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random operators, Wegner estimate, density of states, localization, Schroedinger operators
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\begin{document}


\begin{titlepage}

\begin{center}
  

  {\bf THE INTEGRATED DENSITY OF STATES FOR SOME RANDOM OPERATORS \\
    WITH NONSIGN DEFINITE POTENTIALS}


\vspace{0.3 cm}

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{\bf Peter D.\ Hislop \footnote{Supported in part by NSF grant
DMS-9707049.}} 

\vspace{0.3 cm}

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

\vspace{0.3 cm}

{\bf Fr{\'e}d{\'e}ric Klopp}

\vspace{0.3 cm}

{\ten L.A.G.A, Institut Galil{\'e}e\\
Universit{\'e} Paris-Nord \\
F-93430 Villetaneuse, FRANCE}

\end{center}

\vspace{0.3 cm}

\begin{center}
{\it Dedicated to Jean-Michel Combes on the Occasion of his $60^{th}$
Birthday}
\end{center}

\vspace{0.3 cm}

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

\noindent
We study the integrated density of states of random Anderson-type additive and
multiplicative perturbations of deterministic background operators for
which the single-site potential does not have a fixed sign. Our main
result states that, under a suitable assumption on the regularity of
the random variables, the integrated density of states of such random operators
is locally H{\"o}lder continuous at energies below the bottom of the
essential spectrum of the background operator for any nonzero
disorder, and at energies in the unperturbed spectral gaps, provided
the randomness is sufficiently small. The result is based on a proof
of a Wegner estimate with the correct volume dependence. The
proof relies upon the
$L^p$-theory of the spectral shift function for $p \geq 1$
\cite{[CHN]}, and the vector field methods of \cite{[Klopp]}. 
We discuss the application of this result to \Schr\ operators with
random magnetic fields and to band-edge localization.
 
\vspace{0.5 cm}

\noindent
\today

\end{titlepage}



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

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

In this paper, we combine the results of \cite{[CHN]} and of
\cite{[Klopp]} to prove that, in certain energy regions, the integrated density
of states (IDS) for additive and multiplicative perturbations of
background operators by 
Anderson-type random potentials constructed with nonsign definite
single-site potentials, is locally H{\"o}lder continuous.
To our knowledge, this is the first, general result in dimensions $d \geq 1$
for such random potentials. For one-dimensional Anderson models, 
Stolz \cite{[Stolz]} proved
localization at all energies with no sign restriction on the single-site
potential. He does not, however, obtain a result on the IDS.
Recently, Veselic \cite{[Veselic]} obtained results similar to ours
for additive perturbations
with a special class of nonsign definite potentials included in the
class treated here. 

Our result is based on a {\it Wegner estimate} with the correct
volume behavior. A Wegner estimate is an upper bound on the
probability that the spectrum of the local Hamiltonian $H_{\Lambda}$
(i.e.\ $H$ restricted to a box $\Lambda$ with self-adjoint boundary conditions)
lies within an
$\eta$-neighborhood of a given energy $E$. A good Wegner estimate is
one for which the upper bound depends linearly on the volume $|
\Lambda|$, and vanishes as the size of the energy neighborhood $\eta$
shrinks to zero.  The linear dependence on the volume is essential for
the proof of the regularity of the IDS.
The rate of vanishing of the upper bound as $\eta \rightarrow 0$
determines the continuity of the IDS.

The models that can be treated by this method are described as
follows. We can treat both multiplicative (M) and additive (A)
perturbations of a background self-adjoint operator $H_0^X$, for $X=M$
or $X=A$.  Multiplicatively perturbed operators describe the
propagation of acoustic and electromagnetic waves in disordered media,
and we refer to \cite{[CHT]} for a discussion of their physical
interpretation.  Additive perturbations describe quantum propagation
in disordered media.  For the Wegner estimate, we are interested in
perturbations $V_\Lambda$ of a background operator $H_0^X$, that are
local with respect to a bounded region $\Lambda \subset \R^d$.
Multiplicatively perturbed operators $H_\Lambda^M$ are of the form
\beq H_\Lambda^M = A_\Lambda^{-1/2} H_0^M A_\Lambda^{-1/2}, \eeq where
$A_\Lambda = 1 + V_\Lambda$. We assume that $(1 + V_\Lambda)$ is
invertible (cf.\ \cite{[CHT]} for a discussion of this condition).
Additively perturbed operators $H_\Lambda^A$ are of the form \beq
H_\Lambda^A = H_0^A + V_\Lambda .  \eeq

The unperturbed, background medium in the  multiplicative case is
described by a divergence form operator
\beq
H_0^M = - C_0 \rho_0^{1/2} \nabla \cdot \rho_0^{-1} \nabla \rho_0^{-1/2}
C_0  , 
\eeq
where $\rho_0$ and $C_0$ are positive functions that describe the
unperturbed density and sound velocity. We assume that $\rho_0$ and
$C_0$ are sufficiently regular so that $C_0^\infty ( \R^d)$ is an
operator core for $H_0^M$. 
The unperturbed, background medium in the additive case is described by
a Schr{\"o}dinger operator $H_0$ given by
\beq
H_0^A = ( -i \nabla - A )^2 + W ,
\eeq
where $A$ is a vector potential with $A \in L^2_{loc} ( \R^d )$,
and $W = W_+ - W_-$ is a background potential
with $W_- \in K_d ( \R^d )$ and $W_+ \in K_d^{loc} ( \R^d)$.

The perturbations $V_\Lambda$ that can be treated by the method of Klopp
\cite{[Klopp]} are Anderson-type random potentials.
Let ${\tilde \Lambda}$ denote the lattice points in 
the region $\Lambda$, so that
${\tilde \Lambda} \equiv \Lambda \cap \Z^d$. 
The local perturbation in the Anderson-type model is defined by:
\beq
V_\Lambda (x) = \sum_{i \in {\tilde \Lambda} } \lambda_i ( \omega ) u_i (x
-i - \xi_i ( \omega' ) ),
\eeq
provided the random variables $\xi_i ( \omega' )$, 
modeling thermal vibrations, are uniformly bounded in $i \in \Z^d$.
The functions $u_i$ are compactly supported in a neighborhood of the
origin. They need not be of the form $u_i ( x) = u ( x )$, for
some fixed $u$, since ergodicity plays no role in the Wegner
estimate. However, the proof of the existence and deterministic nature of the
IDS requires ergodicity, so we will make the assumption that 
$u_j (x) = u ( x)$ for those results.   

We will put conditions of the random variables $\lambda_i ( \omega
)$ and the single-site potentials $u_i$. We note that the Wegner
estimate is a local estimate.

\begin{enumerate}
\item[(H1a)]
The self-adjoint operator $H_0^X$ is essentially
self-adjoint on $C_0^{\infty} ( \R^d )$, for $X=A$ and for
$X=M$. The operator $H_0^X$ is semi-bounded
and has an open spectral gap.
That is, there exist constants $- \infty < M_0 \leq C_0 \leq B_- < B_+
< C_1 \leq \infty $ so that $\sigma (H_0) \subset [ M_0 , \infty )$, and 
$$
\sigma( H_0 ) \cap ( C_0 , C_1 )  = ( C_0 , B_- ] \cup [ B_+ , C_1 ).
$$

\item[(H1b)] The self-adjoint operator $H_0^X$ is essentially
  self-adjoint on $C_0^{\infty} ( \R^d )$, and is semi-bounded.

\item[(H2)] The operator $H_0^X$ is locally compact in the sense that
  for any $f \in L^{\infty} ( \R^d )$ with compact support, the
  operator $f( H_0^X - M_1 )^{-1}$ is compact for any $ M_1 < M_0 $.

\item[(H3)] The single-site potential $u_k \in C_0 ( \R^d )$.  For
  each $k \in \Z^d$, there exists a nonempty open set $B_k$ containing
  $k$ so that the single-site potential $u_k \neq 0$ on $B_k$.
  Furthermore, we assume that 
\beq \Sum_{j \in \Z^d} \left\{
    \Int_{\Lambda_1 ( 0 ) } \; | u_j ( x -j ) |^p \right\}^{1/p} < \infty  ,
\end{equation}
for $p \geq d$ when $d \geq 2$ and $p=2$ when $d=1$.

\item[(H4)] The conditional probability distribution of $\lambda_0$,
  conditioned on ${\lambda_0}^{\perp} \equiv \{ \lambda_i \; | \; i
  \neq 0 \}$, is absolutely continuous with respect to Lebesgue
  measure. The density $h_0$ has compact support contained, say, in
  $[m , M ]$, for some constants $( m, M)$ with $- \infty < m < M <
  \infty$. The density $h_0$ is assumed to be locally absolutely
  continuous.

\end{enumerate}

We refer to the review article of Kirsch \cite{[Kr1]} for a proof of
the fact that hypotheses (H3)--(H4) imply the essential
self-adjointness of $H_{\omega}^A$ on $C_0^{\infty} ( \R^d )$ (see
\cite{[CHT]} for the $X=M$ case).
We note that condition (1.6) in (H3) is unnecessary 
if $u_j ( x ) = u ( x )$, 
for some $u \in C_0 ( \R^d )$. Let us also note
that we can take $m = 0$ in (H4) without any loss of
generality.
To see this, let us define a modified background operator for $X=A$ by
${\tilde H}_0 \equiv H_0 + \sum_j  m u_j $, and a modified random potential
by ${\tilde V}_\omega  \equiv \sum_j ( \lambda_j ( \omega ) - m ) u_j $, so
that $H_0 + V_\omega = {\tilde H}_0 + {\tilde V}_\omega$. If we
define random variables ${\tilde \lambda}_j ( \omega ) \equiv 
( \lambda_j ( \omega ) - m )$, then these new random variables 
are distributed with a density
${\tilde h}_0 ( \lambda ) = h_0 ( \lambda + m) $, having
support $[0 , M - m]$.
For the case $X=M$, we can write the prefactor in (1.1) as
$ ( 1 + \sum_j \lambda_j u_j ) = ( 1 + \sum_j m u_j )( 1 + ( 1 +
\sum_j m u_j )^{-1} {\tilde V}_\omega)$, with ${\tilde V}_\omega$ defined above,
and absorb the deterministic factor $( 1 + \sum_j m u_j )$ into the velocity
function $C_0$ in (1.3). 
{\it Therefore, without loss of generality, 
we will assume 
that the random variables are independent, and
identically distributed (iid), with a common density $h_0$
supported on $[0,M]$, for some $0 < M < \infty$.} 
We remark that the results hold in the correlated case
(cf.\ \cite{[CHM1]}).

We remark that a semibounded operator $H_0^X$ always has at least one
open gap $(-\infty,\Sigma_0^X)$, where $\Sigma_0^X \equiv \mbox{inf} \:
\sigma(H_0^X)$. We also note that we always have $\Sigma_0^M = 0$.

The existence of the integrated density of states for additively perturbed, 
infinite-volume ergodic models like (1.2) is well-known. 
A textbook account is found in the lecture notes
of Kirsch \cite{[Kr1]}. 
The same proof applies to the ergodic, multiplicatively perturbed model
(1.1), with minor modifications.
Recently, Nakamura \cite{[Nakamura]} showed the uniqueness of the IDS, 
in the sense that the IDS is independent of the Dirichlet or 
Neumann boundary conditions needed
to define the local operators, in the case of Schr{\"o}dinger operators with
magnetic fields. 
The same proof applies to the multiplicatively perturbed models.
It is interesting to note that the proof uses the 
$L^1$-theory of the spectral shift
function. 

We mention the recent papers of Kostrykin and Schrader
\cite{[KS1],[KS2],[KS3]} in which they define and construct
a spectral shift density (SSD) for random Anderson-type
Schr{\"o}dinger operators with
sign definite single-site
potentials. Among other applications, the SSD gives an alternate
proof of the existence of the IDS for sign definite models.
 
\subsection{Below the Infimum of the Spectrum of $H_0^A$}
\label{sec:below-infim-spectr}

The main result under these hypotheses on the unperturbed operator
$H_0^A$ and the local perturbation $V_\Lambda$, is the following
theorem. We recall that for multiplicative 
perturbations, we have $\Sigma_0^M = \mbox{inf} \; \Sigma^M = 0$, 
where $\Sigma^X 
\equiv \sigma (H_\omega^X )$ almost surely, so these 
results apply only to additive perturbations. 

\vspace{.1in}

\noindent
{\bf Theorem 1.1}. {\em Assume (H1b), (H2), (H3), and (H4).  For any
  $q > 1$, and for any $E_0 \in (-\infty,\Sigma_0^A)$, there exists
  a finite, positive constant $C_{E_0}$, depending only on
  $[\dist(\sigma(H_\Lambda^A), E_0)]^{-1}$, the dimension $d$, and
  $q >1$, so that for any $\eta <  \dist (\sigma(H_0^A), E_0 )$, we have
  \beq \proba\left\{ \dist ( E_0 ,
    \sigma(H_\Lambda^A) ) \leq \eta \right\} \leq C_{E_0} \eta^{1/q} \,
  |\Lambda|\ .  \eeq }

There are several prior results on the Wegner estimate for
multidimensional, continuous Schr{\"o}dinger operators with
Anderson-type random potentials provided the single-site
potential $u$ is sign definite. Kotani and
Simon \cite{[KS]} proved a Wegner estimate with a $| \Lambda
|$-dependence for Anderson models with overlapping single-site
potentials. This condition was removed and extensions were made to
the band-edge case in \cite{[CH1]} and \cite{[BCH]}.
An extension to multiplicative perturbations was made in
\cite{[CHT]}. These methods require a spectral averaging theorem.
Wegner's original proof \cite{[Wegner]} for Anderson models did not 
require spectral averaging. Following Wegner's argument,
Kirsch gave a nice, short proof of the Wegner estimate in
\cite{[Kr2]}, but obtained a $| \Lambda |^2$-dependence.
Recently, Stollmann \cite{[Stollmann]} presented a short, elementary
proof of the Wegner estimate for Anderson-type models with singular
single-site probability distributions, such as H{\"o}lder continuous
measures. He also obtains a $| \Lambda |^2$-dependence.
These proofs, and the proof in this paper, do not require spectral
averaging.
{\it It is not clear, however, how to extend the methods of this paper
  in order to prove a Wegner estimate for singular distributions with
  the correct volume dependence.  \/}

As an immediate corollary of Theorem 1.1, and of the definition of the
density of states, we obtain

\vspace{.1in} 
\noindent
{\bf Corollary 1.1}. {\em 
Assume (H1b)--(H4), and that the model $H_\omega^A$ is ergodic.
The integrated density of states is locally
H{\"o}lder continuous of order $1/q$, for any $q > 1$,
on the interval $(-\infty,\Sigma_0^A )$.
} 

\vspace{.1in}

\subsection{The Case of a General Band Edge and Small Disorder}
\label{sec:gen}

There may also be other open gaps in the spectrum of $H_0^X$. 
To study the regularity of the
density of states in these gaps, we prove a Wegner estimate for
energies in an unperturbed spectral gap under the additional condition
that the disorder is small relative to the size of the gap $G
\equiv (B_+ - B_-)$. In particular, we obtain a good Wegner estimate for 
additive perturbations
$H_\Lambda^A ( \lambda ) = H_0^A + \lambda V_\omega$, 
and for multiplicative
perturbations $H_\Lambda^M ( \lambda ) = 
( 1 + \lambda V_\Lambda )^{-1/2} H_0^M
( 1 + \lambda V_\Lambda )^{-1/2}$, provided
the random potential $V_\omega$
is bounded, and for all $|\lambda|$ sufficiently small.

\vspace{.1in}
\noindent
{\bf Theorem 1.2}. {\it We assume that $H_0^X$ and $V_\omega$ satisfy
(H1a), (H2)--(H4), and 
let $H_\Lambda^A ( \lambda ) \equiv H_0^A + \lambda
  V_\Lambda$, and $H_\Lambda^M ( \lambda ) =
  ( 1 + \lambda V_\Lambda )^{-1/2} H_0^M
  ( 1 + \lambda V_\Lambda )^{-1/2}$.
Let $E_0 \in (B_- , B_+)$ be any energy in the 
unperturbed spectral gap of $H_0$,
and define $\delta_\pm ( E_0 ) \equiv \mbox{dist} \; ( E_0 , B_\pm )$.
We define a constant
$$
\lambda (E_0) \equiv \mbox{min} \; \left( 
  \frac{ (B_+ - B_- ) }{4 \|V_\Lambda \|},
  \frac{1}{4 \|V_\Lambda \|} \left( \frac{ \delta_+ (E_0) \delta_- (E_0) }{2}
  \right)^{1/2} \right) .
$$
 Then, for any $q > 1$, there
  exists a finite constant $C_{E_0}$, depending on $\lambda_0$, the
  dimension $d$, the index $q > 1$, 
  and $[ \mbox{dist} ( \sigma (H_0 )
  , I)]^{-1}$, so that for all $| \lambda | < \lambda (E_0)$, 
  and for all $\eta < \mbox{min} \; ( \delta_- (E_0) , \delta_+ (E_0) ) / 32 $, we have
  \beq 
  \proba\{
  \: \mbox{dist} ( \sigma ( H_\Lambda^X (\lambda) ) , E_0 ) \leq \eta \} 
  \leq C_{E_0} \eta^{1/q} | \Lambda | .  
  \eeq }

With reference to the constant $\lambda (E_0)$, let us note that
the band-edges of the almost sure spectrum of $H_\omega ( \lambda)$ scale
linearly in $\lambda$ (cf.\ \cite{[BCH],[KSS]}), 
and hence are of ${\cal O}( \sqrt{ \delta_\pm (E_0) } )$
from the edges $B_\pm $. Hence, for small coupling constant $|
\lambda| $, our results are valid for energies in bands of size
$[ B_- + C \lambda^2, B_- + C |\lambda| ] \; \cup \; [ B_+ - C
|\lambda |, B_+ - C \lambda^2 ]$. Consequently, the deterministic spectrum
is nonempty in the regions considered. 
As an immediate corollary of Theorem 1.2, and of the definition of the
density of state, we obtain

\vspace{.1in}

\noindent
{\bf Corollary 1.2}. {\em We assume that $H_0^X$ and $V_\omega$
  satisfy (H1a), (H2)--(H4), and let $H_\omega^A ( \lambda ) \equiv H_0^A +
  \lambda V_\omega$, and $H_\omega^M ( \lambda ) =
  ( 1 + \lambda V_\omega )^{-1/2} H_0^M
  ( 1 + \lambda V_\omega )^{-1/2}$. We assume that the models
  are ergodic.  For any proper, open interval $I \subset ( B_-
  , B_+ )$ in the resolvent set of $H_0$, we define $\delta_\pm ( I)
  \equiv \mbox{dist} \; ( I , B_\pm )$. In analogy to Theorem 1.2,
  we define a constant
  $$
  \lambda_0 (I) \equiv \mbox{min} \; \left( \frac{ (B_+ - B_- ) }{4 
   \|V_\Lambda \| } , \frac{1}{4 \|V_\Lambda\|} \left( \frac{ \delta_+ (I) 
    \delta_- (I) }{2} \right)^{1/2} \right) , 
  $$
  so that, for
  $|\lambda|\leq \lambda_0 (I)$, and for any $q >1$,
  the integrated density of states for
   $H_\omega ( \lambda)$ is
  locally H{\"o}lder continuous of order $1/q$, on the interval $I$.  }

We note that we could equally phrase Theorem 1.1 as follows. For any
interval $I \subset ( B_- , B_+)$, there exists a $\lambda_0 (I) > 0$ so that
for any $| \lambda | < \lambda_0 (I)$, the models $H_\Lambda^X (\lambda)$
satisfy a Wegner estimate as in (1.8) with $\eta^{1/q}$ replaced by $|I|^{1/q}$.
We also mention that the constants $\lambda (E_0)$ and $\lambda_0 (I)$ are
not optimal.  

\vspace{.1in}

\subsection{Contents of the Paper}

The contents of this article are as follows. The $L^p$-theory of the
spectral shift function for $p>1$ is reviewed in section 2.
We prove Wegner's estimate, Theorem 1.1, for energies below $\mbox{inf} \; 
\sigma ( H_0^A)$ in section 3 along the ideas of the original argument 
as appearing in \cite{[CHN]}, and incorporating
the work of \cite{[Klopp]}. An application of the theory developed in
section 2 to the single-site spectral shift function
allows us to obtain the correct volume dependence.
In section 4, we prove Theorem 1.2 by adopting the methods of
section 3 using the Feshbach projection method.
Some simple proofs of the trace   
class estimates used in the proof of Wegner's estimate in
sections 3 and 4 are
presented in section 5.
Finally, in section 6, we discuss the application of these ideas to prove the
local H{\"o}lder continuity of the IDS for a family of Schr{\"o}dinger operators
with random magnetic fields, and to prove band-edge localization.

\vspace{.1in}  

\noindent
{\bf Acknowledgements.} We thank Ph.\ Briet, J.\ M.\ Combes,
W.\ Kirsch, A.\ Klein, E.\ Kostrykin,
S.\ Nakamura, R.\ Schrader, K.\ Sinha, 
P.\ Stollmann, and G.\ Stolz for useful discussions. 


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\section{The $L^p$-Theory of the Spectral Shift Function, $p \geq
1$}\label{S.2}

The $L^p$-theory of the spectral shift function for $p \geq 1$ was
developed in \cite{[CHN]}. We briefly recall the essential aspects here.
The $L^1$-theory can be found in the review paper of
Birman and Yafaev \cite{[BY]}, and the book of Yafaev \cite{[Yafaev]}.
Suppose that $H_0$ and $H$ are two
self-adjoint operators on a Hilbert space $\cal H$ having the property
that $V \equiv H - H_0$ is in the trace class. Under these conditions,
we can define the Krein spectral shift function (SSF) $\xi ( \lambda; H ,
H_0 )$ through the
perturbation determinant. Let $R_0 ( z ) = ( H_0 - z)^{-1}$, for $Im \;
z \neq 0$. We then have
\beq
\xi ( \lambda; H , H_0 ) \equiv \frac{1}{\pi} \lim_{ \epsilon \rightarrow
0^+
} \; \mbox{arg} \; \mbox{det} \; ( 1 + V R_0 ( \lambda + i \epsilon )).
\eeq
It is well-known that
\beq
\int_{\R}  \xi ( \lambda ; H , H_0 ) \; d \lambda = Tr V ,
\eeq
and that the SSF satisfies the $L^1$-estimate:
\beq
\| \xi ( \cdot \; ; H , H_0 ) \|_{L^{1}} \; \leq \;  \| V \|_1 .
\eeq
   
Let $A$ be a compact operator on $\cal H$ and let $\mu _j
(A)$ denote the $j^{th}$ singular value of $A$. We say that
$A \in {\cal I}_{1/p}$, for some $p > 0$, if
\beq
\Sum_{j} \; \mu _j (A)^{1/p} < \infty .
\eeq
For $p > 1$, this
means that the singular values of $A$
converge very rapidly to zero. We define a nonnegative functional
on the ideal ${\cal I}_{1/p}$ by
\beq
\| A \|_{1/p} \equiv \left( \Sum_{j} \; \mu _j (A)^{1/p} \right)^p.
\eeq
For $p > 1$, this functional is not a norm but satisfies
\beq
\| A + B \|_{1/p}^{1/p} \; \leq \| A \|_{1/p}^{1/p} + \| B \|_{1/p}^{1/p}.
\eeq
If we define a metric $\rho_{1/p} ( A , B ) \equiv \| A - B
\|_{1/p}^{1/p}$ on
${\cal I}_{1/p}$, then the linear space ${\cal I}_{1/p}$ is a complete,
separable linear metric space.
The finite rank operators are dense in ${\cal I}_{1/p}$
(cf.\ \cite{[BS]}).

Since ${\cal I}_{1/p} \subset {\cal
I}_1$, for all $p \geq 1$, we refer to $A \in {\cal I}_{1/p}$ as being
super-trace class. Consequently, we can define the SSF for a pair of
self-adjoint operators $H_0$ and
$H$ for which $V = H - H_0 \in {\cal I}_{1/p}$. The  main theorem is the
following and we refer to \cite{[CHN]} for the proof.
D.\ Hundertmark and B.\ Simon have recently proved an optimal
$L^p$-bound on the spectral shift function \cite{[HundertmarkSimon]}.

\vspace{.1in}

\noindent
{\bf Theorem 2.1.} {\it Suppose that $H_0$ and $H$ are self-adjoint
operators so that $V = H - H_0 \in {\cal I}_{1/p}$, for some $p \geq
1$. Then, the SSF $\xi ( \lambda; H , H_0 ) \in L^p ( \R )$, and
satisfies the bound
\beq
\| \xi ( \cdot \; ; H , H_0 ) \|_{L^p } \; \leq \; \| V \|_{1/p}^{1/p} .
\eeq
}  

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\section{A Proof of Wegner's Estimate for Energies below 
$\mbox{inf} \; \sigma ( H^A_0 )$ }\label{S.3} 

We give the proof of Wegner's estimate for single-site potentials with
nondefinite sign at energies below $\mbox{inf} \; \sigma ( H^A_0 )$. 
The proof is simpler 
than previous ones as it does not require
spectral averaging, nor does it require the eigenfunction 
localization result of Kirsch,
Stollmann, and Stolz \cite{[KSS]}. 
Following \cite{[Klopp]}, we formulate the Wegner estimate in terms of the
resolvent of $H_\Lambda^A$. If $E_0 < \mbox{inf} \; \sigma ( H^A_0 )$, we have
that $( H^A_0 - E_0 ) > 0$. Consequently, we can write the
resolvent of $H^A_\Lambda$, at an energy $E_0$ in the resolvent set
of $H^A_\Lambda$, as
\beq
R_\Lambda ( E_0 ) = ( H^A_\Lambda - E_0)^{-1} = (H_0^A - E_0)^{-1/2}
    ( 1 + \Gamma_\Lambda ( E_0 ; \omega ) )^{-1} (H_0^A - E_0)^{-1/2} .
\eeq
The operator $\Gamma_\Lambda ( E_0 ; \omega )$ is defined by
\bea
\Gamma_\Lambda ( E_0 ; \omega )& = &  (H^A_0 - E_0)^{-1/2}
V_\Lambda (H^A_0 -
E_0)^{-1/2} \nonumber \\
 & = &  \Sum_{ j \in {\tilde \Lambda}} \lambda_j ( \omega ) (H^A_0 -
  E_0)^{-1/2} u_j (H^A_0 - E_0)^{-1/2} .
\eea
Since $\mbox{supp} \; u_j$ is compact and the sum over $j \in {\tilde
\Lambda}$ is finite, the operator $\Gamma ( E_0 
;\omega_\Lambda )$ is compact, self-adjoint, and uniformly bounded. 
Let us write $\delta$ for $\mbox{dist} ( E_0 , \mbox{inf} \:
\sigma ( H_0^A ) )$.
It follows from (3.1) that
\beq
\| R_\Lambda ( E_0 ) \| \; \leq \; \{ \mbox{dist} \; ( \sigma
(H^A_0), E_0 )
\}^{-1} \; \| ( 1 + \Gamma_\Lambda ( E_0 ; \omega ) )^{-1}  \| 
\leq \delta^{-1} \| ( 1 + \Gamma_\Lambda ( E_0 ; \omega ) )^{-1} \|.
\eeq
It follows from (3.3) that 
\beq
\proba\{ \| R_\Lambda ( E_0 ) \| \leq 1 / \eta \} \geq 
\proba\{ \| ( 1 + \Gamma_\Lambda ( E_0 ; \omega ) )^{-1} \|  \leq \delta / \eta \} .
\eeq   
Consequently, Wegner's estimate can be reformulated as 
\bea
\proba\{ \mbox{dist} \; ( \sigma ( H^A_{\Lambda} ) , E_0 ) < \eta \} 
    & = & \proba\{ \; \| R_\Lambda ( E_0 ) \| > 1 / \eta  \} \nonumber \\
    & \leq & \proba\{ \; \| ( 1 + \Gamma_\Lambda  ( E_0 ; \omega ) )^{-1}  \| \;
          > \; \delta / \eta  \}     \nonumber \\
 & = & \proba\{ \; \mbox{dist} \; ( \sigma ( \Gamma_\Lambda  ( E_0 ; \omega ) )
        , -1 ) < \;  \eta / \delta \} .   
\eea
Hence, it suffices to compute
\beq
\proba\{ \; \mbox{dist} \; ( \sigma ( \Gamma_\Lambda  ( E_0 ;\omega )) , -1  )
 <  \; \eta / \delta \} .
\eeq

The key observation of \cite{[Klopp]} that takes the place of monotonicity
and the eigenfunction localization theorem of
Kirsch, Stollmann, and Stolz \cite{[KSS]}
is the following. We define a vector field $A_\Lambda$  on 
$L^2 ( [ m , M ]^{ \tilde \Lambda } , \Prod_{ j \in {\tilde \Lambda } } 
h_0 ( \lambda_j ) d \lambda_j )$ by
\beq
A_\Lambda \equiv \Sum_{ j \in {\tilde \Lambda }} \; \lambda_j ( \omega )
\frac{ \partial}{ \partial \lambda_j ( \omega ) } .
\eeq
Then, the operator $\Gamma_\Lambda  ( E_0 ;\omega )$ is an eigenvector of
$A_\Lambda$ in that
\beq
A_\Lambda \Gamma_\Lambda ( E_0 ;\omega ) = \Gamma_\Lambda ( E_0 ;\omega ) .
\eeq
It is this relationship that replaces the positivity used in \cite{[CHN]} since,
if $\Gamma_\Lambda  ( E_0 ; \omega )$ is restricted to the spectral subspace
where the operator is smaller than $( - 1 + 3 \kappa / 2)$, we have that
$-\Gamma_\Lambda  ( E_0 ; \omega )$ is strictly positive, and hence
invertible. We will use this below.

The outline of the proof follows Wegner's original argument \cite{[Wegner]}.
We follow the presentation in \cite{[CHN]}.
We work with the compact, self-adjoint operator 
$\Gamma_\Lambda ( E_0 ; \omega)$,
as follows from (3.6). 
As in \cite{[CHN]},
the key estimate on the number of eigenvalues created by
the variation of one random variable is 
obtained by first expressing the quantity in terms of a
spectral shift function corresponding to a perturbation by a single-site
potential, and then by estimating the $L^p$-norm of this
spectral shift function, for $p > 1$.
The proof below uses some of the modifications of Wegner's proof 
\cite{[Wegner]} introduced by Kirsch \cite{[Kr2]}.
We note that this proof of the Wegner estimate does not require 
spectral averaging \cite{[CHM2]}. 

\vspace{.1in}

\noindent
{\bf Proof of Theorem 1.1.}

\vspace{.1in}

\noindent
1. It follows from the reduction given above that we need to estimate
\beq
\proba\{ \; \mbox{dist} \: ( \sigma ( \Gamma_\Lambda ( E_0 ; \omega )) , -1 ) <
 \eta / \delta  \} ,
\eeq
where $\delta = \mbox{dist} \; ( E_0 , \mbox{inf} \; \sigma ( H^A_0 ) )$.
Let $G = ( - \infty , \mbox{inf} \: \sigma ( H^A_0 ) )$ be the unperturbed
spectral gap. Since the local potential $V_{\Lambda}$ is a relatively compact
perturbation of $H^A_0$, the operator $\Gamma_\Lambda ( E_0 ; \omega )$ has
only discrete spectrum with zero the only possible accumulation point.
Let us write $\kappa \equiv  \eta / \delta $. 
We choose $\eta > 0$ small enough so that
$[ E_0 -  \eta  , E_0 +  \eta  ] \subset G$, and so that $[ -1 - 2 \kappa , -1 
+ 2 \kappa  ] \subset \R^{-}$. 
We denote by $I_\kappa$ the interval $[ -1 - \kappa  , -1 + \kappa ]$.
The probability in (3.9) is 
expressible in term of the finite-rank spectral projector
for the interval $I_\kappa$ and $\Gamma_\Lambda ( E_0 ;\omega )$, 
which we write as
$E_{\Lambda} ( I_\kappa )$. Like $\Gamma_\Lambda ( E_0 ; \omega )$,
this projection is a random variable, 
but we will suppress any reference to $\omega$ in the notation.  
We now apply Chebyshev's inequality to the random variable
$Tr ( E_\Lambda ( I_\kappa ) ) $ and obtain
\bea
\proba\{ \; \mbox{dist} \; ( \sigma ( \Gamma_\Lambda ( E_0  ) ) , -1 ) < \kappa
\}
 & = &  \proba\{ Tr ( E_{\Lambda} ( I_\kappa ) ) \geq 1 \}
           \nonumber \\
 & \leq  & \E \{ Tr ( E_{\Lambda} ( I_\kappa ) ) \} .
\eea

\vspace{.1in}

\noindent
2. We now proceed to estimate the expectation of the trace, following
the original argument of Wegner \cite{[Wegner]} as modified by Kirsch
\cite{[Kr2]}.  
Let $\rho$ be a nonnegative, smooth, monotone decreasing function such 
that $\rho ( x ) = 1 $, for $
x < - \kappa /2 $, and $\rho ( x) = 0$, for $ x \geq \kappa /2 $.
We can assume that $\rho$ has 
compact support since $\Gamma_\Lambda (E_0 )$ is lower semibounded,
independent of $\Lambda$. 
As in \cite{[CHN]}, we have
\bea
\E_\Lambda \{ Tr ( E_{\Lambda} ( I_\kappa ) ) \} & \leq & 
      \E_{\Lambda} \{ Tr [ \rho ( \Gamma_\Lambda (E_0) + 1 - 3 \kappa / 2 ) 
          - \rho ( \Gamma_\Lambda (E_0) + 1 +  3 \kappa / 2 ) ] \}  \nonumber \\
 & \leq &  \E_{\Lambda} \left\{ Tr \left[ \int_{- 3 \kappa / 2}^{ 3 \kappa / 2} \; 
    \frac{d}{dt} \rho ( \Gamma_{\Lambda}(E_0) + 1 - t) \; dt \right] \right\} .
\eea
In order to evaluate the $\rho '$ term,
we use the fact that $\Gamma_\Lambda (E_0)$ is an eigenfunction for the
vector field $A_\Lambda$, as expressed in (3.8). We write $\rho '$ as
\bea
A_\Lambda \rho ( \Gamma_\Lambda (E_0) + 1 - t) & = &  \rho ' ( \Gamma_\Lambda
(E_0) + 1 - t)  \; A_\Lambda \Gamma_\Lambda (E_0) \nonumber \\
 & = & \rho ' ( \Gamma_\Lambda (E_0) + 1 - t) \Gamma_\Lambda (E_0) .
\eea
We now note that $\rho ' \leq 0$ (in the region of interest), and that
on $\mbox{supp} \: \rho'$, the operator $ \Gamma_\Lambda (E_0)
\leq ( - 1 + 2 \kappa  )$,
so we obtain
\beq
- \rho' ( \Gamma_{\Lambda}(E_0) + 1 - t ) \leq  - \frac{1}{( 1 - 2 \kappa  )}
\Sum_{k \in {\tilde \Lambda}}
\; \lambda_k \; \frac{ \partial \rho }{ \partial \lambda_k} \; 
( \Gamma_{\Lambda}(E_0) + 1 - t ) .
\eeq
With this estimate, and the fact that $d \rho ( x + 1 - t ) / dt = - \rho ' 
( x + 1 - t )$,
the right side of (3.11) can be bounded above by
\beq
- \frac{1}{( 1 - 2 \kappa )} \; 
\Sum_{ k \in {\tilde \Lambda }} \; \int_{- 3 \kappa / 2}^{ 3 \kappa / 2}
\; \E \{ \lambda_k \: \frac{ \partial }{ \partial \lambda_k} 
\; Tr [ \rho ( \Gamma_{\Lambda}(E_0) + 1 - t ) ] \} \; dt  .
\eeq
In order to evaluate the expectation,
we select one random variable, say $\lambda_k$, with $k \in {\tilde
\Lambda}$, and first integrate with respect to this variable
using hypothesis (H4). The local absolute continuity property is
necessary here because a single term in the sum of (3.13) is not
necessarily positive. Let us suppose that there is a
decomposition
$[ 0 , M ] = \cup_{l = 0 }^{N-1} ( M_l , M_{l+1} )$ so that
$h_0$ is absolutely continuous on each subinterval.
We denote by $\tilde h_0$ the function ${\tilde h_0} ( \lambda ) 
\equiv \lambda h_0 ( \lambda ) $. As ${\tilde h}_0$ is locally
absolutely continuous, we can integrate by parts and obtain
\bea
\lefteqn{ \left|  \int_0^M  d \lambda_k {\tilde h}_0 (\lambda_k)
\frac{\partial }{\partial \lambda_k}
\; Tr \{ \rho (\Gamma_{\Lambda}(E_0) + 1 - t ) - \rho
(\Gamma_{\Lambda}(E_0)^{0,k} +1 - t \} \right| } \nonumber \\ 
  & = & \left| \Sum_{l = 0}^{N-1} \int_{M_l}^{M_{l+1}} d \lambda_k
        {\tilde h}_0 (\lambda_k) \frac{\partial }{\partial \lambda_k} Tr
  \{ \rho ( \lambda_k ) - \rho ( \lambda_k = 0 ) \} \right| \nonumber \\
 & \leq & {\tilde h}_0 (M) | Tr \{ \rho ( \Gamma_\Lambda
         (E_0)^{M ,k} + 1 - t ) - \rho ( \Gamma_\Lambda
         (E_0)^{0,k} + 1 - t)  \} | \nonumber \\
 & & +  \| {\tilde h_0 }' \|_{\infty} 
   \: \sup_{\lambda \in [0,M]} \; | 
  Tr \{ \rho ( \Gamma_\Lambda (E_0)^{\lambda ,k} + 1 - t ) 
  - \rho (  \Gamma_\Lambda (E_0)^{0,k} + 1 - t) \} | , \nonumber
    \\
  & &  
\eea
where $ \Gamma_\Lambda (E_0)^{\lambda ,k}$ is the operator 
$\Gamma_\Lambda (E_0)$
with the coupling constant $\lambda_k$ at 
the $k^{th}$-site fixed at the value $\lambda_k = \lambda$.
Similarly, the value $0$ or $M$ denotes the 
coupling constant $\lambda_k$ fixed at those values. 
Consequently, we are left with the task of estimating
\beq
\frac{ \mbox{max} \: ( \|{ \tilde h_0 }' \|_{\infty}, { \tilde
h_0 }(M) ) }{( 1 - 2 \kappa  )} \;
\Sum_{k \in {\tilde \Lambda}} \int_{- 3 \kappa / 2 }^{ 3 \kappa / 2}
dt \; 
\int_0^M \displaystyle{ \Pi}_{l \neq k}  \; h_0 (\lambda_l) \; d \lambda_l \; 
| Tr \{ D ( k, E_0 , 0 , \lambda_k^+ ) \} | , 
\eeq
where $D ( k, E_0 , 0 , \lambda_k^+ ) $ denotes the operator  
\begin{equation} 
D ( k, E_0 , 0, \lambda_k^+ ) \equiv  \rho( \Gamma_\Lambda (E_0)^{0,k} + 1 - t ) 
 - \rho ( \Gamma_\Lambda (E_0)^{\lambda_k^+ ,k} + 1 - t ) , 
\end{equation}  
and $\lambda_k^+ \in [ 0 , M ]$ denotes the value of the coupling
constant $\lambda_k$ where the maximum in (3.15)
is obtained. We remark that each term in 
(3.16) is easily seen to be trace-class since the operator
$\Gamma_\Lambda (E_0)$ has discrete spectrum with zero the only accumulation
point, and the function $\rho ( x + 1
- t)$ is supported in $x$ in a compact interval away from $0$ for
$t \in [ - 3 \kappa / 2 , 3 \kappa / 2] $.

\vspace{.1in}

\noindent
3. The trace in (3.16) can be rewritten in terms of 
a spectral shift function as follows.
We let $H_1 \equiv  \Gamma_\Lambda (E_0)^{0,k}$ be the unperturbed operator, 
and write,
\bea
\Gamma_\Lambda (E_0)^{\lambda_k^+ ,k} & =  & H_1 + \lambda_k^+ 
(H^A_0 - E_0 )^{-1/2} u_k (H^A_0 - E_0 )^{-1/2} \nonumber \\   
 & = &  H_1 + V .
\eea
Although the difference $V$ is not trace class, the single-site potential
$u_k$ does have compact support. We show in 
section 6 that the difference of
sufficiently large powers of the bounded operators
$H_1 = \Gamma_\Lambda (E_0)^{0 , k}$ 
and $H_1 + V = \Gamma_\Lambda (E_0)^{\lambda_k^+,k}$
is not only in the trace class,
but is in the super-trace class ${\cal I}_{1/p}$, for all $p \geq 1$. 
Specifically, 
let us define the function $g( \lambda ) =  \lambda^{k}$. 
We prove that for $k > pd /2 + 1$, and $p > 1$,
\beq
g(H_1 + V ) - g(H_1 ) \in {\cal I }_{1/p}.
\eeq
The spectral shift function $\xi ( \lambda \; ; H_1 + V , H_1 )$
is defined for the pair $( H_1 , H_1 + V )$ by
\beq
\xi ( \lambda \; ; H_1 + V , H_1 ) = sgn( g'( \lambda )) \;  
\xi ( g ( \lambda ) \; ; g ( H_1 + V ), g(H_1 ) ) .
\eeq 
Recall that both $\rho$ and $\rho ' $ have compact support.
Because of this, and the fact that the difference $\{ g( H_1 + V ) - 
g ( H_1 ) \}$ is super-trace class, 
we can apply the Birman-Krein identity \cite{[BY]} to the trace in (3.16).
This gives 
\bea
\lefteqn{  Tr \{ \rho( \Gamma_\Lambda (E_0)^{\lambda_k^+,k} + 1 - t ) -
     \rho ( \Gamma_\Lambda (E_0)^{0,k} + 1 - t) \}   } \nonumber \\
 & = &  - \int_{ \R } \frac{d }{ d \lambda } \rho ( \lambda + 1 - t) \;
        \xi (   \lambda ; H_1 + V, H_1 ) \; d \lambda \nonumber \\
 & = & - \int_{ \R } \frac{d}{ d \lambda } \rho ( \lambda + 1 - t ) \;
        \xi ( g ( \lambda ); \; g ( H_1 + V ) , g (H_1 ) ) .
\eea
We estimate the integral using the H{\"o}lder inequality and the
$L^p$-theory of section 2.
Let ${\tilde \xi} ( \lambda ) = \xi ( g ( \lambda ); \; g ( H_1 + V
) , g (H_1 ) )$, for notational convenience. Let
$\chi ( x)$ be the characteristic function for the support of $\rho' (x)$ for
$x > 0$,
and we write ${\tilde \chi} ( x ) \equiv \chi ( \lambda
+ 1 - t)$, so that the support of ${\tilde \chi}$ 
is contained in $[ - 1 - 2 \kappa , -1 + 2 \kappa ]$. 
For any $p > 1$, and $q$ such that $\frac{1}{p} + \frac{1}{q} = 1$,
the right side of (3.21) can be bounded above by 
\beq
\left\{ \int | \rho ' |^q \right\}^{1/q}  \; \left\{ \int | {\tilde \xi }
( \lambda ) \; {\tilde \chi} ( \lambda ) |^p \right\}^{1/p} 
 \leq C_0 \kappa^{(1-q)/q} \; \| {\tilde \xi} {\tilde \chi} \|_{ L^p}  ,
\eeq
Here, we integrated one power of $\rho'$, 
using the fact that $- \rho' > 0$ in the region of interest,
and used the fact that $| \rho ' | = {\cal O} ( \kappa^{-1} )$, to obtain
\beq
\left\{ \int | \rho' |^{q-1} \; | \rho' | \right\}^{1/q} 
\leq \kappa^{(1-q)/q} \;
\left\{ - \int \rho' \right\}^{1/q} \leq C_0 \kappa^{(1-q)/q} . 
\eeq
By a simple change of variables, we find
\bea
\| {\tilde \xi } {\tilde \chi} \|_p & = & 
\left\{ \int | \xi ( g ( \lambda ) ; \; 
g( H_1 + V ) , g ( H_1 ) ) |^p \; {\tilde \chi} 
( \lambda ) \; d \lambda \right\}^{1/p}
 \nonumber \\
 & \leq &  C_1 \left\{  
\int_{ \R} | \xi ( \lambda ; g( H_1 
+ V) , g ( H_1) ) |^p \; d \lambda \right\}^{1/p} \nonumber \\
 & \leq & C_1 \; \| g ( H_1 + V) - g( H_1 ) \|_{1/p}^{1/p}.
\eea
We recall that 
\beq
V = \lambda_k^+ (H^A_0 - E_0 )^{-1/2} u_k (H^A_0 - E_0 )^{-1/2} .
\eeq
In particular, the volume of the support of $V$ has order one, and is
independent of $| \Lambda |$.
We prove in section 5 that 
the constant $\| g(H_1 + V) - g(H_1) \|_{1/p}^{1/p}$ depends only on
the single-site potential $u_k$ and $\mbox{dist} ( E_0 , \mbox{inf} \:
\sigma (H^A_0) )$, and is independent of $| \Lambda |$.  
Consequently, the right side of (3.24) is bounded above by $C_0
\kappa^{(1-q)/q}$, independent of $| \Lambda |$. This estimate, equations (3.16)
and (3.21), lead us to the result
\beq
\proba\{ \mbox{dist} \; ( - 1 , \sigma( \Gamma_\Lambda (E_0) ) ) < \kappa \} \leq
C_W \kappa^{ 1 / q} \|g \|_\infty  | \Lambda |,
\eeq
for any $q > 1$. $\Box$


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\section{Internal Gaps for $H_0^X, X=A, \: \mbox{or} \: X=M$}\label{S.4}

Before proceeding with the proof of Theorem 1.2, let us show how to
treat multiplicative perturbations in the same framework as additive
ones.  Recall from (1.1) that the multiplicatively perturbed local
operator $H_\Lambda^M ( \lambda )$ has the form $H_\Lambda^M ( \lambda
) = A_\Lambda^{-1/2} ( \lambda ) H_0 A_\Lambda^{-1/2} ( \lambda )$,
where $A_\Lambda ( \lambda ) = 1 + \lambda V_\Lambda$ is boundedly
invertible provided $0 \leq |\lambda| < \| V_\Lambda \|_\infty^{-1}$.
For computational convenience, we always assume $|\lambda| < ( 2 \|
V_\Lambda \|_\infty )^{-1}$ for multiplicative models.  For ergodic
models, the norm $\| V_\Lambda \|_\infty$ is bounded independently of
$|\Lambda|$.  It follows as in the beginning of section 3 that \beq
\proba \{ dist ( \sigma ( H_\Lambda^M ( \lambda ) ) , E_0 ) < \eta \}
\leq \proba \{ \| ( H_0 - \lambda E_0 V_\Lambda - E_0 )^{-1} \| > 2 /
\eta \} .  \eeq Let us consider the $E_0$-dependent Schr{\"o}dinger
operator $H_0 + \lambda {\tilde V}_\Lambda $, where ${\tilde
  V}_\Lambda \equiv - E_0 V_\Lambda$. The Wegner estimate for
$H_\Lambda^M ( \lambda )$ follows by (4.1) from an estimate for \beq
\proba \{ \; \| ( H_0 + \lambda {\tilde V}_\Lambda - E_0 )^{-1} \| > 2
/ \eta \} .  \eeq The proof given ahead for the additive case now
applies to the Schr{\"o}dinger operator $H_0 + \lambda {\tilde V}_\Lambda$
and, as a consequence, we obtain an estimate of the probability in
(4.2). By (4.1), this allows us to prove a Wegner estimate for the
multiplicative case.  The limitations on the disorder strength
$|\lambda|$ arise from two constraints, as discussed ahead.  The
first, the requirement that the gap remain open after the local
perturbation, implies that $|\lambda| < (B_+ - B_- ) / ( 4 E_0 \|
V_\Lambda \|_\infty )$. So we must first work with $|\lambda| <
\lambda_0^{(1)}$, where $\lambda_0^{(1)} \equiv \mbox{min} \; \{ 1 /
(2 \| V_\omega \|_\infty) , (B_+ - B_- ) / ( 4 E_0 \| V_\Lambda
\|_\infty) \}$. Secondly, the positivity condition (4.19) must hold.
This requires that we define $\lambda_0^{(2)} = (1/( 4 E_0 \|V_\Lambda
\|_\infty))( \delta_+ (E_0) \delta_- (E_0) / 2)^{1/2}$. The constant
in Theorem 1.2, $\lambda (E_0)$, is defined by $\lambda (E_0) =
\mbox{min} \; ( \lambda_0^{(1)} , \lambda_0^{(2)} )$. With this
change, the proof given ahead holds for the multiplicative model.

We now turn to the proof of Theorem 1.2 and, as explained above, we
will give the proof for the additive model $H_\omega ( \lambda ) = H_0
+ \lambda V_\Lambda $.

\vspace{.1in}
\noindent
{\bf Proof of Theorem 1.2.}

\vspace{.1in}
\noindent
1. Let $P_\pm $ denote the spectral projectors 
for $H_0$ corresponding to the components
of the spectrum $[B_+ , \infty )$ and $( - \infty , B_-]$,
respectively, so that $P_+ + P_- = 1$, and $P_+ P_- = 0$. 
We use the Feshbach method to decompose
the problem relative to these two orthogonal projectors.  
Let $H_0^\pm  \equiv P_\pm H_0 $, 
and denote by $H_\pm ( \lambda ) \equiv H_0^\pm + \lambda P_\pm V P_\pm $. 
We will need the various projections of the potential between the 
subspaces $P_\pm L^2 ( \R^d ) $, and we denote
them by $V_\pm \equiv P_\pm  V P_\pm $, and
$V_{+-} \equiv P_+ V P_-$, with $V_{-+} = V_{+-}^* = P_- V P_+ $.
Let $z \in \C$, with $\mbox{Im} z \neq 0$. We can
write the resolvent $R_\Lambda ( z ) = ( H_\Lambda ( \lambda ) - z)^{-1}$ 
in terms of the resolvents of the projected operators 
$H_\pm ( \lambda )$ as follows.
In order to write a formula valid for either $P_+$ or for $P_-$,
we let $P = P_\pm $, $Q = 1- P_\pm $, 
and write $R_P (z) = ( PH_0 + \lambda P V_\Lambda P - zP )^{-1}$. We then have
\beq
R_\Lambda (z) = P R_P ( z ) P  + 
 \{ Q - \lambda P R_P (z) P V_\Lambda Q \}  {\cal G} (z) 
         \{ Q - \lambda Q V_\Lambda P R_P (z)^* P \} ,
\eeq
where the operator ${\cal G}(z)$ is given by
\beq
{\cal G }(z) = \{  Q  H_0 + \lambda Q V_\Lambda Q - zQ 
- \lambda^2 Q V_\Lambda P R_P ( z ) P V_\Lambda Q \}^{-1} .
\eeq

\noindent
2. The choice of $P_\pm $ depends upon whether $E_0$ is located near
$B_\pm $, respectively.  Let us suppose that $E_0 = \mbox{Re} z$ is close
to $B_+$ in that $E_0 > (B_+ + B_-)/2$. In this case, we use formulae
(4.3)--(4.4) with $Q = P_+$ and $P=P_-$.  We define distances
$\delta_\pm (E_0) \equiv \dist ( \sigma ( H_0^\pm ) , E_0 )$, in analogy
with $\delta$ of section 3. If we take $|\lambda| < \delta_- (E_0) / 2
\|V_\Lambda\|_\infty$, for example, then the first term on the right
in (4.3) satisfies the bound \beq \| P_- ( H_0 ^- + \lambda V_- -
E_0P_-)^{-1} P_- \| \; \leq \; 2 / \delta_- (E_0) .  \eeq Let us note
that according to our choice of $E_0$ near $B_+$, we have $\delta_- (
E_0 ) > ( B_+ - B_- ) / 2$.  Consequently, we define a constant
$\lambda_0^{(1)} = (B_+ - B_- )/ 4 \|V_\Lambda\|_\infty$.  The bound
in (4.5), and formulae (4.3)--(4.4), show that the resolvent of
$H_\Lambda ( \lambda )$ has large norm at energies near $E_0$ provided
${\cal G}(E_0)$ be large and provided $| \lambda| < \lambda_0^{(1)}$.
It follows from (4.5) that for $| \lambda| < \lambda_0^{(1)}$,
$$
\| ( P_+ - \lambda P_- ( H_- ( \lambda ) -  E_0P_-)^{-1} P_- V_\Lambda P_+ )\|
\; \leq \; 2 .
$$ 
Following the analysis of the proof of Theorem 1.1, we find that
\bea
\lefteqn{ \proba\{ \; \|  R_\Lambda ( E_0 ) \| \; \leq \; 1/ \eta \} } \nonumber \\
 & \geq &   
    \proba\{  \;  \| ( H_- ( \lambda ) -  E_0P_-)^{-1} \| 
     \; \leq 1/ (2 \eta) \; \mbox{and} \;  
   \|  {\cal G }( E_0 ) \| \; \leq 1 / ( 8 \eta ) \} .  \nonumber \\
  & & 
\eea
Consequently, the probability that $H_\Lambda$ has
spectrum in an $\eta$-neighborhood of $E_0$, where $E_0 > ( B_+ + B_-)/2$,
is bounded above by
\bea
\proba\{ \; \|  R_\Lambda ( E_0 ) \| \; > \; 1/ \eta \}  
 & \leq & \proba\{  \; \| ( H_- ( \lambda )  -  E_0P_-)^{-1} \|
        \; > 1/ (2 \eta) \}  \nonumber \\
  &   & +   \proba\{ \; \|  {\cal G } (E_0 ) \| \; > 1 / ( 8 \eta ) \} .
\eea
In light of (4.5) and (4.7), and the fact that $\eta < \dist( E_0 , B_+) <
\delta_- (E_0)$, we see that for $| \lambda | < \lambda_0^{(1)}$, 
\bea
\proba\{ \mbox{dist} ( \sigma ( H_\Lambda ) , E_0 ) < \eta \}
 &  = & \proba \{ \; \| R_\Lambda ( E_0 ) \| > 1 / \eta \} \nonumber \\
 & \leq &  \proba\{ \; \| {\cal G } ( E_0 ) \| \; > 1 / ( 8 \eta ) \}.
\eea

\noindent
3. We next reduce the estimate of $\proba\{ \; \| {\cal G } (E_0) \| > 
1 / ( 8 \eta ) \}$ to an equivalent
spectral formulation for a certain self-adjoint, compact operator.
Let $R_0^+ ( z ) = ( H_0^+ - z )^{-1}$.
Since $(H_0^+ - E_0) > 0$, the square root $R_0^+ ( E_0 )^{1/2}$ is
well-defined. In analogy with (3.1)--(3.2),
we can write ${\cal G} ( E_0 ) $ as
\beq
{\cal G } ( E_0 ) = R_0^+ ( E_0 )^{1/2} 
( 1 + {\tilde \Gamma}_+ ( E_0 ) )^{-1} R_0^+ ( E_0 )^{1/2} ,
\eeq
where we define ${\tilde \Gamma}_+ ( E_0 )$ by
\bea
{\tilde \Gamma }_+ ( E_0 )  & \equiv &  \lambda R_0^+ ( E_0 )^{1/2}
              V_+ R_0^+ (E_0 )^{1/2}  \nonumber \\
  & &  + \lambda^2 R_0^+ (E_0)^{1/2} 
      V_{+-}( E_0 P_- -  H_- ( \lambda ))^{-1} V_{-+} R_0^+ (E_0)^{1/2} .
  \nonumber \\
  & & 
\eea
Because of the compactness of the support of the local potential, and
hypothesis (H2), the operator ${\tilde \Gamma}_+ ( E_0 )$ is
self-adjoint and compact. Exactly as in the proof of Theorem 1.1, we show that
if $E_0 > ( B_- + B_+ ) / 2$,
\bea
\proba\{ \; \mbox{dist} ( \sigma ( H_\Lambda , E_0 ) < \eta \}
 & = & \proba\{ \| R_\Lambda ( E_0 ) \| \; > 1/ \eta \} \nonumber \\  
 & \leq & \proba\{ \; \| ( 1 + {\tilde \Gamma}_+ ( E_0 ))^{-1} \| \; >
   \delta_+ (E_0) / ( 8 \eta ) \} \nonumber \\
 & = & \proba\{ \: \mbox{dist} ( \sigma ( {\tilde \Gamma}_+ (E_0 ) ) , -1 ) 
    < 8 \eta / \delta_+ (E_0) \} . \nonumber \\
  &  &  
\eea
   
\noindent
4. To estimate the last
probability on the right in (4.11), we proceed as
in (3.10)--(3.11) of the proof of Theorem 1.1. 
Let $\rho \geq 0$ be the function defined in part 2 of the proof
of Theorem 1.1 with $\kappa = 8 \eta / \delta_+ (E_0)$.
In analogy  with (3.11), we must estimate 
\beq
\E_\Lambda \left\{ Tr \left[ \int_{- 3 \kappa / 2}^{ 3 \kappa / 2}
\; \frac{d}{dt} \rho ( {\tilde \Gamma}_+ (E_0) + 1 - t ) dt
\right] \right\} .
\eeq
We do this using the operator $A_\Lambda$ introduced in (3.7),
\beq
A_\Lambda = \Sum_{ j \in {\tilde \Lambda }} \; \lambda_j ( \omega) 
\frac{\partial}{ \partial \lambda_j ( \omega )} .
\eeq
However, unlike (3.8), the operator ${\tilde \Gamma}_+ (E_0)$ is no
longer an eigenvector of $A_\Lambda$. A straightforward calculation
yields instead 
\beq
A_\Lambda {\tilde \Gamma}_+ (E_0) = {\tilde \Gamma}_+ (E_0) + \lambda^2 W(E_0).
\eeq
The remainder term $W(E_0)$ is given by
\beq
W (E_0)  =  R_0^+ (E_0 )^{1/2} V_{+-} R_- ( E_0 ) ( E_0 P_- 
       - H_0^- ) R_- ( E_0 ) V_{-+} R_0^+ (E_0)^{1/2} ,
\eeq
where $R_- ( E_0 )$ is the reduced 
resolvent $( E_0 P_- - H_- ( \lambda ) )^{-1}$.
For $ | \lambda | < \lambda_0^{(1)}$, we easily compute the bound,
\beq
\| W ( E_0 ) \|  \; \leq \; \left(
\frac{4}{ \delta_- (E_0) \delta_+ (E_0) } \right) \|V_\Lambda
\|_\infty^2 .
\eeq
We replace the calculation (3.12) by the Gohberg-Krein formula (cf.\
\cite{[Simon2]} ) that states
\beq
Tr \{ A_\Lambda \rho ( {\tilde \Gamma}_+ (E_0) + 1 - t ) \}  =   
Tr \{ \rho ' ( {\tilde \Gamma}_+ ( E_0) + 1 - t ) A_\Lambda {\tilde \Gamma}_+
 (E_0) \} .
\eeq
In order to evaluate the right side of (4.17), 
we recall that $\rho' ( x + 1 - t )$ has
compact support in $[ -1 - 2 \kappa , -1 + 2 \kappa]$, for any
$t \in [ - 3 \kappa / 2 , 3 \kappa / 2 ]$.
We expand the trace using the
eigenfunctions $\phi_k$ of ${\tilde \Gamma}_+ (E_0)$  satisfying
${\tilde \Gamma}_+ (E_0) \phi_k = E_k \phi_k $, $\| \phi_k \| = 1 $,
and $E_k \in [ -1 -2 \kappa , -1 + 2 \kappa ]$. This gives
\bea
\lefteqn{ Tr \{ \rho' ( {\tilde \Gamma}_+ (E_0) + 1 - t ) A_\Lambda 
{\tilde \Gamma }_+ (E_0) \} } \nonumber \\
 & = & \Sum_k \rho ' ( E_k + 1 - t ) \: \langle \phi_k , A_\Lambda 
       {\tilde \Gamma}_+ (E_0) \phi_k \rangle \nonumber \\
  & = & \Sum_k \rho' ( E_k + 1 - t ) \: \langle \phi_k ,( E_k + 
       \lambda^2 W ( E_0 ) )  \phi_k \rangle   \nonumber \\
  & \geq & \Sum_k \; - \rho' ( E_k + 1 - t ) 
           ( 1 - 2 \kappa - \lambda^2 \|W(E_0)\| ) .
\eea
The second constraint on $|\lambda|$ arises from this expression. 
We have the 
lower bound on the last term in (4.18):
\beq
(1 - 2 \kappa - \lambda^2 \| W(E_0 ) \|)  \geq  
   \left( 1 - \frac{16 \eta }{ \delta_+ (E_0) } - \lambda^2 \| V_\Lambda \|^2
 \frac{4}{ \delta_- (E_0) \delta_+ (E_0) } \right)  .
\eeq
We define $\lambda (E_0)$ of Theorem 1.2 as
\beq
\lambda (E_0) \equiv \mbox{min} \; \left( \lambda_0^{(1)} = 
  \frac{ (B_+ -B_-) }{ 4 \|V_\Lambda \| } ,
   \frac{1}{4 \|V_\Lambda \|}  
   \left( \frac{ \delta_+ (E_0) \delta_- (E_0) }{ 2} \right)^{1/2} \right).
\eeq
So for all $|\lambda| < \lambda (E_0)$, we have a lower bound for (4.18),
\bea
\lefteqn{ Tr \{ \rho' ( {\tilde \Gamma}_+ (E_0) + 1 - t ) A_\Lambda 
 {\tilde \Gamma}_+ (E_0) \} } \nonumber \\
  & \geq & - C_1 Tr \{ \rho' ( {\tilde \Gamma}_+ (E_0) + 1 - t ) \} ,
\eea
for a finite constant $C_1 > 0$.
This estimate replaces (3.13). We are then left with estimating a term 
similar to the one in (3.14) where $\Gamma_\Lambda (E_0)$ has
been replaced by ${\tilde \Gamma}_+ ( E_0 )$. 

Using the notation
of the proof of Theorem 1.1, a variation in the $k^{th}$ coupling
constant results in the operator difference similar to (3.18),  
\bea
{\tilde \Gamma }_+^{\lambda_k^+, k} (E_0) - {\tilde \Gamma
}_+^{0,k} (E_0) & = &
\lambda_k^+ \lambda R_0^+ ( E_0 )^{1/2} P_+ u_k P_+ R_0^+ (E_0 )^{1/2} 
      \nonumber \\
  & &  + \lambda^2 ( \lambda_k^+)^2 R_0^+ (E_0)^{1/2}
      P_+ u_k P_- R_- ( E_0 ) P_- \nonumber \\
  & & \times u_k P_+ R_0^+ (E_0)^{1/2} ,  
\eea 
where $R_- ( E_0 ) \equiv ( E_0 P_- -  H_- ( \lambda ))^{-1}$.
The operator on the right side of (4.22) is compact.
We show in section 5 how to modify the proof in
\cite{[CHN]} to prove that the difference 
of the $l^{th}$-powers of ${\tilde \Gamma }_+^{\lambda_k^+, k} (E_0)$ and
${\tilde \Gamma}_+^{0,k} (E_0)$ is in the super-trace class ${\cal I}_{1/p}$,
for any $p>1$, provided $l > pd / 2 + 1 $.
With this, the proof of Theorem 1.2 continues exactly as in section 3. $\Box$ 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{A Trace Estimate}\label{S.5}  

We present the estimates on additive perturbations
needed in sections 3 and 4. 
The calculations for
multiplicative perturbations can be reduced to those for
additive ones as
discussed at the beginning of section 4.
We let $K_d ( \R^d )$ denote the Kato class of potentials, and  we refer
the reader to Simon's article \cite{[Simon]} for a complete
description.
We recall the main estimate from \cite{[CHN]}, and then show how to apply
it to the present case.
We let $H_0$ be the Schr{\"o}dinger operator
\beq
H_0 = ( -i \nabla - A )^2 + W ,
\eeq
where $A$ is a vector potential with $A \in L^2_{loc} ( \R^d )$, 
and $W = W_+ - W_-$ is a background potential
with $W_- \in K_d ( \R^d )$ and $W_+ \in K_d^{loc} ( \R^d)$. 
We denote by $H = H_0 + V$, for suitable real-valued functions $V$. We are
interested in a bounded potential $V$ with compact support.
The main result on the trace proved in \cite{[CHN]} is the following.

\vspace{.1in}
\noindent
{\bf Proposition 5.1.} {\it Let $H_0$ be as in (5.1), and let
$V_1$ be a Kato-class potential such that $\| V_1 \|_{K_d} 
\leq M_1$. Let $H_1 \equiv H_0 + V_1$, and let $M > 0$ be a sufficiently
large constant given in the proof. Let $V$ be a real-valued, 
Kato-class function supported on $B(R)$, the ball of radius $R>0$ with
center at the origin. Then, for any $p > 0$, we have 
\beq
V_{eff} \equiv (H_1 + V + M )^{-l} - (H_1 + M )^{-l} \in {\cal I}_{1/p} ,
\eeq
provided $l > dp / 2 + 2$. Under these conditions, there exists a
constant $C_0 > 0$, depending on $p, k , H_0, M_1, \| V_1
\|_{K_d},$ and $R$, so that
\beq
\| V_{eff} \|_{1/p} \; \leq \; C_0 .
\eeq
}
\vspace{.1in}

We sketch the proof of a modifications of this theorem needed in the
proofs of Theorems 1.1 and 1.2. We first consider the condition required in
the proof of Theorem 1.1. We replace the resolvent 
$(H_1 + V + M )^{-l}$ by $\Gamma_\Lambda^{b, k} (E_0)^l$,
 and we replace $(H_1 + M )^{-l}$ by $\Gamma_\Lambda^{a, k} (E_0)^l$, 
where the superscript
$(b,k)$ (respectively, $(a,k)$)
denotes the operator $\Gamma_\Lambda (E_0)$
with the $k^{th}$ coupling constant
$\lambda_k$ fixed at the value $b$, respectively, $a$, for any
two values $a , b \in [ 0 , M ]$. We then have,
\beq
\Gamma_\Lambda^{b, k} (E_0)^l - \Gamma_\Lambda^{a, k} (E_0)^l
=  \Sum_{j=0}^{l-1} \; \Gamma_\Lambda^{b, k} (E_0)^{l-j-1}
V_k \Gamma_\Lambda^{a, k} (E_0)^j ,
\eeq
where $V_k$ is defined in (3.18),
\beq
V_k = ( b - a ) (H_0 - E_0 )^{-1/2} u_k (H_0 - E_0 )^{-1/2} .
\eeq
Let us call the difference on the left in (5.4) the effective potential
$V_{eff}$.
Let $J_k \in C_0^\infty ( \R^d )$ be chosen so that $J_k u_k =
u_k$, with $\mbox{supp} \; J_k$ being slightly larger than $\mbox{supp} \; u_k$.
Following the proof in \cite{[Nakamura]}, we first write
$V_{eff}$ as
\beq
V_{eff}  = (b-a) \: \Sum_{j=0}^{l-1} \; [ J_k^{l-j-1} R_0
(E_0)^{1/2} \Gamma_\Lambda^{b, k}
(E_0)^{l-j-1} ]^* u_k [ J_k^j R_0 (E_0)^{1/2} \Gamma_\Lambda^{a, k} (E)^j ].
\eeq
Now for any  $p \in \mbox{supp} \: h_0$, and for any $r \in \N$, we have 
\beq
R_0 (E_0)^{1/2} \Gamma_\Lambda^{p,k} (E_0)^r = ( R_0 (E_0)
V_\Lambda^{p,k} )^r  R_0 ( E_0 ) )^{1/2 }  .
\eeq
Consequently, we can write
the terms in square brackets in (5.6) as
\beq
J_k^r R_0 (E_0)^{1/2} \Gamma_\Lambda^{p, k} (E_0)^r
= J_k^r ( R_0 (E_0 ) V_\Lambda^{p,k})^r R_0 (E_0)^{1/2} .
\eeq
As in \cite{[Nakamura]}, we can commute powers of $J_k$ to the
right and express the term as
\beq
J_k^r ( R_0 (E_0 ) V_\Lambda^{p,k})^r R_0 (E_0)^{1/2} = 
\Sum_{\alpha = 1}^N \: \Pi_{\beta = 1}^r J_{\alpha \beta }
R_0 (E_0 ) B_{\alpha \beta} .
\eeq
Here, the bounded operators $J_{\alpha \beta}$ are
combinations of the derivatives of $J_k$, and hence have
the support contained in the support of $J_k$, and
the operators $B_{\alpha, \beta}$ are uniformly bounded independently of
$| \Lambda |$.
Notice that $J_k V_\Lambda^{p,k} = p u_k + v_k$, where
$v_k \equiv J_k ( V_\Lambda^{p,k}) - p u_k$ has support in a bounded neighborhood
of $u_k$, depending only on the choice of $J_k$ 
and the overlap of the supports
of $u_k$ and $u_n$, for $n \neq k$.  
Consequently, the Lebesgue measure $| J_k V_\Lambda^{p,k} |$
is bounded independent of $| \Lambda|$. We use the basic fact
that $J_k R_0 (E_0) \in {\cal I}_{2q}$, provided $q > d / 4$ 
(cf.\ \cite{[Nakamura]}). The 
${\cal I}_{2q}$-norm depends only on $| \mbox{supp} \; J_k|$, and
is thus independent of $| \Lambda |$. 
It follows from this, standard trace ideal estimates,
and the expansions (5.9) and (5.6),
that each term of the sum on the right side of 
(5.6) is in the super-trace ideal ${\cal I}_{1/p}$, for $p > 1$, provided
$l$ is chosen to satisfy $  l > pd /2 + 1$.  
This lower bound on $l$ differs slightly from Proposition 5.1
due to an extra resolvent factor coming from the definition of
$\Gamma_\Lambda (E_0)$. 
Because of the support properties of $J_{\alpha \beta}$ mentioned above,
the ${\cal I}_{1/p}$-norm is independent of $| \Lambda |$.

We now mention the modifications needed for the proof of Theorem 1.2.
Instead of working with the operators $\Gamma_\Lambda^{p,k} ( E_0 )$, 
we must use the operators ${\tilde \Gamma}_+^{p,k} ( E_0 )$  
defined in (4.10) and (4.22). An equation analogous to (5.4) holds for
the difference of the $l^{th}$ power of these operators. The
potential $V_k$, appearing in the right side of (5.4), is replaced by 
the difference given in (4.22):
\bea
V_k & = & {\tilde \Gamma}_+^{\lambda_k^+ , k } (E_0 ) - 
            {\tilde \Gamma}_+^{0,k} (E_0)  \nonumber \\
  & = &  \lambda_k^+ \lambda R_0^+ (E_0)^{1/2} P_+ u_k P_+ R_0^+ (E_0)^{1/2}
    \nonumber \\
  & & + (\lambda_k^+)^2 \lambda^2 R_0^+ (E_0)^{1/2} P_+ u_k P_- R_- (E_0) 
       P_- u_k P_+ R_0^+ (E_0)^{1/2} . \nonumber \\
  & & 
\eea
Thus, there are two terms that enter into the analog of the 
right side of (5.4), so we write
\beq
{\tilde \Gamma}_+^{\lambda_k^+ , k} (E_0)^l - {\tilde \Gamma}_+^{0,k} (E_0)^l
= V_{eff}^{(1)} + V_{eff}^{(2)} .
\eeq

The first term is identical in form to (5.6), and we can write it as
\beq
V_{eff}^{(1)} = 
\lambda_k^+ \lambda \;
\Sum_{j=0}^{l-1} \; {\tilde \Gamma}_+^{\lambda_k^+,k} (E_0)^{l-j-1} 
\{ R_0^+ (E_0)^{1/2} u_k R_0(E_0)^{1/2} \} {\tilde \Gamma }_+^{0,k} (E_0)^j .
\eeq
As above, let $J_k \in C_0^{\infty} ( \R^d)$ be a smooth
function satisfying $J_k u_k = u_k$, and having
slightly larger support.
Each term in the sum (5.12) can be written as
\beq
[ J_k^{l-j-1} R_0^+ (E_0)^{1/2} {\tilde \Gamma}_+^{\lambda_k^+ ,k} 
(E_0)^{l-j-1}  ]^* u_k [ J_k^j R_0^+ (E_0)^{1/2} {\tilde \Gamma}_+^{0,k}
(E_0)^j ] .
\eeq
Each of the terms in the square brackets has the form, for $r \in \N$,
and $p \in \mbox{supp} \; h_0$,
\bea
J_k^r ( R_0^+ (E_0)^{1/2} {\tilde \Gamma}_+^{p,k} (E_0)^r  ) 
     & = & \Sum_{s=0}^r \; C(r,s) \lambda^{r+s} (-1)^s J_k^r 
       ( R_0^+ (E_0) V_+ )^{r-s} \nonumber \\
  & & \times  ( R_0^+ (E_0) V_{+-} R_- (E_0) V_{-+} )^s  R_0^+ (E_0)^{1/2} 
    \nonumber \\
   & & 
\eea
where $ R_- (E_0) = (H_- (\lambda) - E_0 P_- )^{-1} $.

The second term $V_{eff}^{(2)}$ has the form
\bea
V_{eff}^{(2)} &=& (\lambda_K^+)^2 \lambda^2 \; \Sum_{j=0}^{l-1} \; 
{\tilde \Gamma}_+^{\lambda_k^+ , k } (E_0)^{l-j-1}   \nonumber \\
  &  &  \times \{ R_0^+ (E_0)^{1/2} u_k R_- (E_0)
u_k R_0^+ (E_0)^{1/2} \} {\tilde \Gamma}_+^{0,k} (E_0)^j  . \nonumber \\
  &  & 
\eea
Each term in the sum on the right in (5.15)
can be expanded as in (5.13)--(5.14). The expression
corresponding to (5.13) is obtained by replacing the
$u_k$ appearing there by $u_k R_- (E_0) u_k$. 
The corresponding terms in the square brackets are
the same as in (5.14).  

We now turn to the computation of the 
the super-trace class norms
of the effective potentials in (5.12) and (5.15).
Because of the spectral projectors appearing in these terms, we cannot 
prove a representation for each of these terms as in (5.9). 
Instead, we use the exponential decay of the 
projectors and resolvents appearing 
in (5.14) in a manner similar to that used in \cite{[BCH]}. 
We begin by summarizing the decay estimates that we need.
Let $a , b \in \R^d$ be two distinct points and
let $\chi_a , \chi_b \in C_0^\infty ( \R^d )$ be two
functions localized near $a$ and $b$, respectively, with disjoint supports.
By hypothesis (H1), the operator $H_0$ is assumed to be semibounded from
below with an open spectral gap $G = (B_- , B_-)$.
It follows from the contour integral representation of the
spectral projection, and
the Combes-Thomas estimate on the resolvent (cf.\ \cite{[CT]}
or \cite{[BCH]} ), that for any $\delta > 0$,
there exist two constants $0 < C_\delta , \sigma_\delta < \infty$,
uniform in $a, b \in \R^d$, so that
\beq
\| \chi_a P_- \chi_b \| \; \leq \; C_\delta e^{ - \sigma_\delta \| a - b \| }.
\eeq
This estimate implies that
\beq
\| \chi_a P_+ \chi_b \| \; \leq \; C_\delta e^{ - \sigma_\delta
\| a - b \| } ,
\eeq
since $P_+ = 1 - P_-$, and the supports of $\chi_a$ and $\chi_b$
are disjoint. Of course, when the supports are not disjoint,
the bound on the right side is simply a constant.
Since $E_0 \in G$, the resolvent $R_0 (E_0)$ decays exponentially
when localized between $\chi_a$ and $\chi_b$. It follows from the
argument below, that the operator norms of $\chi_a R_0^+ (E_0)
\chi_b$ and $\chi_a R_- (E_0 ) \chi_b $ 
both exhibit exponential decay when localized between $\chi_a$ and $\chi_b$.

We now prove that for any $q > d / 4$, there exist constant $0 < C_q , \sigma_q 
< \infty$, so that, uniformly in $a , b \in \R^d$, we have   
\beq
\| \chi_a R_0^+ ( E_0 ) \chi_b \|_{2q} \; \leq \; C_q e^{- \sigma_q \| a-b\| }.
\eeq
A similar bound holds for $\chi_a R_- (E_0 ) \chi_b $.
As mentioned above, for any $\chi$ of compact support, the operator
$\chi R_0 (E_0) \in {\cal I}_{2q}$, for $q > d/4$.
Let $\{ \chi_l \; | \; l \in \Z^d \}$ be a partition on unity on $\R^d$ so
that $\chi_l$ is supported in a unit cube centered at $l \in \Z^d$. We then
have
\bea
\| \chi_a R_0^+ (E_0) \chi_b \|_{2q}  & \leq & \Sum_{l \in \Z^d} \; 
   \| \chi_a P_+ \chi_l \| \; \| \chi_l R_0 (E_0) \chi_b \|_{2q} \nonumber \\
 & \leq & \Sum_{l \in \Z^d} \; C_\delta e^{ - \sigma_\delta \|a - l \|} \;
  \| \chi_l R_0 (E_0) \chi_b \|_{2q} \nonumber \\
 & \leq & \Sum_{l \cap b \neq \emptyset} \; 
   C_\delta e^{- \sigma_\delta \| a - l \|} \; 
     \| \chi_l R_0 (E_0) \chi_b \|_{2q} \nonumber \\
  &  & + \Sum_{l \cap b = \emptyset} \; C_\delta e^{ - \sigma_\delta \|a-l\|}
    \; \| \chi_l R_0 (E_0) \chi_b \|_{2q} .
\eea
The notation $l \cap b \neq \emptyset$ means that $\chi_l \chi_b \neq 0$.
The first sum on the right side of the last term in (5.19) is
finite and, with a possible change in weight depending only on
the size of the support of $\chi_b$, it satisfies the bound (5.18).
As for the second sum in (5.19), 
we compute the norm $\| \chi_l R_0 (E_0) \chi_b \|_{2q}$, for
$\chi_l \chi_b = 0$, as follows.
Let ${\tilde \chi}_l^{(1)} \in C_0^1 ( \R^d )$ 
be a function with slightly larger
support than $\chi_l$ and satisfying ${\tilde \chi}_l^{(1)}
\chi_l = \chi_l$.
Let $W( {\tilde \chi}_l^{(1)} ) $ be the commutator $[ H_0, 
{\tilde \chi}_l^{(1)} ]$. This is a first-order operator and
relatively-$H_0$ bounded. Finally, let ${\tilde \chi}_l^{(2)} \in
C_0^1 ( \R^d )$ satisfy $W( {\tilde \chi}_l^{(1)} ) {\tilde
\chi}_l^{(2)} = W( {\tilde \chi}_l^{(1)} )$. We then have,
\bea
\| \chi_l R_0 (E_0) \chi_b \|_{2q} & \leq & \| \chi_l R_0 (E_0) 
     W (\chi_l^{(1)}) R_0 (E_0) \chi_b \|_{2q} \nonumber \\
 & \leq & \| \chi_l R_0 (E_0) W (\chi_l^{(1}) )\|_{2q} \; \| \chi_l^{(2)} 
   R_0 (E_0) \chi_b \| \nonumber \\   
  & \leq & C_0 e^{- \sigma( E_0) \| l - b \| } ,
\eea
where $\sigma( E_0)$ is the exponent in the Combes-Thomas
estimate. 

With these estimates in hand, we now turn to the ${\cal I}_{1/p}$
estimates on $V_{eff}^{(i)}, i = 1 , 2$.
We begin with a term  of $V_{eff}^{(1)}$, given in (5.12).
We write,
\beq
\| V_{eff}^{(1)} \|_{1/p} \; \leq \; C_0 \|J_k^{l-j-1} R_0^+
(E_0)^{1/2} {\tilde \Gamma}_+^{\lambda_k^+ , k} (E_0)^{l-j-1}
\|_o \; \| J_k^j R_0^+(E_0)^{1/2} {\tilde \Gamma}_+^{0 , k} (E_0)^{j}
\|_n ,
\eeq
where $p = 1/o + 1/n$. Each term is now expanded according to
(5.14). The first factor on the right in (5.21) is bounded above
by 
\beq
C \Sum_{s_1 = 0}^{l-j-1} \; \| J_k^{l-j-1} ( R_0^+ (E_0) V_+
)^{l-j-1-s_1} ( R_0^+ (E_0) V_{+-} R_- (E_0) V_{-+} )^{s_1} 
\|_o ,
\eeq
for $0 \leq s_1 \leq l-j-1$. 
We expand each local potential in (5.22) using the definition
(1.5). In this way, we obtain a sum over $(l-j-1+s_1)$
variables, each running over the points of ${\tilde \Lambda }$. 
Let $J$ stand for the $(l-j-1-s_1)$-tuple of indices $J = ( J_1 , \ldots ,
J_{l-j-1-s_1})$, and let $K$ and $L$ stand for $s_1$-tuples of
indices, all taking values in ${\tilde \Lambda }$. 
We also write $\lambda_J$ for the product
$\lambda_{j_1} \ldots \lambda_{j_{l-j-1-s_1}}$, and
similarly for the other index sets. 
A typical element in this sum has the form,
\beq
\begin{array}{l}  
\Sum_{J, K, L} | \lambda_J \lambda_K \lambda_L |  
\; \| J_k R_0^+ (E_0) u_{J_1} \ldots R_0^+ (E_0) u_{J_{l-j-1-s_1}} 
 \\
\times R_0^+ (E_0) u_{L_1} R_-(E_0) u_{K_1} \ldots R_0^+ (E_0) u_{L_s} R_-(E_0)
u_{K_s} \|_o.
\end{array} 
\eeq 

To compute the ${\cal I}_o$ norm of this term, 
we bound each random variable
by $M$, and use the H{\"o}lder inequality for trace norms
repeatedly based on the bound (5.18).
For each index set $X = J, K, L$, and single-site potential $u_{X_i}$, let
$\chi_{X_i}$ be a function of compact support in a region slightly larger than
$\mbox{supp} \; u_{X_i}$, and satisfying $\chi_{X_i} u_{X_i}
= u_{X_i}$. In this way, we obtain an upper bound
on (5.23),
\beq  
\begin{array}{l}
M^{l-j-1+s_1} \Sum_{J , K , L  \in {\tilde \Lambda}}
\; \| J_k R_0^+ (E_0) u_{J_1} \|_{2q} \ldots  \|
\chi_{J_{l-j-2-s_1}} R_0^+ (E_0) u_{J_{l-j-1-s_1}} \|_{2q}  \\ 
\times \|\chi_{J_{l-j-1-s_1}} R_0^+ (E_0) u_{K_1} \|_{2q} \;
 \| \chi_{K_1} R_- (E_0) u_{L_1} \|_{2q} \cdots 
  \| \chi_{K_{s-1}i} R_- (E_0) u_{L_s} \|_{2q'} ,
\end{array}
\eeq
where $2q' = 2qo / ( 2q- (l-j-2+s_1) o)$.
According to (5.18), we require that
$2q' \geq 2q$ in order for the norms to be finite, that is, 
\beq
o > 2q - (l-j-2+s_1)o ,
\eeq
where $0 \leq s_1 \leq l-j-1$. Similarly, the second factor in 
(5.21) can be bounded as in (5.24), provided $n$ satisfies
\beq
n > 2q - (j - s_2 -1)n ,
\eeq
where $0 \leq s_2 \leq j$ is the index from the expansion as in (5.14).
Recalling that $p = 1/o + 1/n$, conditions (5.25)--(5.26) 
require that $l > pd / 2 + 1$. 
Finally, the sum over all the indices $(J, K, L)$
is controlled by the exponential decay of
each term as given in (5.18). It follows that the sums are
bounded independently of $| \Lambda| $.   
This proves that (5.21) is uniformly bounded in $| \Lambda |$
provided $l > pd / 2 + 1 $. 
The proof for $V_{eff}^{(2)}$ is similar.
This implies that the operators $V_{eff}^{(i)} \in {\cal I}_{1/p}, 
i = 1 , 2$, for any $p > 1$, and are bounded uniformly in $|
\Lambda |$, provided we choose $l > pd / 2 + 1 $.
  

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\section{Extensions and Comments on Localization}\label{S.6}  

\subsection{Generalizations: 
Schr{\"o}dinger operators with random magnetic fields}

The methods of this paper can be used to treat a more general
family of random perturbations that includes Schr{\"o}dinger
operators with random magnetic potentials.
We will show that we can treat families of random operators of
the form
\beq
H_\omega( \lambda) = H_0 + \lambda H_{1, \omega} + \lambda^2
H_{2, \omega},
\eeq
provided $| \lambda |$ is sufficiently small,
where $H_0$, a second-order, self-adjoint, partial differential operator,
is a deterministic background operator, 
and the perturbations $H_{j , \omega}, j = 1 , 2$ are symmetric,
relatively-$H_0$ bounded, first-order differential operators.
For the Wegner estimate, it suffices to consider the operators
$H_{j , \omega }$ localized to finite volume regions $\Lambda \subset \R^d$.
We say that a operator $B_\Lambda$ is localized in $\Lambda$
if there exists a constant $0 \leq R < \infty$ so that,   
with $\Lambda_R \equiv \cup_{x \in \Lambda } B_R (x)$,
and for any element $\phi \in C_0^\infty ( \R^d \backslash \Lambda_R )$,
we have $B_\Lambda \phi = 0$.
We make the following assumptions on the random operators
$H_{j, \omega}^\Lambda, j = 1,2$:

\begin{enumerate}
\item[(H5)] The operator $H_{1, \omega}^\Lambda$ is localized in
$\Lambda$, linear in
the random variables $\lambda_j ( \omega)$, and has the form
\beq
H_{1, \omega}^\Lambda = \sum_{j \in {\tilde \Lambda}} \lambda_j ( \omega)
B_j,
\eeq
where the deterministic operators $B_j$ are symmetric, uniformly (in
$|\Lambda|$) relatively-$H_0$
bounded, first-order partial differential operators with coefficients supported
in regions with volumes independent of $|\Lambda|$.

\item[(H6)] The operator $H_{2, \omega}^\Lambda$ is localized in $\Lambda$,
quadratic in the random variables
$\lambda_j ( \omega)$, and has the form
\beq
H_{2, \omega}^\Lambda =\sum_{j,k \in {\tilde \Lambda}} \lambda_j ( \omega)
\lambda_k ( \omega) C_{jk} ,
\eeq
where the deterministic operators $C_{jk} = C_{kj}$ are symmetric, uniformly 
(in $|\Lambda|$) relatively-$H_0$ bounded, first-order partial differential
operators with coefficients supported in regions with volumes independent of
$|\Lambda|$, and such that the support of the coefficients 
of $\Sum_{j \in {\tilde
\Lambda }} C_{jk}$, for each $k \in {\tilde \Lambda}$, are independent
of $| \Lambda |$. 
\end{enumerate}

As the method shows, one can consider operators $H_{2 , \omega}^\Lambda$ 
that depend polynomially on the random variables.

Our primary example is the following.
We consider a vector potential $A_\omega^\Lambda ( \lambda) \equiv
A_0 + \lambda A_\omega^\Lambda$, where $A_0$ is a deterministic
vector potential, and $A_\omega^\Lambda$ is local with respect to
$| \Lambda |$. The background operator $H_0 = ( - i \nabla - A_0)^2$
is assumed to be essentially self-adjoint on $C_0^\infty ( \R^d )$.
The corresponding magnetic Schr{\"o}dinger
operator can be written as
\bea
H_\omega^\Lambda ( \lambda) &  = &  ( - i \nabla - A_\omega^\Lambda
  ( \lambda ) )^2 \nonumber
        \\
  & = &  ( -i \nabla - A_0 )^2 - \lambda \{ ( -i \nabla -
        A_0 ) \cdot A_\omega^\Lambda + A_\omega^\Lambda 
       \cdot ( -i \nabla - A_0 ) \} \nonumber \\
  & & + \lambda^2 A_\omega^\Lambda \cdot A_\omega^\Lambda .
\eea
Comparing with (6.1), we have 
\bea
H_{1, \omega}^\Lambda & \equiv & - \{ ( -i \nabla - A_0 ) \cdot 
A_\omega^\Lambda + A_\omega^\Lambda \cdot ( -i \nabla - A_0 ) \} , 
  \nonumber \\
H_{2, \omega}^\Lambda &  \equiv  &  A_\omega^\Lambda \cdot A_\omega^\Lambda. 
\eea
The local, random vector potential
$A_\omega^\Lambda$ has the Anderson-type form (1.5) with 
vector-valued, single-site potentials
$u_k$. We assume that the single-site
potentials and random variables $\lambda_k ( \omega)$ satisfy
hypotheses (H3) and (H4).  
For this choice, the operators $B_j = - ( -i \nabla - A_0) \cdot u_j -
u_j \cdot ( -i \nabla - A_0 )$, and $C_{jk} = u_j \cdot u_k$. It is clear
that hypothesis (H3) on the $u_j$ imply the locality conditions
of hypotheses (H5)--(H6).
 
As in sections 3 and 4, we will consider two cases: 1) the Wegner estimate
and the IDS near $\mbox{inf} \; \Sigma$, and 2) the Wegner estimate and
the IDS near the internal gaps.

As a concrete example for both of these cases,
we consider a perturbation of the Landau Hamiltonian
$H_0$ on $\R^2$. The background vector potential $A_0$ can be chosen
to be $A_0 (x_1, x_2) = (B_0/ 2) ( 0,  x_1)$. The spectrum of
$H_0$ consists of a discrete family of $E_n (B_0) = (2 n + 1)B_0, n = 0,
1 , \ldots$ of infinitely degenerate eigenvalues.  
We now consider a perturbation $A_\omega$ of Anderson-type, obtained
from (1.5) by summing over all lattice points $\Z^d$.
It is clear that for small $|\lambda|$ the deterministic spectrum $\Sigma$
lies in nonoverlapping bands of width ${\cal O} ( |\lambda| M ) $
about the Landau levels, where $\mbox{supp} \; h_0 = [0 , M]$,
as above (cf.\ \cite{[BrietCornean]} for related results on the spectrum
of magnetic Schr{\"o}dinger operators).  
Hence, we can consider the Wegner estimate and the IDS at energies
near the bottom of the first band and at higher band edges.

Concerning the first case, $\mbox{inf} \; \Sigma$,   
Nakamura \cite{[Nakamura2]} recently proved an upper bound on
the IDS $N(E)$, exhibiting the Lifshitz tail
behavior, near the bottom  of the spectrum for
a general family of Schr{\"o}dinger operators with random magnetic fields. 
Nakamura considered the case of $\lambda = 1$ and $A_0 = 0$ in
(6.4), and $d \geq 2$. 
He assumed that the magnetic field is a random and metrically transitive,
bounded, closed, two-form on $\R^d$,
that is asymptotically clustering in the sense that for any $f,g
\in C_0 ( \R^d)$, the expectation $\E (fg)$
approaches $\E (f) \E (g)$, as the supports of $f$ and $g$
separate. Furthermore, the expectation of the average of the
magnetic field over a unit cell is assumed to be strictly
positive. Under these conditions, Nakamura proved that
\beq
\limsup_{E \rightarrow 0^+} \: \frac{ \mbox{log} \: ( - \mbox{log} \:
N(E) ) }{ \mbox{log} E } \leq - d / 2 .
\eeq
For the special case of a random vector potential described
above, we prove a Wegner estimate and the H{\"o}lder continuity of the IDS
near $\mbox{inf} \; \Sigma$.
{\it It follows from the comments below on localization that Nakamura's
estimate (6.6), and the Wegner estimate, Theorem 6.1, 
prove Anderson localization for a class of random magnetic Schr{\"o}dinger 
operators near the bottom of the deterministic spectrum 
provided $\mbox{inf} \; \Sigma = 0$.} 

As for the situation of internal gaps, 
we can construct examples of families of random \Schr\ 
operators with random vector potentials starting with 
three types of background operators with open internal gaps.
These internal gaps can be proved to remain open 
after a perturbation by random vector potential with weak disorder.
First, for $d=2$, we can consider the Landau Hamiltonian discussed above.
Secondly, pure magnetic Schr{\"o}dinger operators with periodic vector potentials
have been studied by Hempel and Herbst \cite{[HempelHerbst]},  
and by Nakamura \cite{[Nakamura3]}.
These authors prove that there may exist open spectral gaps for
Schr{\"o}dinger operators with strong, periodic magnetic fields.
They give nontrivial examples for which there are open gaps in the spectrum.
Finally, we consider the perturbation of a periodic \Schr\
operator $H_{00} = - \Delta + V_{per}$ by a small vector
potential $\lambda_0 A_0$. It follows
from the results of Briet and Cornean \cite{[BrietCornean]}
that the operator $H_0 ( \lambda) = - ( -i \nabla - \lambda
A_0)^2 + V_{per}$ has open internal gaps provided $|\lambda|$ is taken
sufficiently small. 

We begin with the Wegner estimate for the general family of random
operators (6.1) satisfying  (H5)--(H6) near the bottom of the
deterministic spectrum and near the band edges. 

\vspace{.1in}
\noindent
{\bf Theorem 6.1.} \\ 
\noindent
{\bf a.) Bottom of $\Sigma$.} {\it  
Suppose that the deterministic background
operator $H_0$ satisfies hypotheses (H1b), with $\Sigma_0 = \mbox{inf} \;
\sigma (H_0) > 0$,
and (H2), and that the random processes $H_{j , \omega}^\Lambda , j = 1,2$, 
satisfy (H5)--(H6), with the random variables
satisfying hypothesis (H4).
Let $E_0 < \Sigma_0$, and choose $\eta > 0$ such that $I_\eta \equiv [ E_0 -
2 \eta , E_0 + 2 \eta ] \subset ( - \infty , \Sigma_0)$. 
Then, there exists a constant
$\lambda_0 > 0$, and, for any $q > 1$,
a finite constant $C_W  = C_W ( \lambda_0 , [ \mbox{dist} \; ( \Sigma_0 ,
E_0) ]^{-1},q ) > 0$, such that for all $|\lambda | < \lambda_0$, we have
\beq
\proba \{ \; \mbox{dist} \; ( \sigma (H_{ \Lambda , \omega } ( \lambda )),
E_0 ) \leq \eta \} \; \leq \; C_W \;  \eta^{1/q} \; | \Lambda | .  
\eeq
} \\
\noindent
{\bf b.) Internal Gaps.} {\it Suppose that the deterministic background
operator $H_0$ satisfies hypotheses (H1a) and (H2), and that the
random processes $H_{j, \omega}^\Lambda, j = 1,2$, 
satisfy (H5)--(H6), with the random variables
satisfying hypothesis (H4). Suppose $G = (B_- , B_+ )$
is an open gap in the spectrum of $H_0$. For any $E_0 \in G$,
choose $\eta > 0$ so that the interval $I_\eta = 
[ E_0 - 2 \eta , E_0 + 2 \eta ] \subset G$. Then, there exists a constant 
$ \lambda_0 > 0$, and, for any $q > 1$,
a finite constant $C_W = C_W ( \lambda, [ \mbox{dist} \; (
E_0 , \sigma ( H_0 ) )]^{-1} , q ) > 0$, 
such that for all $| \lambda | < \lambda_0$, we have
\beq
\proba \{ \; \mbox{dist} \; ( \sigma ( H_{ \Lambda , \omega } ( \lambda )),
E_0 ) \leq \eta \} \; \leq \; C_W \; \eta^{1/q} \; | \Lambda| .
\eeq
}

%\vspace{.1in}
\noindent
{\bf Proof.} We follow the proofs of sections 3 and 4. \\ 
\noindent
1. {\bf Bottom of $\Sigma$.} The proof proceeds effectively as in
section 3. The operator that replaces $\Gamma_\Lambda
(E_0 ; \omega )$ in (3.2) of section 3 is
\beq
\Gamma_\Lambda^\lambda ( E_0 ; \omega ) \equiv R_0 ( E_0)^{1/2} (
\lambda H_{1, \omega}^\Lambda + \lambda^2 H_{2 , \omega}^\Lambda ) 
R_0 (E_0)^{1/2},
\eeq
where $H_{j,\omega}^\Lambda$ are defined in (6.2) and (6.3), respectively.
Because of hypotheses (H5)--(H6), we easily find that 
\beq
A_\Lambda H_{1,\omega}^\Lambda = H_{1 , \omega}^\Lambda ,
\eeq
and
\beq
A_\Lambda H_{2, \omega}^\Lambda = 2 H_{2 , \omega }^\Lambda ,
\eeq
so that 
\beq
A_\Lambda \Gamma_\Lambda^\lambda (E_0) = \Gamma_\Lambda (E_0) + \lambda^2
K_\Lambda (E_0) .
\eeq
The bounded operator $K_\Lambda (E_0)$ is defined by
\beq
K_\Lambda (E_0) \equiv R_0 (E_0)^{1/2} H_{2, \omega}^\Lambda R_0 (E_0)^{1/2} . 
\eeq
Let us write $\nu \equiv \| K_\Lambda (E_0) \|$. 
Because of the support of $\rho '$, we need to invert the right
side of (6.12) on the spectral subspace for which $\Gamma_\Lambda^\lambda (E_0)
\leq ( -1 + 2 \kappa)$, where, as in section 3, $\kappa = \eta /
\delta$, for $\delta = \mbox{dist} \; ( E_0 , \sigma (H_0))$. We fix
$\lambda_0$ by the requirement that
$ \lambda_0^2 \nu = ( 1 - 2 \kappa ) / 2$. Thus, for any
$|\lambda| < \lambda_0$, we have
\beq
\| A_\Lambda \Gamma_\Lambda^\lambda (E_0) \rho' ( \Gamma_\Lambda^\lambda
(E_0) - t + 1 ) \| >  ( 1 - 2 \kappa ) / 2 .
\eeq
With these modifications, we arrive at the analogs of (3.15)--(3.16).
In order to apply the results on the spectral shift function,
we let $H_1 \equiv \Gamma_\Lambda^\lambda (E_0)^{0 , k}$ and compute
the analog of (3.18),
\bea
\Gamma_\Lambda^\lambda (E_0)^{\lambda_k^+ , k } & = & H_1 +  
  \lambda \lambda_k^+ R_0 (E_0)^{1/2} B_k R_0 (E_0)^{1/2} \nonumber \\
 & & + 2 \lambda^2 \lambda_k^+ R_0 (E_0)^{1/2} ( \Sum_{j \in {\tilde \Lambda}}
 \lambda_j C_{jk} ) R_0 (E_0)^{1/2},
\eea
where we write $V_\Lambda = \sum_{j \in {\tilde \Lambda }} \lambda_j u_j$.
This is similar to the form of the perturbation 
caused by varying a single coupling
constant appearing in (3.18). The essential point is that 
the first-order operators $B_k$ and $\Sum_{j \in {\tilde \Lambda }}
\lambda_j C_{jk}$, appearing in each term,  
are local operators whose supports are independent of $ \Lambda$.
This insures that the $L^p$-estimate on the
corresponding spectral shift function is independent of $| \Lambda |$.
Consequently, the proof concludes as in section 3.
 
\noindent
2. {\bf Internal Gaps.}
As in section 4, the projectors $P_\pm $ are the spectral projectors for
$H_0$ corresponding to the spectral subspaces $[ B_+ , \infty)$
and $( - \infty , B_-]$, respectively. We consider the case when
$E_0 \in G$ and $E_0 > ( B_+ + B_- ) / 2$. The formulas for the
Feshbach projection method are obtained from (4.3)--(4.4) by
replacing the potential $\lambda V_\Lambda$ by $( \lambda H_{1 ,
\omega}^\Lambda + \lambda^2 H_{2 , \omega}^\Lambda )$. Let the free, reduced
resolvent of $P_\pm  H_0$ be denoted by $R_0^\pm (z) = ( P_\pm H_0 -
P_\pm z)^{-1}$. The resulting formula for $R_{P_-} ( E_0) \equiv R_- (E_0)$,
the first term on the right in (4.3), is 
\bea
R_- (E_0) & = &  R_0^- (E_0)^{1/2}
     \{ 1 + R_0^- (E_0)^{1/2} P_- ( \lambda H_{1 , \omega}^\Lambda   
  \nonumber \\
  &  & +  \lambda^2  H_{2 , \omega}^\Lambda  ) P_- R_0^- (E_0)^{1/2} \}^{-1} 
    R_0^- (E_0)^{1/2},  
\eea
provided the inverse exists. We set $\delta_- = 
\mbox{dist} \; ( \sigma ( H_0^- ) , E_0 ) $. The first factor
on the right in (6.16) exists provided $| \lambda | < \lambda_0^{(1)}$, 
where $\lambda_0^{(1)}$ is fixed by the requirement that
\beq
\lambda_0^{(1)} \delta_-^{-1/2} \{  \| H_{1, \omega}^\Lambda 
R_0^- (E_0)^{1/2} \|
+ \lambda_0^{(1)} \| H_{2, \omega}^\Lambda R_0^- (E_0)^{1/2} \| \} \; < 1 .
\eeq
Similarly, the operator ${\cal G}(E_0)$ can be written, in analogy
with (4.4), (4.9), and (4.10), as
\beq
{\cal G}(E_0)  =  R_0^+ (E_0)^{ 1/2} \{ 1 + {\tilde \Gamma}_+ (E_0) \}^{-1}
R_0^+ (E_0)^{ 1/2} . 
\eeq
The compact, self-adjoint operator ${\tilde \Gamma}_+ (E_0)$ 
has an expansion in $\lambda$ given by
\beq
{\tilde \Gamma}_+ (E_0)  =  \sum_{j=1}^4 \lambda^j M_j ( E_0 ) ,
\eeq
where the coefficients are given by
\bea
M_1 (E_0) & = & R_0^+ (E_0)^{ 1/2} P_+ H_{1 ,\omega}^\Lambda P_+ R_0^+
(E_0)^{ 1/2} , \nonumber \\
M_2 (E_0) & = & R_0^+ (E_0)^{ 1/2} \{ P_+ H_{2, \omega}^\Lambda P_+  \nonumber \\
   & & - P_+ H_{1 , \omega}^\Lambda  P_- R_-(E_0) P_- H_{1 , \omega}^\Lambda   
        P_+ \} R_0^+ (E_0)^{ 1/2} , \nonumber \\
M_3 (E_0) & = & -R_0^+ (E_0)^{ 1/2} \{ P_+ H_{1 , \omega}^\Lambda P_- R_-(E_0) 
    P_- H_{2 , \omega}^\Lambda  P_+ \nonumber \\
  & & + P_+ H_{2 , \omega}^\Lambda P_- R_- (E_0) P_-
    H_{1,  \omega}^\Lambda P_+ \} R_0^+ (E_0)^{ 1/2} , \nonumber \\
M_4 (E_0) & = & - R_0^+ (E_0)^{ 1/2}\{ P_+ H_{2 , \omega}^\Lambda P_- R_-
     (E_0) P_- H_{2 , \omega}^\Lambda P_+ \}  R_0^+ (E_0)^{ 1/2} .
       \nonumber \\
  & & 
\eea
We now compute the action of the vector field $A_\Lambda$,
defined in (4.13), on the operator 
${\tilde \Gamma}_+ (E_0)$. Formulas (6.20) indicate that we need to
compute the action of $A_\Lambda$ on the local perturbations
$H_{j , \omega}^\Lambda, j = 1,2$, and on the resolvent $R_- (E_0)$. 
According to hypotheses (H5)--(H6), the action of $A_\Lambda$ on these
operators is the same as given in (6.10)--(6.11), and
\beq
A_\Lambda R_- (E_0) = - R_- (E_0) \{ \lambda H_{1, \omega}^\Lambda + 2 \lambda^2 
H_{2, \omega}^\Lambda \} R_- (E_0) .
\eeq
Using these results, we obtain
\beq
A_\Lambda {\tilde \Gamma}_+ (E_0) = {\tilde \Gamma}_+ (E_0) 
+ \Sum_{j = 2}^6 \; \lambda^j K_j (E_0) .
\eeq
The remainder terms $K_j(E_0)$ are given by:
\bea
K_2 (E_0 ) &=&   M_2 (E_0) ,  \nonumber \\  
K_3 (E_0) &=& 2 M_3 ( E_0) \nonumber \\
  & &  + R_0^+(E_0)^{1/2} \{ P_+ H_{1, \omega}^\Lambda 
      R_- (E_0) H_{1, \omega}^\Lambda  R_- (E_0) H_{1, \omega}^\Lambda P_+ \} 
      R_0^+(E_0)^{1/2}, \nonumber \\ 
K_4 (E_0) & = & 3 M_4 (E_0) + R_0^+ (E_0)^{1/2}\{ 2 P_+ H_{1, \omega}^\Lambda
   R_- (E_0) H_{2, \omega}^\Lambda R_- (E_0) H_{1, \omega}^\Lambda P_+ 
  \nonumber \\
 & &  +  P_+ H_{1, \omega}^\Lambda 
      R_- (E_0) H_{1, \omega}^\Lambda R_- (E_0) H_{2, \omega}^\Lambda P_+ 
       \nonumber \\
  & & + P_+ H_{2, \omega}^\Lambda R_- (E_0) H_{1, \omega}^\Lambda R_- (E_0)
      H_{1, \omega}^\Lambda P_+  \} R_0^+ (E_0)^{1/2}, \nonumber \\   
K_5 (E_0) & = &  R_0^+ (E_0)^{1/2} \{ 2 P_+ H_{1,\omega}^\Lambda R_- (E_0) 
          H_{2,\omega}^\Lambda R_- (E_0) H_{2, \omega}^\Lambda P_+ \nonumber \\
    & & + 2 P_+ H_{2, \omega}^\Lambda R_- (E_0)H_{2, \omega}^\Lambda R_- (E_0)
       H_{1, \omega }^\Lambda P_+  \nonumber \\
   &  & + P_+ H_{2, \omega }^\Lambda R_- (E_0) H_{1, \omega}^\Lambda R_- (E_0) 
       H_{2, \omega}^\Lambda P_+ \} R_0^+ (E_0)^{1/2} , \nonumber \\   
K_6 (E_0) & = & R_0^+ (E_0)^{1/2} \{ 2 P_+ H_{2, \omega}^\Lambda R_- (E_0) 
   H_{2, \omega}^\Lambda R_- (E_0) H_{2, \omega}^\Lambda P_+ \} 
   R_0^+ (E_0)^{1/2} . \nonumber \\ 
 & &   
\eea
As in part 1 of the proof, we need to compute $\| A_\Lambda {\tilde \Gamma}_+
(E_0) \rho ' ( {\tilde \Gamma}_+ (E_0) - t + 1 ) \|$. 
As in the first part of the proof, 
this requires that we choose $\lambda$ sufficiently
small so that 
\beq
\Sum_{j=2}^6 \lambda^j \; \| K_j (E_0) \| \; < \; (1 - 2 \kappa ) /2 .
\eeq
Let $\lambda_0^{(2)} > 0$ be chosen so that $| \lambda | < \lambda^{(2)}$
guarantees that (6.24) holds. It is clear that $\lambda_0^{(2)}$ 
depends on the gap size, the location of $E_0$ relative to the
gap edges $B_\pm $, and on the relative $H_0$-bounds of
$H_{j, \omega}^\Lambda$.
We now choose $\lambda_0 = \mbox{min} \; 
( \lambda_0^{(1)} , \lambda_0^{(2)} )$. We this choice, we can
continue the proof as in section 4 and arrive at the analog of (4.19).
An examination of (6.19)--(6.20) shows that the effective perturbation
obtained by varying the $k^{th}$ coupling constant has the correct form
so that the trace estimate result of section 5 applies. $\Box$  

\vspace{.1in}
\noindent
{\bf Corollary 6.2.} {\it Let $H_\omega ( \lambda) = H_0 + \lambda
H_{1, \omega} + \lambda^2 H_{2, \omega}$ be a random family of operators
satisfying either 1) hypotheses (H1b), (H2), (H4)--(H6), or 2) hypotheses
(H1a), (H2), (H4)--(H6). Suppose further that the family is ergodic.
Then, for any closed interval $I \subset \R \backslash \sigma (H_0)$,
there exists a constant $0 < \lambda_0 ( I)$ such that
for any $| \lambda| < \lambda_0 (I)$, the
integrated density of states for $H_\omega ( \lambda)$
on $I$ is H{\"o}lder continuous
of order $1/q$, for any $q > 1$. }

\subsection{Localization}

The Wegner estimate plays a key role in the proof of
localization for families of random operators. The Wegner estimate for
nonsign definite potentials proved here can be used to prove band-edge
localization as, for example in \cite{[BCH]} and \cite{[CHT]}, under
some additional assumptions. As the theory is not yet in optimal form,
we indicate the lines of the proof and will return to this in a another 
paper. In order to prove localization, we need to establish an initial
length scale estimate for the resolvent of the local Hamiltonian
$H_\Lambda$. At present, only the method of Lifshitz tails appears to
provide this estimate for the case of nonsign definite single-site
potentials. The standard method (cf.\ \cite{[BCH],[KSS],[vDK]})
depends on the monotonic variation of the eigenvalues of $H_\Lambda^X$
with respect to the
coupling constants that does not hold in the nonsign definite case.
However, there is no satisfactory result for
Lifshitz tails, either at the bottom of the spectrum or at internal
band edges, for the case of nonsign definite single-site potentials.

\vspace{.1in}
\noindent
1. Bottom of the spectrum $\Sigma$.

\vspace{.1in}
\noindent
Let us suppose that the IDS exhibits a weak Lifshitz tail near
$\mbox{inf}\; \Sigma$, in the
sense that 
\beq 
\lim_{ E \rightarrow \mbox{inf} \; \Sigma} ( E - \mbox{inf}
\; \Sigma )^{-N} N(E) = 0,
\eeq
for any $N \in \N$, for 
the models described in this paper, for which
a Wegner estimate holds. Then a standard argument, as in
\cite{[Klopp]}, proves localization below $\mbox{inf} \; \sigma (
H_0^A)$.  Indeed, let $N_\Lambda ( E ) $ be the
number of eigenvalues of $H_\Lambda$ in the interval $[\mbox{inf}
\; \Sigma , E ]  \subset [
\mbox{inf} \; \Sigma , \mbox{inf} \; \sigma (H_0^A)]$. The standard
argument uses the following estimate on the finite 
volume counting function by the IDS: 
\beq
\frac{ N_\Lambda ( E ) }{ | \Lambda |} \leq N ( E ) .
\eeq
>From this, we conclude that the probability that $H_\Lambda$
has no eigenvalues in a small interval of size $\epsilon$ near
$\mbox{inf} \; \Sigma$ is less than $C_N | \Lambda | \epsilon^N$,
for any $N > 0$. This is sufficient to prove an initial length scale estimate
using the Combes-Thomas argument, upon taking $\epsilon$ to depend on the
initial length scale.   

\vspace{.1in}
\noindent
2. Internal gaps. 

\vspace{.1in}
\noindent
The case of internal band-edges is more
complicated.   
We need two results. First, we need weak internal Lifshitz tails (6.25) at the
edges of the internal bands. Secondly, we need an analog of (6.26) in
order to recover information about the finite-volume counting function
near a band edge from the IDS. 

Concerning the first point,
Klopp \cite{[Klopp2]} recently proved
the following. Let $H_0$ be a periodic Schr{\"o}dinger operator and
$V_\omega$ an Anderson-type potential with single-site potentials $u_j
( x ) = u ( x - j)$, and the single-site potential $u \geq 0$ is
bounded with compact support. We assume that $H_0$ has an open
spectral gap $G = ( B_- , B_+ )$. The common density $h_0$ of the random
variables is assumed to be supported in $[0 , M]$, for some $M > 0$,
and $h_0$ vanishes more slowly than an exponential as $ \lambda
\rightarrow 0^+$. We assume that $M$ is 
sufficiently small so that the deterministic spectrum $\Sigma$ of
$H_\omega = H_0 + V_\omega$ 
has an open gap ${\tilde G} = ({\tilde B}_- , B_+)$, for some $B_-
\leq {\tilde B}_-  < B_+$. Klopp proves that 
the IDS $N(E)$ of $H_\omega$ satisfies
\beq
\lim_{E \rightarrow B_+ }
\frac{ log | log ( N(E) - N(B_+ ) ) | }{ log E } = - \frac{d}{2} ,
\eeq
where the limit is taken for $E \geq B_+$,
if and only if the IDS $n(E)$ for the periodic operator $H_0$ is
nondegenerate in the sense that
\beq
\lim_{E \rightarrow B_+ } \frac{ log ( n(E) - n(B_+) ) }{log E} =
\frac{d}{2}.
\eeq
This result is, of course, stronger than the required weak Lifshitz
tails behavior (6.25) near $B_+$. 

This result can be extended to a special case of nonsign definite 
single-site potentials $u$ as follows. 
The periodic Schr{\"o}dinger operator
$H_0$ admits a direct integral decomposition over the 
flat d-torus $T^d$, with fibre
operators $H_0 ( \theta ), \theta \in T^d$. For each $\theta \in T^d$,
the operator $H_0 ( \theta )$ has a compact resolvent and hence
discrete spectrum with eigenvalues $E_j ( \theta )$. Let $B$ be a band
edge of $\sigma ( H_0 )$. At most finitely-many Floquet
eigenvalues satisfy $E_j ( \theta ) = B, \theta \in T^d$. Let 
$\Pi_B$ be the projector for the subspace generated by the
corresponding Wannier functions in $L^2 ( \R^d )$. 
It follows from \cite{[Klopp2]} that, in the small coupling constant
limit, positivity of the potential 
$V_\omega$, and thus of the single-site potential $u$, 
is sufficient, but not necessary, for the proof. Rather, one 
requires that $\Pi_B u \Pi_B \geq 0$.
Hence, the proof of \cite{[Klopp2]} can be modified to accommodate nonsign
definite potentials $u = u_+ - u_-$ provided the negative part $u_-$
is small in the sense that $\Pi_B u_+ \Pi_B \ \neq 0$ and
$ \Pi_B (u_+-u_-) \Pi_B \geq \epsilon \Pi_B u_+ \Pi_B$. This is a condition on
$u$ and the background operator $H_0$.
We remark that one can also apply this argument to Schr\"odinger
operators with random magnetic fields, cf.\ \cite{[Ghribi]}.

To address the second problem, we refer to the recent article of Klopp
and Wolff \cite{[KloppWolff]}. They prove a general result, valid for
all dimensions $d \geq 1$, which provides the analog of (6.26) if
Lifshitz tails are known to exist.
Let $H_\Lambda^P$ be the operator $H_0 + V_\omega$ restricted to a
cube of side length $L$ with periodic boundary conditions. The Wegner
estimates, Theorems 1.1 and 1.2, proved for the local Hamiltonian
$H_0 + V_\Lambda$, also hold for the operator
$H_\Lambda^P$. A version of Proposition 7.1 of Klopp and Wolff, 
that holds even when the single-site potential $u$ is nonsign
definite, implies that for any $\nu > 0$,
\beq
\proba \{ \mbox{dist} ( \sigma ( H_\Lambda^P ) , B_\pm  ) \leq L^{-1} \}
\leq L^{- \nu } .
\eeq
This is sufficient to establish the initial length scale hypothesis
using a Combes-Thomas argument.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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