j]$, we set $\beta(j,\n) =
%({m_3}_j - 2^{n_1-m_1}{n_3}_j)$, (where (a) denotes the fractional part of
%a) and use $(1-\alpha_j)$ factors to reduce the sum. 
This allows us to write the bound 
\begin{equation}
\label{eqn11}
\begin{split}
\sup_{\m \in I}\sum_{\n \in I} |T(\m,\n, E)|^s & \leq
2^{N( sd/2+ 1) } K(d, s)  \left[ \sum_{n_2 \in \Z^d} \sum_{k=1}^d 
\prod_{j=1}^k \frac{1}{|\omega_j (\m , \n ) |^{4s}} \right.  \\
& \left. \sup_{\n,\m \in I} |G_k(\m,\n,E)|^s + 
  \sup_{\n,\m \in I} |G_0(\m,\n,E)|^s  \right] .
\end{split}
\end{equation}
For every $ 1/4 < s <1 $, the $n_2$-sum 
is finite. For such $s$, the above inequality reduces to
\begin{equation}
\label{eqn12}
\sup_{\m \in I}\sum_{\n \in I} |T(\m,\n, E)|^s  \leq
D(N, d, s)  \sup_{k \in \{ 0, 1, \cdots, d\} } \left(  \sup_{\m,\n \in I}
|G_k(\m,\n,E)|^s  \right) .
\end{equation}
It remains to estimate the last term. We do this in the following
lemma. $\Box$

\vspace{.1in}

\begin{lem}
\label{lemma2}
Let $\Psi$ satisfy the conditions of the Hypothesis \ref{hyp1.1}. For $k
\geq 1$, let 
$S_k = \left\{ \frac{2l+1}{2} \; | \;  l = 0, 1, \dots, 4k-1 \right\}$.  Then
the following estimates are valid.
\begin{equation}
\label{eqn13}
\sup_{\m,\n \in I} |G_k(\m,\n,E)|^s \leq  \left\{ \begin{array}{ll} 
   \frac{1}{|E|} & \mbox{if} ~~k = 0 \\
  D(\Psi, a, b, d) \frac{1}{\sqrt{|E|}} \Sup_{j \in S_k} \frac{1}{|E|^j} 
  & \mbox{if} ~~k = 1, \ldots , d ,
\end{array} 
\right.
\end{equation}
where
\begin{equation}
\label{eqn14}
D(\Psi, a, b, d) = C(a, b, d) \sup_{\substack{\alpha = (\alpha_1, \dots,
\alpha_d), 0\leq \alpha_j \leq 4 \\
\beta = (\beta_1, \dots, \beta_d), 0\leq \beta_j \leq 4}}
~\left[ \sup_{\xi}|\widehat{\Psi}^{(\alpha)}(\xi)|
|\widehat{\Psi}^{(\beta)}(\xi)| \right].
\end{equation}
\end{lem}
{\noindent \bf Proof:} \\
\noindent
1. We first note that the derivatives of
$((2^{n_1} \xi)^2 - E)^{-M}$
give the following type of terms, for any positive integer $M$,
\begin{equation}
\label{eqn144}
\begin{split}
\frac{\partial}{\partial \xi_j} \frac{1}{(2^{2n_1}\xi^2 - E)^{M}} &=
\frac{-2M 2^{n_1}\xi_j}{(2^{2n_1}\xi^2 - E)^{M+1}} \\
\frac{\partial^2}{\partial \xi_j^2} \frac{1}{(2^{2n_1}\xi^2 - E)^{M}} &=
\frac{-2M 2^{n_1}}{(2^{2n_1}\xi^2 - E)^{M+1}} +
\frac{4M(M+1) 2^{2n_1}\xi_j^2}{(2^{2n_1}\xi^2 - E)^{M+2}} \\
\frac{\partial^3}{\partial \xi_j^3} \frac{1}{(2^{2n_1}\xi^2 - E)^{M}}  &=
\frac{12M(M+1) 2^{2n_1}\xi_j}{(2^{2n_1}\xi^2 - E)^{M+2}} -
\frac{8M(M+1)(M+2) 2^{3n_1}\xi_j^3}{(2^{2n_1}\xi^2 - E)^{M+3}} \\
\frac{\partial^4}{\partial \xi_j^4} \frac{1}{(2^{2n_1}\xi^2 - E)^{M}}  &=
\frac{12M(M+1) 2^{2n_1}}{(2^{2n_1}\xi^2 - E)^{M+2}} -
\frac{48M(M+1)(M+2) 2^{3n_1}\xi_j^2}{(2^{2n_1}\xi^2 - E)^{M+3}} \\
& ~~~~~ +\frac{16M(M+1)(M+2)(M+3) 2^{4n_1}\xi_j^4}{(2^{2n_1}\xi^2 -
E)^{M+4}}.  \\
\end{split}
\end{equation}

\noindent
2. We write the expression for $G_k ( \m , \n , E), ~k = 1, \dots, d $ as
\begin{equation}
\label{eqn15}
\begin{split}
G_k(\m,\n, E) &= \int_{\RR}d\xi ~e^{i \omega ( \m, \n ) \cdot \xi}
\sum_{\substack{p_1, \dots, p_k = 0 \\ q_1, \dots, q_k = 0 \\ r_1, \dots,
r_k = 0 \\ p_1 + q_1 +r_1 = 4, \\ \vdots \\ p_k+q_k + r_k = 4}}^4
  \prod_{j=1}^k \frac{\partial^{p_j}}{\partial
\xi_j^{p_j}}\overline{\widehat{\Psi_{c(\m)}}}(\xi)
 \prod_{j=1}^k \frac{\partial^{q_j}}{\partial
\xi_j^{q_j}}\widehat{\Psi_{c(\n)}}(2^{n_1 - m_1}\xi)\\
&  ~~~\times ~\prod_{j=1}^k \frac{\partial^{r_j}}{\partial
\xi_j^{r_j}}\left\{\frac{1}{ (2^{n_1}\xi)^2-E }\right\} .
\end{split}
\end{equation}
We now estimate the derivatives using the compactness of the support of
$\widehat{\Psi_{c(\m)}}$ as stated in Hypothesis \ref{hyp1.1}.
Let us define a multiindices $\alpha = (\alpha_1, \dots, \alpha_d), \alpha_j
= 0, \dots, 4$, and $\beta$ (which we take
to make the estimate uniform in $k$, though we do not need it for each
$k$). Let us recall that $| n_1 - m_1 | \leq N$, so the maximum of
four partial derivatives on the second
term $\widehat{\Psi_{c(\n)}}(2^{n_1 - m_1}\xi)$ are bounded by $2^{4N} \|
\Psi_{c(\n)}^{(\alpha)} \|_\infty$.
From these considerations, we obtain from (\ref{eqn15}),
\begin{equation}
\label{eqn16}
\begin{split}
|G_k(\m,\n, E)| &= 2^{4N} \max\{\frac{|B-A|^d}{a^{4d}}, |B-A|^d\} \sup_{\xi\in
[A,B]^d} |\widehat{\Psi_{c(\m)}^{(\alpha)}}(\xi)| 
|\widehat{\Psi_{c(\n)}}^{(\beta)} (\xi)|  \\
& ~~~~\sum_{\substack{\alpha_1 = 0, \dots, r_1, \\ \vdots \\ \alpha_k = 0,
\dots , r_k}} \sup_{\xi \in [A,B]^d} \left| \frac{\prod_{j=1}^k
P_{\alpha_j}(\xi_j)}{(2^{n_1}\xi)^2 -E)^{1 + \alpha_1+\alpha_2
+\dots+\alpha_k}} \right|,
\end{split}
\end{equation}
where for each $\alpha_j = 1, 2, 3, 4$, the polynomials
$P_{\alpha_j}(\xi_j)$ take the
form given in the numerators of the expressions in equation (\ref{eqn144}).

\noindent
3. We now estimate the sum in (\ref{eqn16}). Recall that $E < 0$. 
As we can see from the structure of
the derivative of $( (2^{n_1} \xi )^2 - E)^{-M}$
in (\ref{eqn144}), there are terms of the form
\beq
\left| \frac{ 2^{n_1} \xi_j }{ ((2^{n_1} \xi )^2 - E)^{M+1}} \right| ,
\eeq
that are bounded above by
\beq
\frac{1}{ |E|^{M+1/2}} ,
\eeq 
for any value of $n_1$ and all $\xi_j$. 
There are also terms with free factors of $2^{n_1}$
that are bounded by $1$ for $n_1 < 0$, but that are unbounded for $n_1 >
0$. These free factors of $2^{n_1}$ need to be handled differently. 
We note that each occurrence of a factor $2^{n_1}$, 
without an accompanying $\xi_j$ in the
numerator, comes from taking a derivative of the denominator.  So if
we have an expression like
\beq 
\frac{2^{kn_1} \prod_{j=1}^d (2^{n_j}\xi_j)^{s_j} }{((2^{n_1}\xi)^2 -
E)^M},
\eeq
then, from the computations of derivatives and the explicit formulas
(\ref{eqn144}), it is clear that we must have 
$M \geq k + \sum_{j=1}^d s_j + 1$.
Therefore, we can bound the factors 
$\prod_{j=1}^d (2^{n_j}\xi_j)^{s_j} $ as in the
case of $n_1 <0$ and we will be left with an expression
\beq  
\frac{2^{kn_1}}{((2^{n_1}\xi)^2 - E)^L},
\eeq  
to bound with $L \geq k+1$.
This expression is bounded by $1/a^{2L}$ since 
\beq  
\frac{2^{kn_1}}{((2^{n_1}\xi)^2 - E)^L} =
\frac{2^{kn_1}}{2^{2L n_1}}\frac{1}{(\xi^2 - 2^{-2n_1}E)^L} \leq
\frac{1}{a^{4L}
},
\eeq  
as follows from the facts
that $\xi^2 - 2^{-2n_1}E \geq a^2$, since $E<0$ and $\xi_j^2 \geq
a^2$, for some $j$, and the first factor is bounded by 1 since $L > k$.
Putting these together and absorbing all the constants into a single
constant $C (\Psi, a, b)$, we get the inequality stated in Lemma. $\Box$  

\vspace{.1in}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Random Singular Interaction Models}

Point interaction models have been studied by many authors
because of their rather transparent nature. We refer to the book of
Albeverio, Gesztesy, Hoegh-Krohn, and Holden \cite{[AGHKH]} for an
authoritative account of the subject and a comprehensive bibliography.
We consider one-, two- and three-dimensional models of 
point interactions with random strengths $\lambda_j ( \omega )$,
randomly distributed on the lattice $\Z^d$.
We remark that these are the only dimensions for which these point
interaction models exist.
Formally, these models are described by a  \Schr\
operator having the form
\beq
\label{eq:model}
H_{\omega , \omega' } = - \Delta + \sum_{j \in \Gamma ( \omega') }
\lambda_j ( \omega ) \delta ( x - j ) 
%\nonumber \\
% &=&  - \Delta + \sum_{j \in \Z^d } \lambda_j (\omega ) X_j ( \omega') \delta ( x - j ) 
,
\eeq
where $\{ \lambda_j ( \omega ) \; | \; j \in \Z^d, \omega \in \Omega \}$
is a family of random variables with
probability space $\{ \P , {\cal F} , \Omega \}$. The set $\Gamma (\omega')
\subset \Z^d$ gives the location of the interactions on the lattice.
This set $\Gamma (\omega')$ is determined by 
another family of independent, identically distributed
$\{ 0 , 1 \}$-valued random variables,
$\{ X_j ( \omega' ) \; | \: j \in \Z^d , \omega' \in
\Omega' \}$, 
with $\P \{ X_i ( \omega ' ) = 1 \} = p$, for some $0 < p \leq
1$. For each $\omega ' \in \Omega'$, 
the set $\Gamma ( \omega ' )
= \{ i \in \Z^d \; | \; X_i ( \omega ') = 1 \} \subset \Z^d $ 
indicates the location of the singular potentials.
A simple case is obtained when $X_i ( \omega' ) = 1$, for all
$i \in \Z^d$, corresponding to $p=1$ and $\Gamma ( \omega' ) = \Z^d$.

It is well-known that in $d = 1 , 2, \mbox{and} \: 3$ dimensions, these
models are defined through an explicit formula for the Green's function
(for an alternative definition in the one-dimensional case using boundary
conditions, cf.\ \cite
{[KirschMartinelli1]}). For dimension $d \geq 4$, the delta interactions
cannot be defined in this manner.
One way to see this is to note that the Sobolev
embedding $H^s ( \R^d ) \rightarrow C(\R^d)$
is continuous only if $s > d / 2 $.
Hence, for $s = 2$, only dimensions $d = 1 , 2 , \mbox{and} \;
3$ are permitted (see \cite{[Kr3]} for another discussion of this).
We will let $G_0 ( x , y ; z )$ denote the integral kernel of the
resolvent $R_0 (z) = ( - \Delta - z )^{-1}$, for $z \in \C \backslash [0 ,
\infty )$. Explicitly, the free Green's functions for these dimensions are
\bea
G_0 ( x-y ; z ) & = & \frac{i}{2 \sqrt{z}}  e^{i \sqrt{z} ( x-y ) } ,~~ 
d=1 \label{eq:G1} \\
G_0 ( x-y ; z ) & = & \frac{i}{4} H_0^{(1)} ( \sqrt{z} \|x-y\|) , ~~ d = 2
 \label{eq:G2} \\
G_0 ( x-y ; z ) & = & \frac{1}{4 \pi } \: \frac{ e^{i \sqrt{ z} \| x - y \|
} }{ \| x-y \| } , ~~ d = 3 , \label{eq:G3}
\eea
where $H_0^{(1)}$ is the Hankel function of the first kind,
and the square root is taken with the branch cut along the positive
real axis. It is important to note that for $z = - E $, with $E > 0$,
the free Green's function decays exponentially.

For the general situation of randomly distributed impurities on $\Z^d$,
and random coupling constants, the Green's function for the Hamiltonian
$H_{\omega , \omega'}$, in dimensions $d = 1 ,2, \mbox{and} \;
3$, is defined by
\bea
\label{eq:fullG}
G_{\omega , \omega' } ( x , y ; z )
 & = &   G_0 ( x - y ; z )  \nonumber \\
 &   & + \Sum_{i , j \in
\Gamma (\omega') } G_0 ( x - i ; z ) [ K ( z ; \omega )^{-1}]_{ij}
G_0 ( y - j ; z ) . \nonumber \\
 & &
\eea
The free Green's function $G_0 (x-y; z )$ is given in (\ref{eq:G1}),
(\ref{eq:G2}), and (\ref{eq:G3}), depending to the dimension.
As is clear from (\ref{eq:fullG}),
the random coupling constants enter only into the random matrix
$K ( z ; \omega )$ defined on $\ell^2 ( \Gamma ( \omega
') )$.  The definition of this matrix $K (z ; \omega )$
depends on the dimension through the free Green's function. Its matrix
elements have the
following general form:
\beq
\label{eq:matrix1}
K_{ij} ( z ; \omega ) =  \left( \frac{1}{ \lambda_{j,d} ( \omega
) } - e_d (z)  \right) \delta_{ij}
- G_0 ( i - j ; z ) ( 1 - \delta_{ij} ), ~\mbox{for} ~i , j \in \Gamma (
  \omega ' ). 
\eeq
The energy function $e_d (z)$ is given by
\bea
e_1(z) & = &  - \frac{i}{2 \sqrt{z}}  \\
e_2(z) & = &  \frac{ \mbox{log} \sqrt{-z} }{ 2 \pi } \\
e_3 (z) & = &  \frac{ i \sqrt{z} }{ 4 \pi } ,
\eea
and the effective coupling constant $\lambda_{j,d} ( \omega)$ is given by
\bea
\lambda_{j,1} ( \omega) &=& - \lambda_j ( \omega) \\
\lambda_{j,d} ( \omega) &=& \lambda_j ( \omega), ~~\mbox{for} ~~d=2,3.
\eea

We will use the fact that this operator $K ( z ; \omega )$ 
is related to a generalized, random \Schr\ operator.
On $\ell^2 ( \Gamma ( \omega'))$, we define a kinetic energy
operator $t(z)$ by its matrix elements
\beq
t_{ij}( z) = - G_0 ( i - j ;  z ) ( 1 - \delta_{ij} ), ~i,j \in \Gamma
(\omega').  
\eeq
Note that for $z = - E$, the matrix
elements decay exponentially for $E > 0$. 
We also define a diagonal, local, random potential by
\beq
(vg) ( j ) = \left( \frac{1 }{  \lambda_{j,d} ( \omega )} \right) u(n), ~j \in
\Gamma ( \omega ').
\eeq
Thus, we will study the discrete, generalized \Schr\ operator
$h(z) = t(z) + v$, and the operator $K (-E) = h(-E) - e_d(-E )$, 
for $E>0$, on $\ell^2 (\Gamma ( \omega') )$.

We now make precise our hypotheses on the random potential. We
recall that we are interested in localization at negative energies, that
is, outside of the spectrum of $H_0 = - \Delta$.

\vspace{.1in}

\begin{hyp}
\label{hyp4.1}
The family of random variables  $\{ \lambda_j ( \omega ) \; |
\; \omega \in \Omega, \mbox{and} \; j \in \Z^d \}$ is an
independent, identically distributed
({\it iid}) family of random variables with a common, absolutely continuous
distribution $\mu$ with density $h_0 \in L^\infty ( \R)$.
The support of $h_0$ is the
interval $[-b , -a ]$, for two positive constants $0 < a < b <
\infty$.
\end{hyp}

\vspace{.1in}

\begin{hyp}
\label{hyp4.2}
The random variables $\{ X_j ( \omega' ) \; | \: \omega'
\in \Omega', \mbox{and} \; j \in \Z^d \}$ is a family of independent,
identically distributed $( 0 , 1 )$-valued random variables with
$\P \{ X_j ( \omega' ) = 0 \} < 1 $.
\end{hyp}

\vspace{.1in}

Let us note that there is little loss of generality with the
assumed form of the support of $h_0$. As will become clear, the
values $1 / \lambda_j$ play the role of
effective coupling constants, and determine the spectral
properties of the Hamiltonian.
For dimensions $d=2,3$,
we could take for the support of $h_0$ the
interval $[-b , -a ]$, or $[ c , d ]$, or the union $[-b , -a ]
\cup [ c , d ]$,
for positive constants $0 < a < b < \infty$ and $0 < c < d <
\infty$.
For $d=2,3$, as we will see, there is negative spectrum even for the case
$\mbox{supp} \; h_0 = [ c , d ]$, and an analysis similar to that
given below holds. For dimension $d=1$, we must have an interval in $\R_-$
in the support of $h_0$ in order to have spectrum at negative energies.
To keep things as simple as possible, we will
work with Hypothesis 4.1.  

Before we can state our main results, we need to recall the basic spectral
facts concerning the periodic approximations to the models (\ref{eq:model})
and the deterministic spectrum of the random model (\ref{eq:model}).

\subsection{The Spectra of the Related Periodic Operators}

We briefly review the spectral theory of periodic operators with
point interactions located on the lattice. This theory was
developed by Grossmann, H{\o}egh-Krohn, and Mebkhout
\cite{[GrossmannHoeghKrohnMebkhout]}.
Let us assume that $\Lambda$ is a $d$-dimensional lattice
generated by $d$ independent unit vectors $\{ a_i \; | \;
\langle a_i , a_j \rangle = \delta_{ij} , ~i = 1,2, \ldots, d
\}$,
for $d = 1 , 2 , \mbox{or} \; 3$.
The $\Lambda$-periodic Schr\"odinger
operator with singular interactions and
coupling constant $\lambda \in \R$ has the form
\beq
H ( \lambda ) = - \Delta + \Sum_{j \in \Lambda } \: \lambda \: \delta
( x - j ) .
\eeq
This operator is $\Lambda$-periodic, and hence admits a Floquet
decomposition. Let $C_0$ be a unit cell of $\Lambda$, say, $C_0 = \{
\sum_{j = 1}^d \alpha_j a_j \; | \; - 1/2 < \alpha_j \leq  1/2 \}$.
The dual lattice $\Gamma$ to $\Lambda$ is generated by three
independent vectors $b_i = 1,2, \; \mbox{and} \; 3$, satisfying
$\langle a_j , b_i \rangle = 2 \pi \delta_{ji}$. We denote by
$B_0 \equiv \{ \sum_{j = 1}^d \beta_j b_j \; | \; - 1/2 < \beta_j \leq  1/2
\}$, the unit cell of the dual lattice $\Gamma$, called the
first Brillouin zone.
For example, if $\Lambda = \Z^d$, the generating unit vectors
are $a_i = e_i$, the standard orthonormal vectors, and they
span the unit cube $C_0 = ( - 1/2 , 1/2 ]^d$.
The Brillouin zone is $B_0 = \pi ( -1 , 1 ]^d$,
spanned by vectors $b_j = 2 \pi e_j$. The
operator  $H( \lambda)$ is unitarily equivalent to
a direct integral decomposition
\beq
H( \lambda) \sim \int_{B_0}^{ \oplus} \; {\tilde H}( k ; \lambda  ) ~d^dk ,
\eeq
where each member of
the family of self-adjoint operators ${\tilde H}( k ; \lambda ), k \in B_0$,
acts on $L^2 ( C_0 )$, and has a compact resolvent.
Let $E_n ( k ; \lambda), n = 0 , 1, \ldots, $ be the eigenvalues of
${\tilde H}( k ; \lambda )$ listed in increasing order, including
multiplicities. The $n^{th}$ band $B_n ( \lambda )$ is defined to be $B_n
( \lambda ) =
\cup_{k \in B_0} E_n ( k ; \lambda)$.
The spectrum of $H(\lambda)$ is $\sigma ( H( \lambda )) = \cup_{n
= 0 }^\infty B_n ( \lambda )$.

The spectrum of ${\tilde H}(k ; \lambda)$ can be read off from the formula
for its resolvent. We will explicitly discuss the
three-dimensional case, although the
analysis applies to the one- and two-dimensional cases also.
It is convenient to take the Fourier transform from $L^2 (C_0) \rightarrow 
\ell^2 (\Gamma)$, and to express the resolvent as an operator on $\ell^2 (
\Gamma)$. We denote the unitarily equivalent
operator corresponding to ${\tilde H}(k; \lambda)$
on $\ell^2 ( \Gamma)$ by $H(k ; \lambda)$.
For $k \in B_0$, and $\gamma, \gamma' \in
\Gamma$, we have:
\bea
(H(k, \lambda) - z )^{-1}( \gamma, \gamma') &=& (( \gamma + k)^2
-z)^{-1} \delta_{\gamma, \gamma'} \nonumber \\
 & & - \frac{1}{ (2 \pi)^3} \left( \frac{1}{\lambda} - \alpha_z( k)
\right)^{-1} \; ((\gamma + k)^2 - z)^{-1} \; ((\gamma' + k)^2 -
z)^{-1} . \nonumber \label{eq:reduced} \\
 & &
\eea
The function $\alpha_z (k)$ plays an important role in the
spectral properties of $H(k , \lambda)$. It is defined for $E \in \R$ as
\beq
\label{eq:alpha}
\alpha_E ( k ) \equiv (2 \pi)^{-3} \; \lim_{\omega \rightarrow
\infty}
\left[ \Sum_{ {\stackrel{ \gamma \in \Gamma }{| \gamma + k |
 \leq \omega }}} \; \frac{1}{ | \gamma + k |^2 - E } - 4 \pi
\omega \right] .
\eeq
It follows from the resolvent formula (\ref{eq:reduced}), 
that $E \in \sigma(H(k, \lambda))$ if either
\begin{enumerate}
\item The real energy $E$ is a zero of $(1 / \lambda) - \alpha_E (k) = 0$,
or
\item The energy satisfies $E = ( \gamma + k)^2$, for some $\gamma \in
\Gamma$, and the corresponding pole of the resolvent at this energy is at
least of order two.
\end{enumerate}
Note that the eigenvalues occurring in case (2)
are those of the free Hamiltonian.
It is not too hard to check that $\mbox{inf}
\; \{ \sigma(H(k,\lambda)) \; k \in B_0 \}$ is given by the smallest solution
$E_0(k, \lambda)$ in case (1).
An analysis of the function $\alpha_E (k)$
reveals that one always has $E_0 (0 , \lambda) = \mbox{inf}
\; \{ \sigma(H(k,\lambda)) \; k \in B_0 \} < 0$,
for any $\lambda \neq 0$.

The spectrum of $H(\lambda)$ is the union of the range of each band
function $E_n ( k ; \lambda )$, over $k \in B_0$. 
It is shown in \cite{[GrossmannHoeghKrohnMebkhout]}
that the first band function $E_0 ( k ;
\lambda)$ takes its
minimum at $k = 0$, and its maximum at $k_0 = \sum_{j = 1}^3 \:
(1/2)b_j $, where, as above, the $b_j$ span the Brillouin
zone. For $\lambda \neq 0$, we know that $\mbox{inf} \; \sigma (H(\lambda)) 
= E_0 ( 0 ; \lambda ) < 0$.
It also follows that the upper edge of the first band, given by $E_0 ( k_0
; \lambda)$, is negative, provided $1 /\lambda < \alpha_0 (k_0)$.
It is easy to check that for the square lattice $\Lambda = \Z^d$,
the constant $\alpha_0 ( k) = 0$, for all $k \in B_0$.

\vspace{.1in}
\noindent
{\bf Theorem 4.1. \cite{[GrossmannHoeghKrohnMebkhout]}}
{\it The spectrum of $H(\lambda)$ is
absolutely continuous. Let $E_0 (k ; \lambda )$ be the first band function
for $H(k, \lambda)$, and let $\alpha_0 ( k), k \in B_0$, be the
constant defined in (\ref{eq:alpha}) with $E = 0$.
The minimum of $E_0 (k; \lambda )$ occurs at $k =
0$, and the maximum occurs at $k = k_0 \equiv (1/2) ( b_1 , b_2 , b_3 ) $.
The lower edge of the first band satisfies
$E_0 ( 0; \lambda ) = \: \mbox{inf} \; \sigma ( H (
\lambda)) < 0$, for any $\lambda \neq 0$. 
If, furthermore, $1 / \lambda < \lambda_0 ( k_0 )$,
then the upper edge of the first band satisfies $E_0 ( k_0; \lambda ) < 0$.
For all $\lambda$, the spectrum of $H(\lambda)$ is
given by
\beq
\sigma( H(\lambda) ) =
[ E_0 ( 0 ; \lambda ) , \infty ) , ~~\mbox{if} ~~ 1 / \lambda \geq
\alpha_0 (k_0) ,
\eeq
and
\beq
\sigma( H(\lambda) ) = [ E_0 (0; \lambda ) , E_0 ( k_0 ; \lambda ) ] \cup [
0 ,
\infty ) , ~~\mbox{if} ~~ 1 / \lambda < \alpha_0 (k_0).
\eeq
The two-dimensional case is similar. For the one-dimensional case,
the spectrum has a band structure with an infinite number of open gaps
except for $\lambda = \infty$. The bottom of the spectrum is negative if
and only if $\lambda < 0$. }

\vspace{.1in}

In summary, in two- and three-dimensions, 
we have $\sigma ( H(\lambda))
\cap ( - \infty , 0 ) \neq \emptyset$, for all $\lambda \neq 0$.
If further, $1 / \lambda < \alpha_0
(k_0)$, then there is one band of negative spectrum separated
from the rest of the spectrum $[0 , \infty)$.
For $d=1$, the $\sigma ( H(\lambda))
\cap ( - \infty , 0 ) \neq \emptyset$, provided $\lambda < 0$.
We remark that if there are $N$ point interactions per unit cell for
$d=2,3$, then there are at most $N$-bands, 
possibly separated from an unbounded half-line
of continuous spectrum.
It is of particular importance for us to compute the dependence of the
$\mbox{inf} \: \sigma ( H(\lambda ))$
on $\lambda$.
We recall that the parameter $\lambda \in \R$ plays different roles for the
models with $d=2,3$ and $d=1$. For the one-dimensional case, $\lambda$ is
the coupling constant and the weak interaction limit is $\lambda = 0$. On
the other hand, the weak coupling limit for the models with $d=2,3$ is
$\lambda = \infty$. In this sense, the parameter $\lambda$ plays the role
of the inverse of the coupling constant for $d=2,3$. 

\vspace{.1in}
\noindent
{\bf Proposition 4.2.} {\it Let $E_{0,d} (0 ; \lambda)$, for $d=1,2,
~\mbox{and} ~3$, denote the bottom of the spectrum for the
periodic, point interaction Hamiltonian $H( \lambda )$.
This is the minimum of the first band function.
For $d=3$ and $\lambda < 0$, there exists a finite constant $ C_{0,3}
> 0$ so that we have
\beq
- \left( \frac{4 \pi}{ \lambda} \right)^2 \leq 
E_{0,3} (\lambda ) \leq - \left( \frac{4 \pi}{ \lambda} \right)^2 +
{\cal O} ( e^{- C_{0,3} / | \lambda | })  < 0 ,   
\eeq
as $\lambda \rightarrow 0^-$. For $d=2$ and $\lambda < 0$, there
exists a finite constant $ C_{0,2} > 0$ so that we have
\beq
-   4 e^{4 \pi / | \lambda| } \leq E_{0,2} (\lambda) \leq 
-   4 e^{4 \pi / |\lambda| } + 
    {\cal O} ( e^{- C_{0,2} / | \lambda | })  < 0 ,
\eeq
as $\lambda \rightarrow 0^-$.
Finally, for $d=1$ and $\lambda < 0$, there exists a
finite constant $C_{0,1} > 0$ so that the ground state energy 
$E_{0,1} ( \lambda)$ satisfies  
\beq
E_{0,1} ( \lambda ) =  - \mathcal{O} (| \lambda |),
\eeq 
as $\lambda \rightarrow 0^-$.}

\vspace{.1in}
\noindent
{\bf Proof.} 
We prove the upper bound for $\lambda < 0$ using the variational principle.
We give the proof for $d=3$. The same method is used to prove the result
for $d=2$. Let $\psi_\lambda$
be the $L^2 (\R^3)$-normalized 
eigenfunction for the one $\delta$-function potential at zero,
\beq
\psi_\lambda (x) = | \lambda |^{-1/2} \frac{e^{- 4 \pi \|x\| / |\lambda|}}{
\|x\|}.
\eeq
Let $\chi_0 \in C_0^\infty ( \R^d)$ be a smooth function so that $0 \leq \chi_0
\leq 1$, with $\chi_0 = 1$ in a neighborhood of the origin, and
$\mbox{supp} ~u$ is sufficiently small so that contains no other point of
$\Z^d$.
We define a cut-off function $\chi (x) \equiv \sum_{j \in \Z^d \backslash
\{0 \}} \; \chi_0 ( x - j)$, so that $1 - \chi $ vanishes in a neighborhood
of each lattice point except the origin.
The function $\Psi_\lambda \equiv (1 - \chi) \psi_\lambda \in L^2 ( \R^d)$
is in the domain of $H(\lambda)$. 
We use this function $\Psi_\lambda$
as a trial function for the ground state of $H(\lambda)$.
There exist finite constants $B_0 , B_1 > 0$, depending only on $| \mbox{supp}
~\nabla \chi |$, so that 
obtain
\beq
\langle \Psi_\lambda , H(\lambda) \Psi_\lambda \rangle 
 \leq  - \left( \frac{4 \pi}{\lambda} \right)^2  
 + \sum_{j \in \Z^3 \backslash \{0\} } 
\frac{B_0 }{\|j\| | \lambda|^2} e^{- B_1 \|j\| / |\lambda|} .
\eeq
We estimate the nonnegative sum from above by an integral and obtain
\beq
\langle \Psi_\lambda , H(\lambda) \Psi_\lambda \rangle \leq  - \left(
\frac{4\pi}{\lambda} \right)^2 +
\lambda A_0  e^{- A_1 / |\lambda| } ,   
\eeq
for finite constants $A_0 , A_1 > 0$.
It follows from this and the variational principle that as $\lambda
\rightarrow 0^-$, 
\beq
E_0 (0; \lambda)  = \inf \sigma ( H(\lambda))
\leq - \left( \frac{4 \pi}{ \lambda } \right)^2 + {\cal O} ( e^{- C_0 / |
\lambda | }).
\eeq
For the lower bound, 
we study the relation for $\alpha_E (0)$, for $E < 0$. If we approximating the
sum in (\ref{eq:alpha}) by an integral, 
and then take the limit $\omega \rightarrow
\infty$, we obtain a function ${\tilde \alpha}_E (0)$ given by
\beq
{\tilde \alpha}_E (0) = - \frac{ \sqrt{ |E|} }{ 4 \pi} .
\eeq
This function pointwise
dominates $\alpha_E (0)$, for $E < 0$. Since both of these functions are
monotone increasing functions of $E$, 
we find that the solution ${\tilde E}_0 (0; \lambda)$ of
$1 / \lambda = {\tilde \alpha}_E (0)$, for $\lambda < 0$, satisfies
\beq
{\tilde E}_0 (0; \lambda) \leq E_0 (0; \lambda).
\eeq
It follows from this and the definition of ${\tilde \alpha}_E (0)$
that 
\beq
{\tilde E}_0 (0 ; \lambda ) = - \left( \frac{4 \pi}{ \lambda } \right)^2 ,
\eeq
proving the result. For the case $d=1$, the bottom of the first band
$E_{0,1} (0 ; \lambda ) = - k^2$, where
$k > 0$ satisfies the Kronig-Penney relation $1 = cosh k + (| \lambda | /
2k) sinh k$. The estimate follows from analysing the root of this equation.
$\Box$

\subsection{The Deterministic Spectrum of the Random Family}

We are interested in proving the existence of localized states
at energies outside of the spectrum of $H_0 = - \Delta = [ 0 , \infty)$
caused by the singular interactions.
To this end, we must show that the ergodic models (\ref{eq:model}) 
have deterministic
spectrum, including almost sure
spectrum at negative energies.
The existence of a deterministic spectrum
$\Sigma$ for the model (\ref{eq:model})
with $\Gamma ( \omega' ) = \Z^d$, for
$d = 1 , 2 , 3$, was proven in \cite{[KirschMartinelli1]} for the
case of {\it iid} random coupling constants.
They also determined the nature of $\Sigma$ as a set.

\vspace{.1in}
\noindent
{\bf Theorem 4.3.\ \cite{[KirschMartinelli1]}} {\it Assume hypothesis
4.1 on the model (\ref{eq:model}), and that $\Gamma ( \omega' ) = \Z^d$, 
for $d = 1 , 2, 3$.
For any $\lambda \in \mbox{supp} \: h_0$, let $H ( \lambda) $ be
the related periodic operator defined as in subsection 4.1. Then,
we have
\beq
\Sigma = \overline{ \bigcup_{ \lambda \in \mbox{supp} \: h_0 } \; 
\sigma ( H ( \lambda )) }.
\eeq
For $d=3$, let $E_0 ( k ; \lambda )$ be the first band function 
for $H (k, \lambda) $.
We recall that $E_0 ( k_0 ; \lambda ) < 0$ if $ 1 / \lambda <
\alpha_0 (k_0)$, where $k_0 \in B_0$ is the point for which $E_0 ( k ;
\lambda)$ assumes its maximum. With $\mbox{supp} \: h_0 = [ -b , -a]$, for
$0 < a < b$, we have
\beq
\Sigma = [ E_0 ( 0 ; -1/a ) , E_0 ( k_0 ; - 1/b ) ] \cup [ 0 , \infty ) ,
\eeq
and, if $- 1/b  < \alpha_0 (k_0)$, we  have $E_0 ( k_0 ;
-1/b ) < 0$. A similar result holds for $d=2$. For $d=1$, a similar result
holds provided we replace $1/a$ by $a$ and $1/b$ by $b$. }

\vspace{.1in}

The deterministic spectrum $\Sigma$ of the random family
$H_{\omega , \omega'}$, described in (\ref{eq:model}) for random 
$\Gamma ( \omega')$, was determined by
Albeverio, H{\o}egh-Krohn, Kirsch, and Martinelli
\cite{[AlbeverioHoeghKrohnKirschMartinelli]}.
These authors discovered the surprising fact that the
deterministic spectrum $\Sigma$ does not depend upon the nature
of the random set $\Gamma ( \omega' )$ on which the family
of random variables $\{ X_j ( \omega' ) \}$ takes the value one,
provided that the probability that this set is nonempty is nonzero.

\vspace{.1in}
\noindent
{\bf Theorem 4.4.\ \cite{[AlbeverioHoeghKrohnKirschMartinelli]} }
{\it In addition to the Hypotheses 4.1 and 4.2, 
on the model (\ref{eq:model}), we assume that $\P \{ X_0 ( \omega' ) = 0 \} <
1$. Then, precisely the same characterization
of the almost sure spectrum $\Sigma$ as in Theorem 4.3 holds in this case.}

\subsection{Resolvent Estimates}

We now apply these results to the model (\ref{eq:model}). 
Because of the form of the potential energy term $v$ in the discrete 
Schr\"odinger-like operator $h ( -E ; \omega ) = t (-E) + v$, for $E > 0$,
it is convenient to introduce a
random variable $\sigma_k ( \omega ) \equiv 1 / \lambda_k ( \omega )$. Recall
that the random variable $\lambda_k ( \omega )$ is distributed with
a density $h_0$ having support $ [ - b, - a ]$, for some
$0 < a < b < \infty$. 
Consequently, the random variable $\sigma_k ( \omega )$ is distributed
with a density ${\tilde h}_0 ( \sigma ) = \sigma^2 h_0 ( 1 / \sigma )$, so
that ${\tilde h}_0$ is supported on $[ - 1/ a, - 1 / b ]$.

\vspace{.1in}
\noindent
{\bf Proposition 4.5.} {\it Let $K (-E ; \omega ) = t(-E) +
v - e_d(-e)$, for $E > 0$, 
be the discrete, self-adjoint operator on $\ell^2 ( \Gamma
(\omega'))$, for any $\omega' \in \Omega'$, defined in (88)--(89).
Let $\gamma_d (E) = \sqrt{E}$, for $d=2,3$, and $\gamma_1 (E) = 1$.
Then, under Hypotheses 4.1--4.2, for any $0 < s < 1$, there exists an 
energy $E_1^{(d)} = E_1^{(d)} (K_s) > 0$, and a constant $C_s > 0$, 
both independent of $\omega'$, so that for all $E > E_1^{(d)}$, for
$d=2,3$, and for $0 < E < E_1^{(1)}$,
condition (13) holds, and 
\beq
\E \{ | [ K ( -E ; \omega )^{-1}]_{ij} |^s \} \: \leq \: 
C_s e^{ - s \gamma_d (E) \| i - j \| } ,
\eeq  
for all $i,j \in \Gamma ( \omega' )$. The dependence of $E_1^{(d)}$ on
$K_s$
is given by
\beq
E_1^{(d)} = \mathcal{O} ( ( K_s C_d )^{\delta_d (s)} ), 
\eeq
where $\delta_1 (s) = 2 / (s-2)$,
$\delta_2 (s) = 2/ (5s + 2)$, and $\delta_3 (s) = 2 / (18s+3) $. The finite
constant $C_d > 0$ is defined in the proof.}

\vspace{.1in}
\noindent
{\bf Proof.} We apply the method of proof of
Theorem 2.1 to the discrete operator $h( -E ) = t (-E) + v$, at
energy $e_d (-E)$, for $E>0$. 
Let $P_n$ be the rank-one projection onto the site $n \in \Z^d$.
We must verify condition (13) at energy $e_d(-E)$
for $A_0 = t ( -E ), ~E > 0$, 
where $t (z)$ is the 
operator on $\ell^2 ( \Gamma (\omega') )$ with matrix elements
\beq
t_{ij}(z) = -G_0 ( i-j ; z )( 1 - \delta_{ij}) .
\eeq
This kinetic energy operator is zero on the diagonal and
for $z = - E , ~E> 0$, the off-diagonal terms decay exponentially with weight 
$\sqrt{ E }$. 
As $t(z)$ depends only on $\| i - j \|$,
it acts as a convolution operator
on $\ell^2 ( \Gamma ( \omega') )$.
We estimate the operator norm using Young's inequality and find
\beq
\label{tbound}
\| t( - E) \| \; \leq \; \| t(-E)\|_{\ell^1 ( \Gamma ( \omega')} \; \leq \;
\frac{C_d }{E^{\alpha_d} } ,
\eeq
where $C_d$ is a finite, positive constant, and $\alpha_1 = 1,~~\alpha_2 =
1$, and $\alpha_3 = 3/2$. 
To prove exponential decay,
we treat the cases of dimension $d=2,3$ and dimension $d=1$ separately,
since for $d=2,3$ we consider $\lambda \rightarrow - \infty$ and $-E << 0$, 
whereas for $d=1$, we consider $\lambda \rightarrow 0^-$ and $-E < 0$ and
small. 
For $d=2,3$, let $\Delta_d \equiv \; \mbox{dist} ( e_d( -E) , \sigma (
t(-E))$. It follows from (\ref{tbound})
and the definition of $e_d(z)$, (84)--(85), that
\beq
\Delta_2 = \mathcal{O} ( \log \sqrt{E} ), ~~\mbox{and} ~~\Delta_3 =
\mathcal{O} (\sqrt{E} ) .
\eeq
Consequently, the gap increases with $E \rightarrow \infty$.  
%$i , j \in \Z^d$, and for $d=2,3$, we have
%\beq
%\| P_i \: ( t(-E) - e_d (-E) )^{-1} P_j \|^s 
%= |e_d (-E) |^{-s}  | [ {\tilde t} ( -E ) + 1 ]^{-1}_{ij} |^s , 
%\eeq
%where ${\tilde t} ( -E ) \equiv ( e_d(-E) )^{-1} t ( -E)$.
%Let $\Delta \equiv \; \mbox{dist} ( -1 , \sigma ( {\tilde t} ( -E
%) ) )$. Because of the form of $\tilde{t} ( -E )$, there exists an energy
%$E_0 > 0$, depending on the dimension $d=2,3$,
%so that for all $E > E_0$, we have $\Delta > 1/2$.
%To compute $E_0$, we use an a priori estimate on the spectral
%radius of ${\tilde t} (-E)$
%given by its norm. For $d=3$, this is
%\beq
%\| {\tilde t} (-E) \| \; \leq \; \frac{ e^{- \sqrt{E} } }{
%\sqrt{E} } ,
%~~ \mbox{for} ~~ E > 0.
%\eeq
%It follows form this that we can take $E_0 = 1$.
%For $d=2$, the Green's function (79) and the form of $e_2(-E)$, (84), can
%be used to show that we can also take $E_0 = 1$.
Under these conditions, we can apply the Combes-Thomas method 
\cite{[CT]} as developed by Aizenman in the appendix to 
\cite{[Aizenman]} for the case of exponentially-decaying off-diagonal terms
with a large gap $\Delta_d$. 
We conclude that for any $0 < \delta < 1$, there exist finite constants
$C_d(\delta) > 0$ so that
\bea
\| P_i \;  ( t (-E) - e_d (-E) )^{-1} P_j \|^s & = & 
  | \langle i | ( t ( -E ) - e_d (_E))^{-1} | j \rangle |^s
  \nonumber \\
 &  \leq & 
   \frac{C_d(\delta)}{ (E^{\beta_d} \Delta_d )^{2s}}  
   e^{- a s \sqrt{E}  \| i-j \| } ,
\eea
where $\beta_2 = 5 / 4$ and $\beta_3 = 4$.
We use estimate (\ref{tbound}) to evaluate $M_0$ defined in (22). 
For $d=2,3$, and for any $\gamma = \delta ' \sqrt{E}$, with
$0 < \delta' < \delta $, we have,
\bea
\label{eq:msum}
M_0^{(d)}
( s,E,\gamma) & \leq & \frac{C_d(\delta)}{ (E^{\beta_d} \Delta_d )^{2s}} 
\left\{ \sup_{i \in \Gamma ( \omega')} \Sum_{j \in \Gamma (\omega')} 
e^{ - s ( \delta - \delta') \sqrt{E} )  \|i-j\| } \right\} \nonumber \\
 & \leq & \frac{C_d(\delta, \delta')}{ (E^{\beta_d} \Delta_d )^{2s} E^{d/2}} .  
\eea
For the case $d=1$, the estimate (\ref{tbound})
and the form of $e_1 (-E) = \mathcal{O} (E^{-1/2})$ show that the
gap $\Delta_1 = \mathcal{O} ( E^{-1/2} )$. The appropriate Combes-Thomas
estimate \cite{[Aizenman]} is
\bea
\| P_i \; ( t (-E) - e_1 (-E) )^{-1} P_j \|^s &=&
| \langle i | ( t ( -E ) - e_1 (_E))^{-1} | j \rangle |^s
\nonumber \\
 &  \leq &
 C_1(s) E^{s/2} e^{- s \delta (1- \delta) E  \| i-j \| } .
\eea
Calculating $M_0^{(1)} ( s,E,\gamma)$, with $\gamma = \delta' E$,
for any $0 < \delta' < \delta ( 1 - \delta)$, we obtain the bound
\beq
\label{eq:msum2}
M_0^{(1)} ( s,E,\gamma) \; \leq C_1 ( \delta, \delta' , s) E^{s/2 - 1} .
\eeq
%and for $d=1$, we obtain
%\beq
%\label{eq:msum2}
%M_0 ( s,E,\gamma) \leq \left( \frac{1}{ 2 \sqrt{ E}} \right)^s \left\{
%\sup_{i \in \Gamma ( \omega')} \Sum_{j \in \Gamma (\omega')}  e^{-
%( \sqrt{E} s - \gamma ) \|i-j\| } \right\}.
%\eeq
%The supremum and sum over $\Gamma (\omega')$ can be replaced
%by the supremum and sum over $\Z^d$ because of positivity.
%We see that we must require
%\beq
%\gamma < \frac{ c  \Delta_1 }{f+\Delta}, ~~d=2,3; ~~\gamma < \sqrt{E}, ~~d=1 .
%\eeq
%Let us define $\kappa_s > 0$ by
%\beq
%\label{eq:energy}
%\kappa_s \equiv \frac{ c s \Delta }{f+\Delta} - s \gamma ~d=2,3; ~~\kappa_s
%\equiv s ( \sqrt{E} - \gamma ) , ~d=1 ,
%\eeq
%and assume that $E > E_0$, so that $\Delta > 1/2$.
%Under this condition, and with $E > E_0$,
%the sums in (\ref{eq:msum})--(\ref{eq:msum2}) are
%uniformly bounded by a constant
%$C_d > 0$, depending on dimension, and independent of $\omega'$. 
%It follows that
%if the energy satisfies $E > E_0$, then
%\bea
%M_0 ( s,E,\gamma) & \leq & 4^s C_d e_d (-E)^{-s} , ~~d=2,3 \nonumber \\
%M_0 ( s,E,\gamma) & \leq & 2^{-s} C_1 E^{-s/2}, ~~d=1 .
%\eea
Finally, the condition $K_s M_0 ( s,E,\gamma) < 1$, and the estimates
(\ref{eq:msum})--(\ref{eq:msum2}) yield the result.  $\Box$

\vspace{.1in}

We can now state the main theorem on the exponential decay
of small moments of the Green's function at negative energies.
Since the free Green's functions exhibits locally integrable
singularities for $d = 2 , 3$, we let $L ( \|x \| )$ denote the
local singularity. We have $L(\|x\|) = | \log (\|x\| ) |$, for $d = 2$,
and $L( \|x\| ) = \|x \|^{-1}$, for $d=3$.
For $i \in \Z^d$ so that $\|x-i \| < 1$, we define ${\cal L }_s ( \|x\| )
\equiv \Sum_{i : \|x-i \| < 1} L(\|x-i\|)^s $. 

\vspace{.1in}  
\noindent
{\bf Theorem 4.7.}
{\it Let $G_\omega ( x , y ; -E)$, for $E > 0$,
be the Green's function for the random interaction
Hamiltonian defined in (\ref{eq:model}). Let $\chi_0$
be a function of compact support. Under Hypotheses 4.1--4.2, 
for any $0 < s < 1$, there exists a finite constant 
$A_s > 0$, independent
of $x , y \in \R^d$, 
but depending on $\mbox{supp} \: \chi_0$, so that
for Lebesgue almost every $E > E_1$ (see Proposition 4.5), 
and for any $0 < \gamma (E)  < \gamma_d (E)$,
the expectation of the Green's function satisfies the estimate:
\beq
\label{eq:est}
\E \{ | G_\omega ( x , y ; -E ) \chi_0 (y ) |^s \} 
  \; \leq  \; A_s {\cal C} ( x,y ) e^{- s \gamma (E)  \| x - y \| } ,
\eeq
where the singular coefficient is given by 
\beq
{\cal C} (x,y) = ( L(\|x-y\|)^s + {\cal L}_s (\|x\|) + {\cal L}_s (\|y\|) 
 + {\cal L}_s (\|x\|) {\cal L}_s (\|y\|) ) .
\eeq
}

\vspace{.1in}
\noindent 
{\bf Proof.} 
This is a direct consequence of the exponential decay of the free Green's
function $G_0$ and the estimate on $K ( -E ; \omega )$ given in 
Proposition 4.5. We fix $\omega' \in \Omega'$. All the estimates are
uniform in $\omega'$, as in the proof of Proposition
4.5. From definition (81), we have
\bea
\label{eq:expect}
\lefteqn{ \E \{ | G_\omega ( x , y ; -E )|^s \chi_0 (y) \} }
\nonumber \\
 & \leq & | G_0 ( x-y ; -E )|^s \chi_0 (y) \nonumber \\   
  &  &       + \Sum_{i,j \in \Gamma (\omega')} \: |G_0 (x-i; -E )|^s \: 
     \E \{ | K^{-1} ( -E ; \omega )_{ij} |^s \}
      \: | G_0 ( y-j ; -E) |^s \chi_0 (y) . \nonumber \\
  & &   
\eea
The expectation on the last line is evaluated in Proposition 4.5. 
For notational convenience,
let $\mu \equiv s \gamma_d (E)$, and ${\tilde \mu} \equiv s \gamma
(E)$, where $0 < \gamma (E) < \gamma_d (E) $.
As in Proposition 4.5, we can replace the double sum in (\ref{eq:expect}) by 
a double sum over $\Z^d$.
We divide the double sum in (\ref{eq:expect}) into four terms depending
upon whether $\| x - i \| $ or $\| y-j \|$ are less than or
greater than one.
For the sum over $(i,j)$ with $\|x-i \| > 1$ and $\|y-j \| > 1$,
we must control the sum
\beq
\Sum_{i,j} e^{ - \mu \|x-i\|} e^{- \mu \|i-j\| } e^{- \mu \|y - j\| }
   \chi_0 ( y ) e^{ {\tilde \mu} \|x-y \|} .
\eeq
Because $\chi_0$ has compact support, we can bound the sum above by
\beq
\left( \sup_y  e^{ ( {\tilde \mu} +  \mu ) \|y \| } \chi_0 ( y )
\right) \; \left( \Sum_{i,j} e^{- \delta \| x-i\| } e^{- \delta \|i-j\|} 
     e^{ - \delta \|j \| } \right) ,
\eeq
where $\delta = \mu - {\tilde \mu} > 0$. This sum is finite and
independent of $\|x\|$. For the most singular term with $\|x-i \| < 1$ and $\|
y - j \| < 1$, we have
\beq
B_s \; \Sum_{i' , j' } L ( \|x-i' \|)^s \; L ( \|y-j'\| )^s  = B_s {\cal L}_s
(\|x\|) \; {\cal L}_s ( \|y\| ),
\eeq
for some finite constant $B_s > 0$.
This is a locally integrable function. Its $L^1$-norm depends only
on $| \Lambda_0 |$ and $| \mbox{supp} \: \chi_0 |$.
Similarly, the other two terms, involving $\|x-i\| < 1$ and $\|y-j\| > 1$, for
example, are bounded by
\beq
\left( C_s \; \Sum_{i'} L( \|x-i' \| )^s e^{2 \mu \|i'\| } \right) \; 
\left( \Sum_{j : \|y-j\| > 1} 
e^{- \mu \|x\| } e^{- \mu \|j\| } e^{- \mu \|y-j\|} \chi_0(y) e^{ {\tilde \mu }
\|x-y \| } \right) ,
\eeq
that is easily seen to satisfy the bound
on the right in (\ref{eq:est}). 
These estimate prove the bound (\ref{eq:est}). $\Box$ 

In order to use the results on the perturbation of singular spectra
in \cite{[CH1]}, we need to prove the following estimate on the
localized resolvent:
\beq
\| R_\omega ( -E ) \chi_0 \| \leq \infty ,
\eeq
for almost every $\omega$ and Lebesgue almost every $- E \in 
( - \infty , -E_1) $, where $E_1 > 0$ is determined in Proposition 4.5.

\vspace{.1in}
\noindent
{\bf Proposition 4.8.} {\it Under the conditions stated above,
we have, for any function $\chi_0$ of compact support,   
\beq
\| R_\omega ( -E) \chi_0 \|_{HS} \; < \: \infty ,
\eeq
almost surely, for Lebesgue almost every $E > E_1$, and where $HS$ 
denotes the Hilbert-Schmidt norm, and $E_1$ is determined in Proposition
4.5.}

\vspace{.1in}
\noindent
{\bf Proof.}  
Let $\chi_0$ be a function with compact support, and let $\Lambda_0 
\subset \R^d$ be a compact region so that $\mbox{dist} \: ( \Lambda_0 , 
\mbox{supp} \: \chi_0 ) > 1$.
Let $\Z_0^d$ denote those lattice points so that
the translates $\{ \Lambda_0 + i \} $ satisfy
$\mbox{dist} \: ( \{ \Lambda_0 + i \} ,  \mbox{supp} \: \chi_0
) > 1 $. The complement of this set of lattice points, $ ( \Z_0^d )^c
\equiv \Z^d \backslash
\Z^d_0$, consists of finitely-many points depending only on $| \mbox{supp}
\: \chi_0|$.
For any $x \in \Lambda_0$, estimate (\ref{eq:est}) implies that
\bea
\Sum_{i \in \Z_0^d } \E \{ | G_\omega ( x+i , y ; -E ) \chi_0 (y) |^s \}
 & \leq  & A_s \: \Sum_{i \in \Z_0^d}
 e^{- s \gamma (E) \| i - ( y - x ) \| } \nonumber \\
 & \leq & A_s e^{ s \gamma (E) \| x - y \| } .
\eea
Similarly, we estimate the sum over the complementary set
of points using the form of the local singularities given in Theorem
4.7:
\bea
\label{eq:est2}
\lefteqn{ 
\Sum_{ i \in ( \Z_0^d )^c } \E \{ | G_\omega ( x + i , y ; -E ) \chi_0 (y)|^s 
\} } \nonumber \\
& \leq & C_s ( | \mbox{supp} \: \chi_0 | ) \{ L ( \| x-y \| )^s + 
{\cal L}_s ( \|x\| ) + {\cal L}_s ( \|y \| ) + {\cal L}_s ( \|x\| ){\cal
L}_s ( \|y \| ) \} \chi_0 ( y ) . \nonumber \\
 && 
\eea
Interchanging the sum and expectation in (\ref{eq:est2}), we find that for all 
$x \in \Lambda_0$, and for all $y \in 
\mbox{supp} \chi_0$, and for almost every $\omega \in \Omega$, there
exists a finite, positive constant $C( \Lambda_0 , \chi_0, \omega ) $ such that
\beq
\Sum_{i \in \Z_0^d} | G_\omega ( x + i , y ; -E ) \chi_0 (y) |^s 
\; \leq  \; C( \Lambda_0, \chi_0 , \omega ) < \infty .
\eeq
We obtain a similar result for the sum over $( \Z_0^d )^c$, with
the singularity occurring explicitly in the upper bound.
Using the fact that for real $a_j > 0$, the finiteness of 
$\sum_j a_j^s < \infty$, for $0 < s < 1$, implies 
$\sum_j a_j^2 < \infty$, we obtain
\beq
\label{eq:est3}
\Sum_{i \in \Z_0^d}  
| G_\omega ( x + i , y ; -E ) \chi_0 ( y) |^2 \; \leq
\;  C( \Lambda_0 , \chi_0, \omega) < \infty,
\eeq
for almost every $\omega \in \Omega$, and for any $x \in \Lambda_0$, and 
$y \in \mbox{supp} \: \chi_0$, with $\Lambda_0$ disjoint
from $\mbox{supp} \: \chi_0$. 
Again, for the sum over $( \Z_0^d)^c$, we obtain a finite upper bound
with the explicit form of the local singularity as in (\ref{eq:est}).
We now use the following elementary lemma.

\vspace{.1in}

\noindent
{\bf Lemma 4.9.} {\it If $f \geq 0$ is finite for each $x \in K \subset \R^d$,
compact, then $\int_K f (x) ~dx < \infty$. 
}

\vspace{.1in}

\noindent  
As a consequence, we have that 
\beq
\int_{\Lambda_0} \: \Sum_{i \in \Z_0^d} 
| G_\omega ( x+i , y ; -E ) \chi_0 (y) |^2 ~dx
< \infty .
\eeq
Now let $\Lambda_0 = \Lambda_1 ( 0)$,
and let $\chi_1$ be its characteristic function.
We denote by $\chi_j$ the translated characteristic function
$\chi_1 ( x + j )$.
Integrating over $y$ and changing variables, it follows from
(\ref{eq:est2})--(\ref{eq:est3}) 
that
\bea
\lefteqn{ \int_{\mbox{supp} \chi_0 } dy \Sum_{i \in \Z^d}
\; \int_{\Lambda_0} dx 
| \chi_1 ( x+i ) G_\omega (x,y; -E ) \chi_0 (y) |^2  } \nonumber \\
 & \leq & \int_{\mbox{supp} \chi_0 } dy \Sum_{i \in \Z_0^d} \int_{\Lambda_0} dx 
| \chi_i (x) G_\omega (x,y; -E)|^2 \nonumber \\
 & & + C_s \int_{\mbox{supp} \chi_0 } dy \Sum_{i \in (\Z_0^d)^c} 
\int_{\Lambda_0} dx \: | \chi_i (x) \left\{ 
L( \|x-y\| )^s + {\cal L}_s ( \|x\| )
 + {\cal L}_s ( \|y\| ) \right. \nonumber \\
 && \left. + {\cal L}_s ( \|x\| ){\cal L}_s ( \|y\| ) \right\}|^2 \nonumber \\
 & \leq & \infty ,
\eea
that follows from explicit integration of the local singularity and
(\ref{eq:est3}).
$\Box$






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\renewcommand{\thechapter}{\arabic{chapter}}
%\renewcommand{\thesection}{\thechapter}

%\setcounter{chapter}{5}
%\setcounter{equation}{0}

\section{Spectral Averaging and the Perturbation of Singular
Spectra for the Random Singular Model}\label{S.5} 

In order to complete the proof of localization at negative
energies for the random singular model,
we need to prove a spectral averaging result (implying
Kotani's trick) for these models, and a result on the
perturbation of singular spectra.  
Since we will be working at negative energies, local Hamiltonians
$H_\Lambda$ are
defined as follows. For any bounded, regular, open region $\Lambda \subset
\R^d$, we let $H_\Lambda$ be the operator formally defined in (77) by
restricting the sum to lattice points in ${\tilde \Lambda} \cap \Gamma
(\omega')$, where ${\tilde \Lambda} = \Lambda \cap \Z^d$. 
The exact definition is supplied by the resolvent formula (81) with the
summations restricted to this set.
It will be useful to write the resolvent of $H_\Lambda$, for $E > 0$, as
\beq
R_\Lambda ( - E ) = R_0 ( -E ) + R_0 ( -E ) P_\Lambda^*
K_\Lambda^{-1} ( -E) P_\Lambda R_0 ( -E ) ,
\eeq
where $P_\Lambda : H^2 ( \R^d ) \rightarrow \ell^2 ( { \tilde \Lambda }
)$ is defined as follows.
Since $d \leq 3$, each element of $H^2 ( \R^d )$ has a
continuous representative. For $u \in H^2 ( \R^d )$,
we define $P_\Lambda$ as
\beq
(P_\Lambda u) (i) \equiv  u(i), ~~\mbox{for} ~~ i \in {\tilde \Lambda },
\eeq
and zero otherwise. Note that we have,
\beq
\| P_\Lambda u \|_{ \ell^2 ( { \tilde \Lambda } )}^2 =
\sum_{k \in { \tilde \Lambda } } | u(k) |^2 \leq \|u\|_{L^2 (\R^d)^2}.
\eeq

\vspace{.1in}
\noindent
{\bf Proposition 5.1.} {\it Let $H_\Lambda ( \lambda)$ be the local
Hamiltonian defined through the resolvent formula (81)
with the summations over ${\tilde \Lambda} \cap \Gamma(\omega')$, and
with all but the $\lambda \equiv
\lambda_k$, for $k \in {\tilde \Lambda}$, fixed.
Let $I_\eta = [ E - \eta , E+ \eta ] \subset \R^{-}$ be a negative energy 
interval. Let $E_\Lambda ( I_\eta , \lambda )$
be the spectral projector for $H_\Lambda ( \lambda)$ and the
interval $I_\eta$. There exists a finite, positive constant $C_0 > 0$,
depending on $E$, so that  
\beq
\left| \int h_0 ( \lambda ) E_\Lambda ( I_\eta , \lambda ) ~d \lambda
\right|  \leq
C_0 \: ( \sup_{ \lambda } | \lambda^2 h_0 ( \lambda ) | ) \: | \eta |
.
\eeq
}

\vspace{.1in}
\noindent
{\bf Proof.} We follow the proof in \cite{[CHM]}
using a differential inequality. 
Let $\phi \in H^2 ( \R^d)$. By Stone's formula, we have
\beq
\int h_0 ( \lambda ) ~d \lambda \langle \phi, 
E_\Lambda ( I_\eta , \lambda ) \phi \rangle = \lim_{\epsilon
\rightarrow 0} \int h_0 ( \lambda ) ~d \lambda \; \int_{I_\eta}
\: Im \: \langle \phi, R_\Lambda ( E+ i \epsilon ) \phi \rangle ~dE.
\eeq
Using the local version of 
the resolvent formula (130), and the analyticity of $R_0
(z)$ for $z$ in a neighborhood of $I_\eta \subset \R^{-}$, 
we can write the inner
product on the right side of (135) as
\beq
\langle P_\Lambda R_0 ( E - i \epsilon ) \phi , K_\Lambda^{-1} ( E + i
\epsilon ) P_\Lambda R_0 ( E + i \epsilon ) \phi \rangle .
\eeq
The vector $R_0 ( z ) \phi$ is an analytic function for $z$ in a
neighborhood if $I_\eta$. Hence, it suffice to prove that 
for any $\xi \in \ell^2 ( {\tilde \Lambda} )$,
\beq
\Sup_{\epsilon \rightarrow 0 }
\left| \int h_0 ( \lambda ) ~d \lambda \langle \xi , Im K_\Lambda^{-1} 
( E + i\epsilon ) \xi \rangle \right| \leq
C_1 \; \| \xi \|^2  .   
\eeq
We recall that
\bea
K_\Lambda ( E + i\epsilon ) & = & ( t ( E + i \epsilon ) + v
- e_d ( E + i\epsilon ) ) | \ell^2 ( {\tilde \Lambda} \cap \Gamma
  (\omega')) \nonumber \\
 & \equiv &  ( h_\Lambda ( E + i\epsilon ) - e_d ( E + i\epsilon))
  | \ell^2 ( {\tilde \Lambda} \cap \Gamma
  (\omega')) .
\eea
We write $h_\Lambda (z; \sigma) = t_\Lambda (z) + v_\Lambda$ 
and $K_\Lambda (z)$
for the operators acting on $\ell^2 ( {\tilde \Lambda} \cap \Gamma (\omega'))$.
We change variable from $\lambda$ to $\sigma \equiv 1 / \lambda$,
so that $v_{ij} = \sigma_i \delta_{ij}$. This is convenient
because
\beq
\dot{h} ( z ; \sigma ) \equiv \frac{ d h( z ; \sigma ) }{ d \sigma } =
\delta_{ij} .  
\eeq
We will use this parametrization the integral (134). To apply the
differential inequality method, we let $\delta > 0$, and define
\beq
F ( \epsilon , \delta ) 
\equiv \int \sigma^{-2} {\tilde h}_0 ( \sigma ) ~d \sigma \; \langle \xi , 
( h_\Lambda ( E + i \epsilon; \sigma ) - e_d( E + i\epsilon ) -
i \delta \dot{h} ( E + i\epsilon ; \sigma ) )^{-1} \xi \rangle.
\eeq
As in \cite{[CHM]}, we obtain an a priori estimate
on $F ( \epsilon , \delta )$. Because of the dependence of the
kinetic energy $t(z)$ on $\epsilon$, the form of this estimate is
slightly different. Let $M( \sigma , \delta, \epsilon )$ denote
the matrix occurring in the inner product in (139). We then have, 
for any $\delta_m \in \ell^2 ( {\tilde \Lambda} )$, and $\epsilon > 0$ small
enough, 
\bea
\| M( \sigma , \delta, \epsilon ) \delta_m \| & \geq &  Im
\langle \delta_m , M( \sigma , \delta, \epsilon ) \delta_m
\rangle \nonumber \\
 & \geq & \langle \delta_m , \overline{M} \{
- Im t_\Lambda (E+i\epsilon) + Im e_d ( E + i \epsilon
  ) + \delta \dot{h} \} M \delta_m \rangle \nonumber \\
  & \geq & C_0 ( \delta - \epsilon ) \| M( \sigma , \delta,
\epsilon ) \delta_m \|^2 ,
\eea
since one can check that the imaginary parts of the
expression are ${\cal O} ( \epsilon )$.
Differentiating $F ( \epsilon ,
\delta )$ with respect to $\delta$, we find
\beq
\frac{ d F ( \epsilon , \delta ) }{ d \delta } = 
\int \sigma^{-2} {\tilde h}_0 ( \sigma ) ~d \sigma \frac{d}{d
\sigma} \; \langle \xi ,
( h_\Lambda ( E + i \epsilon; \sigma ) - e_d( E + i\epsilon ) -
i \delta \dot{h} ( E + i\epsilon ; \sigma ) )^{-1} \xi \rangle.
\eeq
We now integrate by parts and obtain a new estimate
\beq
\left| \frac{ d F ( \epsilon , \delta ) }{ d \delta } \right|
\; \leq \; \left\| \frac{d }{ d \sigma } \left( \sigma^{-2}
{\tilde h}_0 ( \sigma ) \right) \right\|_\infty \; C_0 \; ( \delta -
\epsilon )^{-1},
\eeq
for a constant $C_0$ uniform in $\epsilon > 0$. Integrating this 
identity, we find an improved bound
\beq
| F ( \epsilon , \delta ) | \; \leq C_1| log (\delta - \epsilon)
| + C_2 .
\eeq
Iterating this procedure one more time proves the uniform
boundedness of $F ( \epsilon , \delta )$. The proposition now
follows as in \cite{[CHM]}. $\Box$  

\vspace{.1in}

This result implies that if one can prove that the singular
continuous spectrum has Lebesgue measure zero, then, almost
surely, it has spectral measure zero. To prove the former
statement, that the family of operators has singular continuous
spectra with Lebesgue measure zero, we need to show that the
argument of \cite{[CH1]} can be modified for these
singular models.
We do this in two steps. First, the result of Simon-Wolff
\cite{[SW]} and the estimate (109) in Proposition 4.5 proves that
for almost every energy $E \in I$, and with probability one,
\beq
\sup_i \sum_j ~| [K ( - E ; \omega )^{-1}]_{ij} |^2 < \infty ,
\eeq
Let $I_0$ be the subset of $I$ of full measure for which the sum in
(144) is finite with probability one. Second, for any function $\chi$
of compact support, the resolvent formula,
\beq
R(z)\chi = R_0 (z) \chi + R_0 (z) P^* K^{-1} (z) P R_0 (z) \chi ,
\eeq
and Proposition 4.5, imply that, 
with probability one,  
\beq
\sup_{\epsilon \rightarrow 0 } \| R ( E + i \epsilon ) \chi \| < \infty ,
\eeq
for $E \in I_0$.
We use this result as the starting point for the analysis of the
perturbation of singular spectra as in \cite{[CH1]}. 

\vspace{.1in} 

\noindent
{\bf Proposition 5.2.} {\it
For any $k \in \Z^d$, consider the Hamiltonian $H(\lambda; k)$ obtained
from $H_\omega$ by varying the coupling constant at site $k \in \Z^d$.
Then, we have that $\sigma ( H(\lambda; k)) \cap I$ is purely pure point
almost surely for Lebesgue almost every $\lambda \in \R$. }

\vspace{.1in}

\noindent
{\bf Proof.} We begin with the operator form of the identity
(81):
\beq
R(z) = R_0 (z) + R_0 (z) P^* K^{-1} (z) P R_0 (z) ,
\eeq
where $P : H^2 ( \R^d ) \rightarrow \ell^2 ( \Z^d )$ is a bounded operator. 
We consider a variation of the coupling constant $\lambda_k
\rightarrow {\tilde \lambda}_k \equiv \lambda_k + \lambda$,
and look at the difference of the
two resolvents that we label $R_{\lambda_k}(z)$ and $R_{{\tilde
\lambda}_k} (z)$,
\bea
R_{\lambda_k}(z) - R_{{\tilde\lambda}_k} (z) & = & R_0 (z) P^* 
\{ K^{-1}_{\lambda_k} (z) - K^{-1}_{{\tilde\lambda}_k} (z) \} P 
 R_0 (z) \nonumber \\
 & = & \frac{- \lambda }{ \lambda_k {\tilde
\lambda}_k } \; R_0 (z) P^* K^{-1}_{{\tilde\lambda}_k} (z)
 \cdot \delta_k  \cdot K^{-1}_{\lambda_k} (z) P R_0 (z)
\nonumber \\
  & = & R_0 (z) P^* K ( z ) P R_0 (z),
\eea
where $\delta_k$ is the projection onto the site $k \in \Z^d$,
and the operator $K(z) : \ell^2 ( \Gamma (\omega')) \rightarrow \ell^s (\Gamma
(\omega'))$  has the form
\bea
K(z) & = & K^{-1}_{\lambda_k} (z) - K^{-1}_{{\tilde\lambda}_k}
     (z) \nonumber \\
 & = & \frac{-\lambda }{ \lambda_k {\tilde \lambda}_k } 
    K^{-1}_{{\tilde\lambda}_k} (z) \cdot \delta_k  
     \cdot K^{-1}_{\lambda_k} (z)  .
\eea
We write (149) as
\beq
K^{-1}_{{\tilde\lambda}_k}(z)
\left( 1 - \frac{\lambda}{ {\tilde \lambda}_k \lambda_k}
K^{-1}_{\lambda_k} (z) \cdot \delta_k \right)
  = K^{-1}_{\lambda_k} (z) .
\eeq
Since the second factor on the left in (150) is a finite-rank operator,
we can apply Fredholm theory to analyze the inverse.
Let us take $z = E + i \epsilon$, for $\epsilon > 0$, with $E \in I_0$. 
It follows from the existence of the limit in (146) that
$n-\lim_{\epsilon \rightarrow 0} K^{-1}_{\lambda_k} (E+i \epsilon)
\equiv K^{-1}_{\lambda_k} (E+i0)$ exists and is a compact operator.
As in \cite{[CH1]},
it follows that the boundary-value is not invertible at energy $E$
if and only if there
exists a $\psi_E \in \ell^2 ( \Z^d))$ so that
\beq
\{ h_{{\tilde \lambda}_k} (E) - e_d(E) \} \psi_E = 0.
\eeq
The set of such energies is countable. It is easy to check that such
energies are zeros of $K^{-1}_{{\tilde\lambda}_k} (z)$. It follows
from the resolvent formula that such an energy is an eigenvalue of
$H({\tilde\lambda}_k)$.
Consequently, there exists a subset $I_\lambda
\subset I_0$, with $| I_\lambda | = |I_0| = |I|$,
for which the factor is invertible. From this, we conclude
that there is a finite $C_\lambda (E) > 0$ so that
\beq
\sup_{\epsilon >0} ~\| R_{{\tilde\lambda}_k} ( E + i \epsilon ) \chi \|
\leq C_\lambda (E) < \infty .
\eeq
This proves that there is no absolutely continuous spectrum for
$H(\lambda)$ in $I$ and that the singular continuous spectrum lies in a set
of Lebesgue measure zero. Proposition 5.1 now implies the result. $\Box$   

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\section{Exponential Localization and Dynamical Localization}\label{S.4}

Control over the expectation of the small moments of the Green's function
implies several results for the system. In this section, we first
prove exponential localization for the random singular interaction model.
This includes the proof that the eigenfunctions are exponentially localized 
in space. 
We then study the dynamics of the system and prove that the time evolution
of moments of the position operator remain bounded in time.
The completion of the proof of the first part of
Theorem 1.2, the existence of pure point spectrum, follows as in
\cite{[CH1]}.

We need one further estimate in order to prove exponential localization.
Let $\chi$ be a function of compact support and let $\chi_n$ be its
translate by $n \in \Z^d$. Then, with probability one and for almost every
$E \in I$, there exist finite constants $C(E) > 0$ and $\gamma(E) >0$,
so that we have
\beq
\label{expo1}
\| \chi_n R(E + i0) \chi \| \leq C(E) e^{- \gamma(E) \|n\| } .
\eeq
The proof of this follows from the resolvent formula, (145), the
exponential decay of the free resolvent at negative energies, and
Proposition 4.5.

\subsection{Exponential Localization}

We first prove that given a single-site probability distribution $\mu_0$
with support on $[-b , -a]$, there exists an energy interval $I$ near the
bottom of the spectrum for which $\sigma (H_\omega) \cap I \neq \emptyset$
and is pure point with probability one. 
According to Theorem 2.1, we must establish an interval of energies for
which (13) holds. 
The constant $K_s$, appearing in the inequality (13),
is a function of $\mu_0$.
We compute $K_s$ as a function of the endpoints of the support of the
density $h_0$, for the $iid$ case when $d \mu_n ( \lambda )
= h_0 ( \lambda ) d \lambda$, for all $n \in \Z^d$.
The computation of $K_s$ follows from \cite{[Aizenman]}, with an
improvement in \cite{[KrishnaSinha]}.

\vspace{.1in}
\noindent
{\bf Proposition 6.1.} \cite{[KrishnaSinha]}.
{\it  Let $d \mu (x) = h_0 (x) dx$
be the probability measure for the
random variable $\lambda_0$.
Suppose that for some $q > 0$, we
have $Q \equiv \int h_0^{1+q} (x) ~dx < \infty$, and for some $0 <
\tau \leq 1$, we have $B \equiv \int |x|^{\tau} d\mu (x) <
\infty$. Then, for any $0 < s < [ 1 + (1/ \tau ) +( 1/
q)]^{-1}$, we have
\beq
K_s = B^{s / \tau} ( 2^{1 + 2s} + 4) \left[ B^{1 - s/ \tau } +
B^{s / \tau } C \left( Q , \frac{s}{1-\frac{2s}{\tau}} , q
\right)^{\frac{\tau - 2 s}{\tau}} \right] ,
\eeq
where the constant $C ( Q , \frac{s}{1-\frac{2s}{\tau}} , q)$
is given by
\beq
C \left( Q , \frac{s}{1-\frac{2s}{\tau}} , q \right) = 1 +
\frac{ s ( 2^q Q )^{ 1 / ( 1 + q )}}{ \frac{q }{ q+1 } - s } .
\eeq
}


\vspace{.1in}

\noindent
We now combine this result with the estimates of sections 4 and 5.

\vspace{.1in}

\noindent
{\bf Theorem 6.2.}
{\it Let us consider the random singular model (77) satisfying
Hypotheses 4.1--4.2.
We recall that $\mbox{supp} ~h_0 = [ -b , -a]$, for $0 < a < b$,
and that $\inf \Sigma = E_{0,d} ( 0 , - 1/a) = - \mathcal{O} (a^2)$, 
as $a \rightarrow \infty$, for $d=2,3$, and 
$\inf \Sigma = E_{0,d} ( 0 , - a) = - \mathcal{O} (a)$,
as $a \rightarrow 0$, for $d=1$. 
For $d=2,3$ and $a > 1$, there exists an energy $E_1 (a) > 0$,
defined in Proposition 4.5,
so that if we define the closed interval $I \equiv [ E_0 ( 0 , - 1/a) , E_1
(a) ]$, we have $\Sigma \cap I \neq \emptyset$ and
$\Sigma \cap I$ is pure point with exponentially decaying eigenfunctions.
For $d=1$, for $0 < a < 1$ sufficiently small, there
exists an energy $E_1 (a) > 0$, with $E_1 (a) = \mathcal{O}(a^\alpha)$,
for any $1 \leq \alpha < 2 / (2 - s)$, for $s < 1/3$, so that there is
exponential localization on the interval $[ E_0 ( 0 , -a) , E_1 (a) ]$
almost surely.}

\vspace{.1in}
\noindent
{\bf Proof:}
It follows from Proposition 6.1 and Hypothesis 4.1 that
$K_s \leq A  a^{1 + \tau}$,
for $0 < \tau \leq 1$,
and for a constant depending on $\|h_0\|_\infty$ and
$s$. We can take $\tau = 1$, provided
$s < 1 / ( 3 + \epsilon)$, for any $\epsilon > 0$.
We recall from Proposition 4.5, that condition (25) $M_0^{(d)} K_s < 1$,
is satisfied provided $E > \mathcal{O} ( (K_s C_d )^{\delta_d (s)} )$. On the
other hand, we also need $- E > \mbox{inf} ~\Sigma$. By Proposition 4.2,
the bottom of the spectrum scales like $\mathcal{O} (a^2)$, as $a
\rightarrow \infty$ (even faster in the case $d=2$). 
Upon evaluating $K_s$ as a function of $a$,
these two conditions require that
we have $a^2 > C_d^{\delta_d (s)} a^{2 \delta_d (s)}$, or $2(1 -
\delta_d (s)) > 0$, for large $a > 0$. 
As $\delta_d(s) < 1$, this condition is
satisfied. 
It follows that for large $a > 0$, the conclusion of Theorem 2.4 holds for
the interval $I = [ \inf \Sigma , - E_0 (a) ]$, and by Proposition 4.8 and
section 5, we have pure point spectrum on
this interval.  
For $d=1$, we consider energies $E(a) = \mathcal{O} ( a^\alpha )$, for $\alpha
\geq 2$, since these energies satisfy $\inf \Sigma \leq -E (a) < 0$. 
For simplicity, let us $\tau = 1$. Condition (25), that $M_0^{(1)} K_s < 1$, 
requires that $a^{1 + \alpha((s-2)/2)} > 0$, as $a \rightarrow 0$. 
In this case, we have $1 \leq \alpha < 2 / (2 - s)$, and
localization for any $0 < E  = \mathcal{O} (a^\alpha)$ 
almost surely provided $a > 0$ is small enough. 
The proof of exponential decay of the eigenfunctions follows
from (\ref{expo1}) and the proof in \cite{[CH1]} . $\Box$
 
\vspace{.1in}
\noindent  
We next turn to the question of exponential decay of eigenfunctions
corresponding to eigenvalues in $I$.

\vspace{.1in}
\noindent
{\bf Proposition 6.3.} {\it Let $- E < -E_1$, where $E_1 > 0$ is determined
in Proposition 4.6, be an eigenvalue of the
Hamiltonian $H_\omega$ with normalized eigenfunction $\psi_{-E ,
\omega}$, satisfying $\| \psi_{-E , \omega} \|_{L^2 ( \R^d )} = 1$. Then
there exist finite, positive constants $C_{-E , \omega} > 0$, and $\mu
( -E , \omega ) > 0$, so that 
\beq
\| e^{ \mu ( -E , \omega ) \|x\| } \psi_{-E , \omega} \|_{L^2 (
\R^d )} \; \leq \; C_{-E , \omega} ,
\eeq
that is, the eigenfunction decays exponentially at infinity.}


\vspace{.1in}
\noindent
{\bf Proof.} In order to prove exponential decay of the eigenfunctions
corresponding to negative eigenvalues of $H_\omega$, we need to
obtain the following type of estimate. Let $\chi_0$ be a function
of compact support around the origin, and let $\chi_{x_0}$ be
a function of compact support around $x_0 \in\R^d$. We must prove
for almost every $\omega$, there exists a finite constant
$C_\omega > 0$, so that, uniformly with respect to $\epsilon > 0$, we have 
\beq
\| \chi_{x_0} R_\omega ( -E + i \epsilon ) \chi_0 \| \; \leq \; C_\omega
e^{ - \mu \| x_0 \| }.
\eeq
As in section 2, we begin with an estimate on the localized
Green's function $\chi_{x_0} (x) G_\omega ( x , y ; -E ) \chi_0
(y)$. It follows from Proposition 2.4 that
\beq
\E \{ | \chi_{x_0} (x) G_\omega ( x , y ; -E ) \chi_0 (y) |^s \} 
\; \leq \; C_s e^{ - \mu \|x_0 \| } ,
\eeq
Consequently,  for almost every $\omega$, there exists a constant
$C_\omega > 0$ so that
\beq
\Sum_{i \in \Z^d} \: e^{ \mu \|x_0 \| }
| \chi_{x_0} ( x+i) G_\omega ( x+i , y ; -E
) \chi_0 (y) |^s \; \leq \; C_\omega .
\eeq
Since the support of $\chi_{x_0}$ is compact, the sum over $i \in
\Z^d$ is finite. As in section 2, we may replace the exponent $s$
by the power 2, and obtain
\beq
\Sum_{i \in \Z^d} \: e^{ \mu \|x_0 \| }
| \chi_{x_0} ( x+i) G_\omega ( x+i , y ; -E
) \chi_0 (y) |^2 \; \leq \; C_\omega ,
\eeq
almost surely. This implies that 
\beq
\Sum_{i \in \Z^d} \;  | \chi_{x_0} ( x+i) G_\omega ( x+i , y ; -E
) \chi_0 (y) |^s \; \leq \; C_\omega e^{ - \mu \|x_0\| } ,
\eeq
almost surely.     
This puts us in the situation analogous to (2.4). Following the
same Hilbert-Schmidt argument as there, we arrive at the result
\beq
\| \chi_{x_0} R_\omega ( -E + i \epsilon ) \chi_0 \| \; \leq \; 
C_\omega e^{ - \mu \|x_0\| } ,
\eeq
almost surely. We now follow the argument in section 3 of
\cite{[CH1]} to prove the proposition. $\Box$ 


\subsection{Dynamical Localization}

Dynamical localization \cite{[DeBievreGerminet]} is the statement
that the $q^{th}$-power of the position operator
in a state evolving according to the time evolution generated by
$H_\omega$, and localized in energy to an interval of energy in
the localization regime, remains bounded in time with probability
one. For $q = 2$, this implies the absence of diffusion. 
In particular, let $\phi \in L^2 ( \R^d )$ be
normalized with compact support. Let $I \subset ( - \infty , 0)$
be an interval for which $I \cap \Sigma_{pp} \neq \emptyset$, and let
$E_\omega (I)$ be the corresponding spectral projector for $I$
and $H_\omega$. Dynamical localization (at order $q$) 
means that for almost all $\omega$,
\beq
\sup_{t>0} \: \| \: \|x\|^{q/2} e^{-i H_\omega t } E_\omega (I)
\phi \|_{HS} < \infty .
\eeq

\vspace{.1in}
\noindent
{\bf Theorem 6.4.} {\it Under hypotheses (H1)--(H2), the
family of random operators $H_\omega$ exhibits dynamical
localization for any $q \in \N$. That is, with probability one, 
for any $\phi \in L^2_0
( \R^d)$, we have
\beq
\sup_{t>0} \: \| \: \|x\|^{q/2} e^{-i H_\omega t } E_\omega (I)
\phi \|_{HS} < \infty .
\eeq
}

\vspace{.1in}
\noindent
{\bf Proof.} 
Let us consider two compactly supported functions $\chi_{x_0}$
and $\chi_{y_0}$, localized near two different points $x_0, y_0
\in \R^d$. From section 4, we have
\beq
\E \{ | \chi_{x_0} (x) G_\omega ( x,y ; -E ) \chi_{y_0} (y) |^s
\} \; \leq \; C ( s ) e^{ - \mu \| x_0 - y_0 \| },
\eeq
for finite, positive constants $C(s), \mu > 0$.
As in the proof of Proposition 6.3, we pass  
to an almost sure statement:
\beq
\Sum_i \:  e^{ \mu \| x_0 - y_0 \| } \chi_{x_0} (x+i) | G_\omega (
x+i ,y ; -E ) \chi_{y_0} (y) |^s \; \leq \; C ( s , \omega ) |
\mbox{supp}\: \chi_{x_0} | ,
\eeq
for some finite, positive constant $C(s, \omega)$.
Replacing the exponent $s$ by $2$ in the sum, we obtain
\beq
| \chi_{x_0} (x ) G_\omega ( x ,y ; -E ) \chi_{y_0} (y) |^2 
\; \leq \; C ( s , \omega ) e^{ - \mu \| x_0 - y_0 \| } ,
\eeq
almost surely.   
Applying the same type of Hilbert-Schmidt analysis as above, we
obtain
\beq
\| \chi_{x_0} R_\omega ( -E + i \epsilon ) \chi_{y_0} \| \; \leq
\; C ( s ) e^{ - \mu \| x_0 - y_0 \| }.
\eeq
Here, the constants are uniform on an interval $I$ in
$\Sigma_{pp}$. By functional analytic methods, this implies that
for any real-valued function $f$, we have the almost sure bound
\beq
\| \chi_{x_0} f( H_\omega ) E_\omega (I) \chi_{y_0} \|_{HS} \; \leq
\; C ( s ) e^{ - \mu \| x_0 - y_0 \| }.
\eeq
Replacing $\chi_{y_0}$ by a function $\phi$ of compact support,
and using the exponential decay in $x_0$, we easily conclude that
\beq
\sup_{t>0} \| \: \|x\|^{q/2} e^{- i H_\omega t} E_\omega (I) \phi \|_{HS}
 \; \leq \; \infty ,
\eeq
almost surely. $\Box$ 




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\section{The Wegner Estimate and the IDS for 
the Random Singular Model}\label{S.7}  

The existence of the density of states measure for random,
singular \Schr\ operators requires a modification of the standard
proof. We first sketch this proof. Next, 
we prove a Wegner estimate valid for any subinterval of negative
energy. This establishes the local Lipschitz continuity of the
integrated density of states (IDS) for these models.
As a consequence, the density of states measure has a positive density
at all negative energies. 

\vspace{.1in}
\noindent
{\bf Theorem 7.1.} {\it Let $E_0 > 0$ and consider $\eta > 0$ so
that the interval $I_\eta \equiv [-E_0 - \eta ,-E_0 + \eta ]
\subset ( - \infty , 0)$. There exists a constant $C_W ( E_0 , d)$,
depending on the dimension $d$ and $E_0^{-1}$, so that
\beq
\P \{ \; \mbox{dist} \: ( \sigma( H_\Lambda ) , -E_0 ) < \eta \} 
\leq C_W ( E_0 , d) | \Lambda | \eta .
\eeq
}

\vspace{.1in}
\noindent
{\bf Proof.} 
We consider the resolvent of $H_\Lambda$ for
energies $-E \in I_\eta \subset ( -\infty , 0)$, with $E>0$.
We write the resolvent of $H_\Lambda$ as
in section 5,
\beq
R_\Lambda ( - E ) = R_0 ( -E ) + R_0 ( -E ) P_\Lambda^*
K_\Lambda^{-1} ( -E) P_\Lambda R_0 ( -E ) ,
\eeq
where $P_\Lambda : H^2 ( \R^d ) \rightarrow \ell^2 ( { \tilde \Lambda }
)$ is defined in section 5. 
Let us choose $\eta < |E_0| / 2 $, and define $\delta \equiv \|
P_\Lambda R_0 (-E_0 ) \|_{ L^2 ( \R^d ) , \ell^2 ( { \tilde
\Lambda } )} $.
Since $R_0(z)$ is analytic in a neighborhood of $I_\eta$, it is
clear from (170) that the singularities of $R_\Lambda (-E)$, for $-E
\in I_\eta$, coincide
with those of $R_0(-E) P_\Lambda^* K_\Lambda^{-1} (-E) P_\Lambda
R_0(-E)$, for $-E \in I_\eta$.  
Specifically, it follows from formula (170) for the
resolvent that
\bea
\P \{ \; \mbox{dist} ( \sigma ( H_\Lambda) ,-E_0 ) \geq \eta \} & = &
\P \{ \| ( H_\Lambda + E_0 )^{-1} \| \; \leq \; 1 / \eta \} \nonumber
\\
& \geq & \P \{ \| K_\Lambda ( -E_0 )^{-1} \| \; \leq \;
 1 / ( 2 \delta^2 \eta ) \}  , \nonumber \\
 & & 
\eea
Consequently, we have
\bea
\P \{ \; \mbox{dist} ( \sigma ( H_\Lambda) ,-E_0 ) < \eta \}  
 & \leq & \P \{ \| K_\Lambda ( -E_0 )^{-1} \| \; > \;
1 / ( 2 \delta^2 \eta ) \}  \nonumber  \\
 & \leq & \P \{ \; \mbox{dist} ( \sigma ( h_\Lambda (-E_0 ))
,  e_d (- E_0 )  ) <  2 \eta \delta^2 \} . \nonumber
\\
 & & 
\eea
The theorem will then follow from the following Proposition 7.2.
$\Box$.


\vspace{.1in}
\noindent
{\bf Proposition 7.2.} {\it Under the conditions of Theorem 7.1,
we have
\beq
\P \{ \; \mbox{dist} ( \sigma ( h_\Lambda (-E_0 ))
,  e_d (-E_0 )  ) <  2 \eta \delta^2 \} 
 \; \leq \; C_W( E_0, d) \eta \delta^2 | \Lambda | ,
\eeq
for some constant $C_W ( E_0, d)$ depending on the dimension $d$ and 
$|E_0|^{-1}$.  
}

\vspace{.1in}  
\noindent
{\bf Proof.} Let us denote by $N = | {\tilde \Lambda} |$, the number
of lattice points in $\Lambda$, and the interval
$J_\eta \equiv ( e_d(-E_0 ) - 2 \eta \delta^2 ,
 - e_d(-E_0 ) + 2 \eta \delta^2 )$. The operator
$h_\Lambda ( -E_0 )$ is an $N \times N$-matrix with diagonal terms
$h_\Lambda ( -E_0 )_{ii} = \sigma_i ( \omega )$, and off-diagonal
terms $h_\Lambda ( -E_0 )_{ij} = G_0 ( i - j ; -E_0 )$, for $i \neq
j$, and $i , j \in {\tilde \Lambda}$. Let $E_\Lambda ( J_\eta )$
be the spectral projector for $h_\Lambda ( -E_0)$ and the interval
$J_\eta$. We follow the proof of
\cite{[CombesHislopNakamura]}. It suffices to bound
$\E \{ Tr ( E_\Lambda ( J_\eta ) ) \}$, where the expectation is
with respect to the random variables $\lambda_j ( \omega ) =
1 / \sigma_j (\omega)$ in ${\tilde \Lambda }$, and
the trace is on $\ell^2 ( {\tilde \Lambda } )$. 
As in \cite{[CombesHislopNakamura]}, we find the estimate
\beq
\E \{ Tr ( E_\Lambda ( J_\eta ) ) \} \; \leq \; \E \left\{ \Int_{-
3 \eta \delta^2 / 2}^{ 3 \eta \delta^2 / 2 } \; Tr \{ - \rho ' (
h_\Lambda ( E_0 ) - e_d(-E_0)  - t ) \} ~dt \right\}
.
\eeq
The Gohberg-Krein formula in this case is
\bea
\lefteqn{ \Sum_{k \in {\tilde \Lambda }} \; Tr \left\{
\left( \frac{\partial}{\partial \sigma_k} \right) \rho ( h_\Lambda (- E_0 ) - e_d(-E_0) 
 - t ) \right\} } \nonumber \\ 
 & = & \Sum_{k \in {\tilde
\Lambda }} \; Tr \left\{ \rho ' ( h_\Lambda (-E_0 ) - e_d(- E_0
) - t )  \; \left( \frac{ \partial h_\Lambda (-E_0 ) }{
\partial \sigma_k } \right) \right\} .
\eea
We see that 
\beq
\left[ \frac{ \partial h_\Lambda ( -E_0 ) }{\partial \sigma_k } \right]_{jk} =
\delta_{jk}  ,
\eeq
so we obtain
\beq
Tr \{ - \rho ' ( h_\Lambda (-E_0 )  - e_d(- E_0 ) - t
) \} \; \leq \; - \Sum_{k \in {\tilde \Lambda }} \; Tr \left\{
\frac{\partial}{\partial \sigma_k} \rho ( h_\Lambda (-E_0 ) -
e_d(- E_0) - t ) \right\} .
\eeq
We note that the variation of the random variable
$\sigma_k$ is a rank one perturbation and that it is positive.
We now take the expectation. Let $\E'$ denote the expectation
with respect to the random variables $\sigma_j ( \omega )$, for $j \neq k$.
Let $H_0 (x) = x^2 h_0(x)$, the density for the random variable
$\sigma_k$. 
As in \cite{[CombesHislopNakamura]}, we integrate with respect to the
random variable $\sigma_k$ and obtain
\bea
\lefteqn{ \Sum_{k \in{\tilde \Lambda}} \E' 
\left\{ \int \sigma_k^2 {\tilde h}_0 ( \sigma_k)
 \; \frac{\partial}{\partial \sigma_k} Tr [ \rho ( h_\Lambda (-E_0 ) -
e_d(-E_0) - t ) ] \right \} } \nonumber \\
 & \leq & \| H_0 \|_\infty \; \Sum_{k \in {\tilde \Lambda}} \;
  \E' \left\{ Tr [ \rho ( h_\Lambda^+ (-E_0) - e_d(-E_0)  
  -t ) \right.  \nonumber \\
  & & \left.  - \rho( h_\Lambda^- (-E_0) - e_d(-E_0)   - t ) ] 
  \right\} \nonumber \\    
 & \leq & C_0 \| H_0 \|_\infty |\Lambda| . 
\eea
We used the positivity and the bound on the spectral shift function
\beq
0 \leq \xi ( \lambda ; h_\Lambda^+ (-E_0) , h_\Lambda^- (-E_0) )  \; \leq 1 , 
\eeq
since, as mentioned above, the perturbation is of rank one. Using
this estimate and performing the $t$-integration in (174) gives the
result. $\Box$ 

\vspace{.1in}

\noindent
{\bf Corollary 7.3.} {\it Under 
Hypotheses 4.1 and with $\Gamma = \Z^d$, the IDS for the
random singular family is Lipschitz continuous at all negative energies.}

\vspace{.1in}

It is clear from this discussion that the eigenvalues of $H_\Lambda$ in 
near $E_0 < 0$ correspond with the eigenvalues of the
matrix $K_\Lambda (E)$. We state this in the following proposition.

\vspace{.1in}
\noindent
{\bf Proposition 7.4.} {\it The energy $E_j < 0$ is an eigenvalue
of $H_\Lambda$ if and only if $0$ is an eigenvalue of the $|
{\tilde \Lambda} | \times |{\tilde \Lambda} |$-matrix 
$K_\Lambda ( E_j )$.}

\vspace{.1in}
\noindent
{\bf Proof.} The spectrum of $H_\Lambda$ on the negative real
axis is discrete and finite. Let $\gamma (J_\eta) $ be a contour
about $J_\eta$ in $\C$. From the resolvent formula (170),
and the analyticity of the free resolvent on $\C \backslash \R$,
it follows that the kernel of the spectral projector for
$H_\Lambda$ and the interval $J_\eta$ is given by the contour
integral
\beq
E_\Lambda ( J_\eta ) ( x , y ) = \frac{1}{2 \pi i} \:
\Sum_{i,j \in {\tilde \Lambda }} \int_{\gamma (I_\eta) } G_0 (
x-i ; z )
K_\Lambda^{-1} (z)_{ij}
G_0 (y-j;z ) ~dz .
\eeq
The only singularities of the integrand inside of the contour
come from the negative real values at which
$K_\Lambda (z)$ has a pole. These values $-E_j$, with $E_j > 0$,
are precisely the values 
for which the matrix $K_\Lambda (-E_j)$ is not invertible. $\Box$


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\section{Scattering Theory for Singular Interactions
on Hypersurfaces}\label{S.8}

In this section, we consider point interaction models in dimensions $d = 2 , 3$
for which the random subset $\Gamma ( \omega')$
of occupied sites lies in a 
sublattice of $\Z^d$ of codimension at least one. 
This means that for $d = 3$,
the impurities are located on either a subset of some
hyperplane $\Z^2$, or of some line $\Z$, and for $d=2$, the
impurities are located on a subset of
a line $\Z$. We prove that
in any of these cases, the random family $H_{\omega , \omega '}$, that we 
sometimes write as $H_{\omega , \omega'}^{(d,j)}$ for emphasis, has
absolutely continuous spectrum on $[ 0 , \infty )$, and localized states
in some negative energy interval, with probability one.  
Although we strongly suspect that the spectrum is purely absolutely continuous
at positive energies, we do not have an analog of the Jaksic-Last 
\cite{[JaksicLast]} result for these models.  

The models we study here are defined by the Green's function formula (81),
where the sum is taken over the points $\Gamma ( \omega') \subset \Z^{d -
j}$, where, for $d =3$, we can take the codimension $j = 1, ~\mbox{or} ~2$, 
and for $d=2$, we have the codimension $j=1$.
Because the Laplacian $H_0$ is Euclidean invariant, we can assume that for
$d=3$, the set $\Gamma ( \omega ' )$ is a subset
of either the $\{x_3=0\}$-plane, for $j = 1$, or a subset of
the line $x_2=x_3 = 0$, for $j=2$.
Similarly, for $d=2$, we will assume that $\Gamma ( \omega')$ is a
subset of the line $x_2 = 0$.

The proof of the existence of absolutely continuous spectrum at positive
energies follows the general lines of \cite{[HundertmarkKirsch]}.
Because the definition of the random Hamiltonian is given by the Green's
function, it is convenient to work with certain functions of the
Hamiltonian and appeal to the invariance principle for wave operators (cf.\
\cite{[ReedSimon3]}).
Let $\phi$ be the function $\phi (x) = ( x + M_0 )^{-1}$.
We will prove the existence of the wave operators  
\beq
\label{eq:wo}
\Omega_\pm \equiv s-\lim_{t \rightarrow \pm \infty} 
e^{i t \phi ( H_{\omega , \omega ' } )} \; e^{ - i t \phi ( H_0 ) } ,
\eeq
on certain states in $L^2 ( \R^d)$, and where, as before, $H_0 = - \Delta $.
The states for which the limit exists 
are localized in momentum space so as to have outgoing
momentum localized away from
the hyperplane of codimension $j$. For example,
for $(d,j)=(3,1)$, we choose states with the third
component of momentum $p_3$ localized in a positive
set away from zero. Such states propagate away
from the $\{x_3=0\}$-plane where the
impurities are located. 
The existence of the wave operators on these states
follows from the fact that the
effective potential is localized on a submanifold of $\R^d$.
The existence of these wave operators implies that $[ 0,\infty)
\subset \sigma_{ac} ( H_{\omega, \omega'})$, with probability one. 

We prove the existence of the wave operators (\ref{eq:wo}) by considering
\beq
\label{eq:rwo}
( \Omega_\pm - 1 ) \psi  =  \lim_{t \rightarrow \pm \infty }
\; \int_0^t e^{i s \phi ( H_{\omega , \omega' }) }
\left( \phi ( H_{\omega , \omega'} )
- \phi ( H_0 ) \right) e^{ - i s \phi ( H_0) } ~\psi \; ds ,
\eeq
for certain $\psi$.
Because of our choice of $\phi$, the kernel of (\ref{eq:rwo})
can be written as
\beq
\phi ( H_{\omega , \omega' } ) - \phi ( H_0 ) =
\phi ( H_0 ) P^* K_{\omega , \omega ' }^{-1} ( - M_0 ) P \phi ( H_0 ) ,
\eeq
as follows from the resolvent formula (81).
The projection $P : H^2 ( \R^d ) \rightarrow \ell^2 ( \Gamma ( \omega '
))$, given by pointwise evaluation, is a bounded operator, and
its bound is independent of $\omega'$.
This follows from the boundedness of the trace map $T: H^2 (\R^d)
\rightarrow H^{3/2} (\R^{d-1})$, and the Sobolev embedding theorem,
so that there is a finite constant $C_{d,j} > 0$, 
depending on $(d,j)$ so that for
$u \in H^2 ( \R^d)$,
\beq
\| P u \|_{\ell^2 ( \Gamma (\omega'))} \; \leq \; C_{d,j} \|u \|_{H^2 (
\R^d)}.
\eeq
Let us note an important property of this map. 
If $f \in C_0^\infty ( \R^d)$
has the property that $\mbox{supp} \: f \cap \Gamma ( \omega ' ) =
\emptyset$, for all $\omega ' \in \Omega '$, then $P f = 0$.
We have the following result of the method of stationary phase.

\vspace{.1in}

\noindent
{\bf Proposition 8.1.} {\it
Let $x_\perp$ be a component of $x$ in a direction away
from the hyperplane containing the impurities, and let $p_\perp$
be the corresponding component of momentum.
Let $g \in C_0^\infty ( \R)$
be a cut-off function with $\mbox{supp} ~g \subset [ \alpha ,
\beta]$, for $0 < \alpha < \beta < \infty$. Let $\chi_a$ be a spatial
cut-off function that vanishes outside of a neighborhood of
the hyperplane of width $a > 0$. 
Then, for any integer
$M$, and for any $\psi \in C_0^\infty ( \R^d)$, we have
\beq
\label{eq:sp}
\| \chi_a e^{-i \phi (H_0) t} g(p_\perp ) \psi \| \; 
\leq \frac{C_M(\psi;g) }{ (1+ |t|)^M },
\eeq
for some finite constant $C_M (\psi;g) >0$. 
}

\vspace{.1in}

\noindent
{\bf Proof.}
This is an application of the method of stationary phase. By the Fourier
transform, we can write
\beq
\label{eq:stph}
\chi_a (x_\perp)( e^{-i \phi (H_0) t} g(p_\perp) \psi )(x)= \chi_a (x_\perp)
\int_{\R^d} e^{ i \Phi (k,x,t)} g (k_\perp) \hat{\psi} (k) ~d^dk,
\eeq
where $\Phi ( k , x , t ) = k \cdot x - t \phi (k^2)$ is the phase function
appearing in right side of (\ref{eq:sp}).
A simple calculation gives
\beq
(\nabla_k \Phi )(k,x,t) = x + \frac{2kt}{(k^2 + M_0 )^2 } ,
\eeq
so that the properties of the cut-off functions imply that
\beq
| ( \nabla_k \Phi )_\perp (k,x,t) \chi_a (x_\perp) g (k_\perp) | 
\; \geq \; ~\mbox{max} \left( \frac{2 \alpha t}{(\beta^2 + M_0 )^2 } , 
\frac{a }{2} \right) \chi_a (x_\perp ) g (k_\perp) .
\eeq
Consequently, we can write
\beq
e^{i \Phi } \chi_a g(k_\perp) = \frac{-i}{\partial_{k_\perp} \Phi } 
( \partial_{k_\perp} e^{i \Phi}) \chi_a g(k_\perp ) ,
\eeq
and substitute this into (\ref{eq:stph}). Integrating by parts $M$-times,
using the fact that
\beq
\partial_{k_\perp} ( \partial_{k_\perp} \Phi )^{-l}  \chi_a g(k_\perp) =
{\cal O}( |t|^{-l} ),
\eeq
we obtain the result. $\Box$

\vspace{.1in}

\noindent
{\bf Theorem 8.2.} {\it 
For any $\psi \in C_0^\infty ( \R^d)$, and cut-off function $g$ as in
Proposition 8.1, we have
\beq
\Omega^\pm g(p_\perp) \psi = \lim_{t \rightarrow \pm \infty} 
e^{i t \phi ( H_{\omega , \omega ' } )} \; e^{ - i t \phi ( H_0 ) } g(p_\perp )
\psi .
\eeq
Consequently, the interval $[0 , \infty)$ belongs to the absolutely
continuous spectrum of $H_{\omega, \omega'}$ for almost every $(\omega ,
\omega') \in \Omega \times \Omega'$. }

\vspace{.1in}

\noindent
{\bf Proof.}
We denote by $g(p_\perp)$ and by $\chi_a ( x_\perp)$ the same cut-off functions 
appearing in Proposition 8.1, where $x_\perp$ is a component
$x$ in a direction away from the hyperplane containing the impurities, and $p_\perp$
is the corresponding component of the momentum.
We let $\psi \in C_0^\infty ( \R^d)$.
As in (183)--(185) above, we study 
\beq
\label{eq:rwo2}
( \Omega_\pm - 1 ) g(p_\perp) \psi  =  \lim_{t \rightarrow \pm \infty }  
\; \int_0^t e^{i s \phi ( H_{\omega , \omega' }) } 
\left( \phi ( H_{\omega , \omega'} )
- \phi ( H_0 ) \right) e^{ - i s \phi ( H_0) } g(p_\perp) \psi \; ds .
\eeq
Because of our choice of $\phi$, the kernel of (\ref{eq:rwo2})
can be written as 
\beq
\phi ( H_{\omega , \omega' } ) - \phi ( H_0 ) =
\phi ( H_0 ) P^* K_{\omega , \omega ' }^{-1} ( - M_0 ) P \phi ( H_0 ) ,
\eeq
as follows from the resolvent formula (81).
Because the projection $P : H^2 ( \R^d ) 
\rightarrow \ell^2 ( \Gamma ( \omega ' ))$ 
vanishes for functions localized away from the hyperplane of codimension
$j \geq 1$ containing the impurities, we
have $P = P( \chi_a + (1 - \chi_a) ) = P \chi_a$.
Furthermore, it follows from Proposition 4.5 that $\| K_{\omega,
\omega'}^{-1} ( -M_0 ) \|$ is finite with probability one. Hence, we can write
\beq
\| ( \Omega_\pm - 1 ) g(p_\perp ) \psi \| \leq 
\lim_{t \rightarrow \pm \infty } \; \int_0^t ~ds ~\| \chi_a e^{ - i s \phi
( H_0) } g(p_\perp ) \psi \| ,
\eeq
and this limit is finite by Proposition 8.1. It now follows from the
intertwining property of the wave operators, and the absolutely continuous
spectrum of $\phi (H_0)$, that 
the spectrum of $\phi (H_{\omega, \omega'})$, and hence of $H_{\omega
, \omega'}$, has nontrivial absolutely continuous spectrum containing $[0 ,
\infty)$, with probability one. 
$\Box$

\vspace{.1in}

Having established the existence of absolutely continuous spectrum for the
Hamiltonians with impurities located on subsets of sublattices
of dimension $(d-j)$, we now prove the
existence of localized states at negative energies with probability one.
We first need to establish the existence of
spectrum at negative energies, with probability one, as was done
in sections 4.2--4.3 for the case $j=0$.
Let us unify the notation. For permissible pairs $(d,j)$, we write
$H^{(d,j)}_{\omega}$
for the Hamiltonian with impurities located on each point of the full 
sublattice $\Z^{d-j}$ of
$\Z^d$ of codimension $j$.
We recall that we write
$H^{(d,j)}_{\omega, \omega'}$
for the full Hamiltonian with impurities located on the random subset 
$\Gamma ( \omega')$ of
the sublattice $\Gamma ( \omega') \subset \Z^{d-j}$.
We begin by discussing the spectrum of the periodic
Hamiltonians obtained
by fixing all of the coupling constants on the points of $\Z^{d-j}$
at a constant value $\lambda \in {\mbox supp} ~h_0$.
We denote these Hamiltonians by $H^{(d,j)} ( \lambda )$.
The results on the spectra of these periodic operators are due to Grossmann,
Hoegh-Krohn, and Mebkhout \cite{[GrossmannHoeghKrohnMebkhout]}, 
and can also be found in the text \cite{[AGHKH]}.
The case $(d,j)=(3,1)$ models a monomolecular monolayer, and the case
$(d,j)=(3,2)$ provides a model of a straight polymer.
The structure of the spectrum of $H^{(d,j)} (\lambda)$ is quite
similar to the spectrum of the operator $H(\lambda)$ in the $j=0$ case.

For $(d,j) = (3,1)$, the periodic impurities lie on the plane $x_3 =0$. Let
$\Gamma_{(3,1)}$ be the
two-dimensional translation group with Brillioun zone $B^{(3,1)}_0$ spanned
by vectors $(b_1, b_2)$.
Let $k_0^{(3,1)} = - (1/2) ( b_1 + b_2 )$ be a distinguished vector in
$B^{(3,1)}_0$.
For the case $(d,j) = (3,2)$,
the periodic impurities lie on a line $x_3 = x_2 =0$, and the symmetry
group $\Gamma_{(3,2)}$ is the group of 
translations by lattice vector $a \hat{e}_1$, for some $a >0$.
Finally, for the case $(d,j) =(2,1)$, we take the periodic
impurities to lie on the line $x_2 = 0$, and the one-dimensional group of
translations $\Gamma_{(2,1)}$ with lattice vector 
$a \hat{e}_1$, for some $a > 0$.
The distinguished vector in the Brillouin zone
in this case is $k_0^{(2,1)} = - \pi / a$.

For each case $(d,j)$, there is a constant $\alpha_{(d,j)}$, analogous
to $\alpha_E (k)$ in (93), that determines if there is a band of negative
spectrum below zero.
The definition of this constant depends on $(d,j)$, and will be given
explicitly below. For all $(d,j)$, the spectrum of $H^{(d,j)} (\lambda)$ is
absolutely continuous. The spectrum of the periodic
operators has the following structure:
\beq  
\sigma ( H^{(d,j)} (\lambda)) = \left\{ \begin{array}{cc}
  
  [E_0^{(d,j)} (0; \lambda ) , \infty ), & \mbox{if} ~~1 / \lambda \geq 
\alpha_{(d, j)} \\
 
 [E_0^{(d,j)} (0; \lambda ) , E_0^{(d,j)} (k_0^{(d,j)} ; \lambda ) ] 
  \cup [ 0, \infty ), & \mbox{if} ~~1 / \lambda < \alpha_{(d,j)} 
                 \end{array} 
                \right.
\eeq 
We note that we always have $E_0^{(d,j)} ( 0; \lambda ) < 0$, for all
values of $\lambda \in \R$.
As in the three-dimensional case with $j=0$, there is a band of negative
spectrum provided the coupling constant
has a value below a certain value given by the following constants:
\bea
\alpha_{(3,1)}& = & \frac{1}{8 \pi^2} ~\lim_{\omega \rightarrow \infty} \left[
\sum_{ \stackrel{\gamma \in \Gamma_{(3,1)} }{  |\gamma + k_0^{(3,1)} | \leq \omega }}
\frac{ |B_0 |}{ |\gamma + k_0^{(3,1)}|} - 2 \pi \omega \right] , \nonumber \\ 
\alpha_{(3,2)} & = & - log 2 / 2 \pi a , \nonumber \\
\alpha_{(2,1)} & = &  \frac{1}{2 \pi} [ \Psi (1) + \log 2] \nonumber \\
  & & + \frac{1}{(2 \pi)^2} ~\lim_{\omega \rightarrow \infty} \left[ \sum_{\stackrel{ \gamma \in \Gamma_{(2,1)}}{
  | \gamma - \pi / a|}} \frac{a}{| \gamma - \pi / a|} - 2 \pi \log \omega \right],
\eea
where $\Psi$ is the digamma function.
It is important to note that $\inf \sigma ( H^{(d,j)} (\lambda ) ) =
E_0^{(d,j)} ( 0; \lambda )$ has the same
asymptotics as given in Proposition 4.2, equation (96).

We now discuss the deterministic spectrum of the random operators
$H^{(d,j)}_\omega$.
For $\gamma \in \Gamma_{(d,j)}$, let $U_\gamma$ be the corresponding
unitary operator. We have
the usual covariance property $U_\gamma H^{(d,j)}_\omega U_\gamma^{-1} =
H^{(d,j)}_{\omega_\gamma}$,
where the map $\omega \rightarrow \omega_\gamma$ acts ergodically on the
probability space by translations.
It follows from the general theory \cite{[Kr1]} that there is a subset
$\Omega_0 \subset \Omega$,
with measure one, and a closed subset $\Sigma \subset \R$, 
so that $\sigma( H^{(d ,j)}_\omega ) = \Sigma$,
for $\omega \in \Omega_0$.
Furthermore, the set $\Sigma$ can be constructed from the spectra of the
periodic operators $H^{(d,j)} ( \lambda)$, as 
\beq
\Sigma = \overline{\bigcup_{\lambda \in {\mbox supp}~h_0 } ~\sigma ( H^{(d,j)} (
\lambda ) ) }.
\eeq
As a consequence, we have a result analogous to (105) of Theorem 4.3 for
$\Sigma$. In particular, there is a negative
energy component with $\inf \Sigma = E_0^{(d,j)} (0 ; -a) < 0$, if $\mbox{supp}
~h_0 = [-b, -a ]$, for some $0 < a < b$. 
It also follows from the comment above that $E_0^{(d,j)}
( 0 ; -a ) = {\cal O} ( 1/a^2)$.

It remains to note that these results are valid for the family of random operators
$H_{\omega, \omega'}^{(d,j)}$ with impurities located on the subset $\Gamma ( \omega')$
of the sublattice $\Z^{d-j}$. The probability space is the product space $\Omega \times
\Omega'$, with the product measure. As in the discussion of the family
$H^{(d,j)}_\omega$, the translations group $\Gamma_{(d,j)}$ acts ergodically on this
probability space and, using the same notation as above, we have $U_\gamma H^{(d,j)}_{\omega, \omega'}
U_\gamma^{-1} = H^{(d,j)}_{\omega_\gamma, {\omega'}_\gamma }$, for $\gamma \in \Gamma_{(d,j)}$.
It follows that this family of operators has a deterministic spectrum $\Sigma$.
We now determine the nature of this set $\Sigma$ 
following the paper of Albeverio, et.\ al. 
\cite{[AlbeverioHoeghKrohnKirschMartinelli]}. 
As in \cite{[AlbeverioHoeghKrohnKirschMartinelli]}, 
we identify the random singular potential 
with the collection of pairs 
$\{ ( X_k (\omega'), \lambda_k ( \omega) ) \; | \;
X_k (\omega') \in \{ 0 , 1 \},  \mbox{and} ~\lambda_k ( \omega ) 
\in \mbox{supp} ~h_0, ~\mbox{with} ~k \in \Z^{d-j} \}$. 
We call a deterministic potential $V(x) = \sum_{k \in \Z^{d-j} : X_k = 1 } 
\lambda_k \delta ( x - k )$ {\it admissible} if the collection of
pairs satisfies $\{ ( X_k , \lambda_k ) \; | k \in \Z^{d-j},
~\mbox{with} ~X_k \in \{0,1 \} , ~\mbox{and}
~\lambda_k \in \mbox{supp} ~h_0 \}$. If an admissible potential
is periodic under some $\Z^{d-j}$-translation subgroup, then
the potential is called {\it periodic}, in analogy with the discussion above.
This means that the corresponding set $\{ ( k , \lambda_k ) \}$
is periodic under some subgroup of $\Z^{d-j}$. 
We will denote these periodic Hamiltonians by
${\tilde H}_P^{(d,j)}$, and note that 
they correspond to $H_{\omega,\omega'}^{(d,j)}$
for certain choices of $(\omega, \omega')$. 
We let ${\cal P}$ denote all such periodic potentials.
The proof that the deterministic spectrum for $H_{\omega , \omega'}^{(d,j)}$
is the same as that for $H_\omega^{(d,j)}$ relies on two propositions
as in \cite{[AlbeverioHoeghKrohnKirschMartinelli]}. 

\vspace{.1in}

\noindent
{\bf Proposition 8.3.} {\it Let $\Sigma$ be the almost sure spectrum of
the family $H^{(d,j)}_{\omega , \omega'}$, and let $H^{(d,j)}_P$, 
with $P \in {\cal P}$, be a Hamiltonian
with a periodic potential as described above. Then, we have
\begin{enumerate}
\item $\sigma ( H^{(d,j)}_{P} ) \subset \Sigma$;
\item The almost sure spectrum $\Sigma$ is given by
\[ 
\Sigma = \overline{ \bigcup_{P \in {\cal P}} \sigma ( H^{(d,j)}_P )} .
\]
\end{enumerate}
}

\noindent
The proof of Proposition 8.3
depends on the following version of the key Proposition 2 
of \cite{[AlbeverioHoeghKrohnKirschMartinelli]}.

\vspace{.1in}
\noindent
{\bf Proposition 8.4.} {\it Let $X \subset \Z^{d-j}$ be an arbitrary set. Recall that
$H^{(d,j)} ( \lambda)$ is the periodic operator with impurities on the 
hyperplane $\Z^{d-j}$, and all coupling constants 
equal to $\lambda \in ~\mbox{supp} ~h_0$. 
Let $H_X^{(d,j)} (\lambda)$ be the operator 
obtained by setting the coefficient $\lambda = 0$
for those sites in $X$. 
We then have $\sigma ( H_X^{(d,j)} (\lambda) ) \subset \sigma ( H^{(d,j)} (\lambda))$.}

\vspace{.1in}

The proof of the main assertion follows by proving that each periodic potential
has the required spectral properties. 
As a consequence, we have the following theorem on localized states at negative energy.

\vspace{.1in}
\noindent
{\bf Theorem 8.5.} {\it Let $H_{\omega , \omega'}^{d,j}$ be the random Hamiltonian with impurities
located on a subset of the lattice $\Z^{d-j}$. Then, under Hypotheses 4.1 and 4.2, there exists
an energy $E_0 > 0$ so that $\Sigma \cap ( - \infty , - E_0 ] \neq \emptyset$
and is pure point
almost surely with exponentially decaying eigenfunctions. The model
exhibits dynamical localization in the interval $( - \infty , - E_0 ]$.}

\vspace{.1in}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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