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Wegner estimates, random operators, localization, integrated density of states, spectral shift function
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\begin{document}


\begin{titlepage}

\begin{center}

{\bf THE WEGNER ESTIMATE  \\
AND THE INTEGRATED DENSITY OF STATES FOR SOME RANDOM OPERATORS}


\vspace{0.2 cm}

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{\bf J.\ M.\ Combes \footnote{
D\'epartement de Math\'ematiques, Universit\'e de Toulon et du
Var, 83130 La Garde, France.}$^,$\footnote{
Supported in part by CNRS and NATO Grant
CRG-951351.} \\ P.\ D.\ Hislop \footnote{
Mathematics Department, University of Kentucky, Lexington KY
40506-0027 USA.}$^,$\footnote{
Supported in part by NSF grant DMS-9707049
and NATO Grant CRG-951351.}
}

\vspace{0.2 cm}

{\ten Centre de Physique Th\'eorique\footnote{
Unit\'e Propre de Recherche 7061} \\
CNRS Luminy Case 907 } \\
{\ten F-13288 Marseille Cedex 9 France }

\vspace{0.2 cm}

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

\vspace{0.2 cm}

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

\vspace{0.2 cm}

{\bf Shu Nakamura \footnote{Supported in part by JSPS grant Kiban
B 09440055.}}

\vspace{0.2 cm}

{\ten Graduate School of Mathematical Sciences \\
University of Tokyo \\
3-8-1, Komaba, Meguro-ku, Tokyo 153--8914, Japan}


\end{center}

\vspace{0.3 cm}

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

\noindent
The integrated density of states (IDS) for random operators is
an important function describing many physical characteristics of
a random system. 
Properties of the IDS are derived from the Wegner estimate
that describes the influence of finite-volume perturbations 
on a background system.
In this paper, we present a simple proof of the Wegner estimate
applicable to a wide variety of random perturbations
of deterministic background operators. The proof yields the
correct volume dependence of the upper bound.
This implies the local H\"older continuity of
the integrated density of states at energies in the unperturbed
spectral gap. The proof depends on the 
$L^p$-theory of the spectral shift function (SSF), for $p \geq
1$, applicable to pairs of self-adjoint operators whose difference is in
the trace ideal ${\cal I}_p$, for $0 < p \leq 1$.
We present this and other results on the SSF due to other authors.
Under an additional condition of the
single-site potential, local H\"older continuity is proved at all
energies. Finally, we present extensions of this work to random
potentials with nonsign
definite single-site potentials.

\vspace{0.5 cm}

\noindent
\today

\end{titlepage}



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

Much progress has been made in the study of random systems describing the
propagation of electrons and classical waves in randomly perturbed media.
In this paper, we concentrate on the Wegner estimate and
on some recent results concerning the 
integrated density of states for random operators 
on $\R^d$, for $d \geq 1$.
The Wegner estimate also plays a 
key role in the proof
of localization for random systems, but we will not discuss
localization here, and refer the reader to
various references \cite{[BCH],[CHT],[FK1],[FK2],[Klopp],[KSS],[vDK]}. 
The Wegner estimate is a fine analysis of the effect of finite-volume, random
perturbations $V_\Lambda$, for a bounded region $\Lambda \subset \R^d$, 
on the spectrum of a
self-adjoint operator $H_0$, 
describing the background, unperturbed, situation. 
More specifically,
a {\it Wegner estimate} is
an upper bound on the probability that the spectrum of 
the local Hamiltonian $H_{\Lambda}$
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 properties
of the IDS. The rate of vanishing of the upper bound as $\eta \rightarrow 0$
determines the continuity of the IDS.

We present a new, simple proof of 
a good Wegner estimate applicable to random operators
with some additional conditions on the single-site potential.
This proof uses more directly the ideas of Krein, Birman, and Simon
than the proof in \cite{[CHN]}. 
As in \cite{[CHN]}, the proof employs
$L^p$-estimates on the
spectral shift function related to the single-site perturbation.
This result allows us to prove exponential localization and the
local H\"older continuity of the integrated density of states for more
models than previously known.

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 $X=A$.
Additively perturbed operators describe electron propagation,
and multiplicatively perturbed operators describe the propagation of
acoustic and electromagnetic waves.
We refer to \cite{[CHT]} for
a further discussion of the physical interpretation of these operators.
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 $A_\Lambda = 1 + V_\Lambda$ is assumed 
to be 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\"odinger 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 $W_+ \in K_d^{loc} ( \R^d)$.

In this note, we will
limit ourselves to Anderson-type perturbations. These methods can
also be used to treat the breather-type perturbations, and we
refer the reader to \cite{[CHN]} for the details.
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
small enough so that one of the conditions 
(H3), (H3a), (H3b), or (H3c) (given ahead) holds. 
To simplify the
discussion in this paper, however, we take $\xi_i ( \omega ') = 0$.
The functions
$u_i$ are nonzero and 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. 
When the sum in (1.5) extends over all
the lattice points $\Z^d$, we write $V_\omega$ for the potential
and $H_{\omega}^X$, with the operator for $X=A$ 
given by (1.2) with $V_\Lambda$
replaced by $V_\omega$, and similarly for (1.1) in the case $X=M$.  

The hypotheses for the models are listed here. The first two (H1)
and (H1a) concern the spectrum of the unperturbed operator
$H_0^X$. The second is a local compactness condition on $H_0^X$. 
There are four different conditions on the single-site potentials
$u_j$. Finally, there are two possible conditions on the random
variables $\lambda_k ( \omega)$.
We note that the Wegner
estimate is a local estimate so that for a finite region $\Lambda \subset
\R^d$, only a finite number of single-site potentials are involved.
We denote the ball of radius $R > 0$ about the origin by $B(R)$, and
by $\Lambda_r (k) = \{ x \in \R^d \; | \; | k_j - x_j | < r / 2 , j = 1,
\ldots , d \}$, the cube of side length $r > 0$, centered at $k$.

\begin{enumerate}
\item[(H1)]
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[(H1a)]
The self-adjoint operator $H_0^X$ is essentially
self-adjoint on $C_0^{\infty} ( \R^d )$, and $H_0^X$ is semi-bounded with
$\sigma ( H_0^X ) \subset [ M_0 , \infty )$, for some $M_0 > - \infty$.

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

\item[(H3)] The single-site potentials $u_k , k \in \Z^d$,
are nonzero. For the Anderson-type model (1.5), we assume that there exists
$R > 0$ so that $u_k \in C_0 ( B(R))$, and that $u_k \geq 0$ 
for each $k \in \Z^d$.
Furthermore, we assume that the family $\{ u_k \; | \; k \in \Z^d \}$
is equicontinuous. 

\item[(H3a)] In addition to (H3), we
assume that there exists $\epsilon_1 > 0$ so that $u_k \geq \epsilon_1$ 
on $\Lambda_1 ( 0)$. 

\item[(H3b)] In addition to (H3),
we assume that there is a nonempty subset $B \subset \Lambda_1 (0)$
so that $\mbox{supp} \; u_k \subset B$.

\item[(H3c)] The single-site potentials $u_k \in C_0 ( \R^d )$.
For each $k \in \Z^d$, there exists a nonempty open set $B_k$
containing the origin 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
$[m , M ]$, for some constants $( m, M)$
with $- \infty < m < M < \infty$. The density $h_0$ satisfies
$\| h_0 \|_{\infty} < \infty $, where the sup norm is defined
with respect to the probability measure $\P$.

\item[(H4a)] In addition to (H4),
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 these hypotheses 
imply the essential self-adjointness
of $H_{\omega}^A$ on $C_0^{\infty} ( \R^d )$ (see \cite{[CHT]} for
the $X=M$ case).
As stated in hypothesis (H4), 
we will assume that the random variables are independent, and identically
distributed, but the results hold in the correlated case, and in the
case that the supports of the single-site potentials are not necessarily 
compact (cf.\ \cite{[CHM2],[KSS2]}). 

Our main results under these hypotheses on the unperturbed operator
$H_0^X$, and the local perturbation $V_\Lambda$,
concern two cases depending upon whether or not the
single-site potentials are sign-definite.
For the case of sign-definite single
site potentials, hypotheses (H3), (H3a), or (H3b),
our main theorem is the following.

\vspace{.1in}

\noindent
{\bf Theorem 1.1}. {\em Assume (H1), (H2), (H3), and (H4).
For any $E_0\in G = (B_-,B_+)$, for any $q > 1$,
and for any $\eta < \frac{1}{2}\, \dist (E_0 , \sigma(H_0^X) )$,
there exists a finite constant $C_{E_0}$, depending on
$[\dist(\sigma(H_0^X), E_0)]^{-1}$, the dimension $d$, and $q > 1$, such that:
\beq \P \left\{ \dist ( E_0 , \sigma(H_\Lambda^X) )
\leq \eta  \right\} \leq C_{E_0}\eta^{1/q} \, |\Lambda|\ .
\eeq
If, in addition, 
that the single-site potential satisfies (H3a), then the result (1.7)
holds for $q = 1$ and
for $H_\omega^X \; | \; \Lambda$, with Dirichlet boundary conditions
on the boundary of $\Lambda$, and for any $E_0 \in \R$.
}   

\vspace{.1in}

There are several prior results on the Wegner estimate for
multidimensional, continuous Schr\"odinger operators with Anderson-type
potentials constructed from fixed-sign, single-site potentials. Kotani and
Simon \cite{[KS]} proved a Wegner estimate with a $| \Lambda
|$-dependence for Anderson models with overlapping single-site
potentials satisfying (H3a). 
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],[Faris],[FK1],[FK2]}. 
These methods require a spectral averaging theorem (cf.\ \cite{[CHM]} and
references therein).
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 that are assumed to be simply
H\"older continuous. He also obtains a $| \Lambda |^2$-dependence.
These proofs, and the proof in this paper, do not require spectral
averaging. 

An immediate consequence of Theorem 1.1 concerns the IDS.
In order to discuss the IDS, we need to assume that the model is
ergodic. For example, we can take $u_j = u$, for all $j \in
\Z^d$. Let $\Sigma$ denote the deterministic spectrum of the
family $H_\omega$.

\vspace{.1in}

\noindent
{\bf Theorem 1.2}. {\em 
Assume (H1), (H2), (H3), and (H4), and that the model is ergodic.
The integrated density of states is
H\"older continuous of order $1/q$, for any $q > 1$,
on the interval $(B_-,B_+)$.
If, in addition, 
we assume that the single-site potential satisfies (H3a), then the integrated
density of states is locally Lipschitz continuous on $\Sigma$.} 

\vspace{.1in}

Concerning the second case of nonsign-definite single-site
potentials, hypothesis (H3c), our main results are not quite as
general. The first results concern energies below the bottom of the
spectrum of $H_0^A$, and are given in Theorem 4.1 and Corollary 4.2.
For the case of energies in an unperturbed spectral gap of $H_0^X$,
we must suppose that the disorder is sufficiently small. The main 
results for this case are given in Theorem 4.3.
These are, however, the first general results for nonsign-definite 
single-site potentials. Some related results concern the IDS 
for magnetic \Schr\ operators with unbounded Gaussian random potentials
studied by Hupfer, Leschke, M\"uller, and Warzel \cite{[HLMW]}. They
prove a Wegner estimate for these models and that the IDS is absolutely
continuous at all energies. Veseli\'c \cite{[Veselic]} recently considered
the nonsign-definite case for a restricted class of Anderson-type potentials
that we discuss at the end of section 4.

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 multiplicatively perturbed model (1.1) with
minor modifications.  
Recently, Nakamura \cite{[Nakamura]} showed the uniqueness of the IDS, 
in the sense that it is independent of
Dirichlet or Neumann boundary conditions, in the case of Schr\"odinger
operators with magnetic fields. The same proof applies to the
multiplicatively perturbed model.  
It is interesting to note that the proof uses the $L^1$-theory of the
spectral shift function.
Another proof of the uniqueness of the IDS for \Schr\ operators with magnetic
fields is given by \cite{[DIM]}.

The contents of this paper are as follows. The $L^p$-theory of the
spectral shift function (SSF) for $p>1$ is developed in section 2.
We also give a summary of other estimates on the SSF.
We give a simple proof of Wegner's estimate in section 3.
This proof is different from the proof on \cite{[CHN]} and especially
transparent.  In section 4, we extend the results to Anderson-type potentials
with nonsign-definite single-site potentials following
\cite{[HislopKlopp]}.

\vspace{.1in}  

\noindent
{\bf Acknowledgements.} We thank W.\ Kirsch, A.\ Klein,
V.\ Kostrykin, R.\ Schrader, B.\ Simon,
K.\ Sinha, P.\ Stollmann, 
and G.\ Stolz for useful discussions. 

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

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

The $L^p$-theory of the spectral shift function for $p \in [1 ,
\infty ]$ can be viewed as an interpolation between the two
well-known cases of $p = 1$ and $p = \infty$.
Let us recall the $L^1$ and $L^\infty$-theory, 
which 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 separable
Hilbert space $\cal H$ having the property
that $V \equiv H - H_0$ is in the trace class. We denote by $\| V
\|_1$ the trace norm of $V$. 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
At the other extreme, $p = \infty$, we
recall that for a perturbation $V$ of rank
$K$, the SSF is essentially bounded and satisfies the bound
\beq
\| \xi ( \cdot ; H , H_0 ) \|_{L^\infty} \; \leq \; K .
\eeq
In particular, for rank one perturbations, we have
\beq
| \xi ( \lambda; H , H_0 ) | \leq 1 .
\eeq
This implies that $\| \xi ( \cdot ; H , H_0 ) \|_{L^\infty} \;
\leq K$ for finite-rank perturbations $V$.

Let us now consider the cases $1 < p < \infty$ (cf.\
\cite{[Simon2]}).
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 $p \geq 1$, if
\beq
\Sum_{j} \; \mu_j (A)^{1/p} < \infty .
\eeq
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 $V = H - H_0 \in {\cal I}_{1/p}$. Our main theorem is the
following.

\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
}  

Notice that this theorem provides the correct estimates for the
endpoints $p = 1$ and $p = \infty$, where we take $1 / \infty =
0$, and that the bound on the right side of (2.9) in this case is a
constant depending only on the rank of $V$.
In this sense, Theorem 2.1 is an interpolation theorem for the SSF in
$L^p$-spaces for $p \in [0 , \infty]$.
The proof of Theorem 2.1 follows the same lines as the proof for the
trace class case as found in, for example, Yafaev \cite{[Yafaev]}.
This bound was recently improved by Hundertmark and Simon
\cite{[HundertmarkSimon]}.

\vspace{.1in}

\noindent
{\bf Theorem 2.2. \cite{[HundertmarkSimon]}}
{\it Suppose that $H_0$ and $H$ are
self-adjoint operators so that $V = H - H_0 \in {\cal I}_1$.
Let $F : [0 , \infty ) \rightarrow \R^+$ be a nonnegative, convex
function with $F(0) = 0$. Then, the SSF $\xi ( \lambda; H , H_0 )$
satisfies the bound
\beq
\int_{\R} \; F ( | \xi ( \lambda ; H , H_0 ) | ) \; d\lambda 
\; \leq \; \Sum_{j=1}^\infty \; [ F( j) - F(j-1) ] \; \mu_j ( V).
\eeq
}

\vspace{.1in}
\noindent
If one takes $F(t ) = t^p , p \geq 1$ in Theorem 2.2, one obtains
\beq
\int_{\R} | \xi ( \lambda ; H , H_0 ) |^p ~d \lambda = \Sum_j ( j^p - (j-1)^p )
\mu_j ( V) .
\eeq
The bound is better than the bound in Theorem 2.1,
and provides an optimal upper bound for the $L^p$-norm of the SSF.

Other integral bounds on the SSF were obtained by Pushnitski
\cite{[Pushnitski]}. Among them, we mention the following result
concerning \Schr\ operators. 
We recall that for unbounded operators, such as \Schr\ operators,
the SSF is defined through the invariance principle.
Suppose that $H_0$ and $H$ are two self-adjoint operators and $g
: \R \rightarrow \R$ is a function so that $[ g(H) - g(H_0)]
\equiv V_{eff} \in {\cal I}_1$. Then, we define the SSF
for the pair $(H_0 , H)$ by
\beq
\xi ( \lambda ; H , H_0 ) \equiv \; \mbox{sgn} (g') \; \xi (
g( \lambda ) ; g(H) , g(H_0) ) .
\eeq

\vspace{.1in}

\noindent
{\bf Theorem 2.3. \cite{[Pushnitski]}} {\it Let $d \geq 3$. 
Suppose that $H_0 = - \Delta$
and $H = H_0 + V$, where the potential $V \geq 0$ and satisfies
the bound
\beq
V(x) \leq C_0 ( 1 + \| x\|)^{- \rho}, ~~\mbox{for} ~~\rho > d.
\eeq
Then, there exists a finite constant $C_1 \geq 0$, such that for
any nonnegative, monotone decreasing function $f$, we have
\beq
\int_0^\infty \; \xi ( \lambda; H , H_0) f( \lambda) ~d\lambda
\; \leq \; 
C_1 \int_0^\infty \; \lambda^{(d/2) - 1} f(\lambda) ~d\lambda \;
\int_{\R^d} V(x) ~dx .
\eeq
}

\vspace{.1in}

In addition to these integral bounds on the SSF, we would like to
mention the pointwise bound of A.\ V.\ Sobolev \cite{[Sobolev]}.

\vspace{.1in}

\noindent
{\bf Theorem 2.4. \cite{[Sobolev]}} {\it Suppose that $H_0$ and $H$ are
self-adjoint operators so that $V = H - H_0 \in {\cal I}_1$.
Also suppose that
\beq
\lim_{\epsilon \rightarrow 0^+} \| |V|^{1/2} ( H_0 - \lambda - i
\epsilon )^{-1} |V|^{1/2} \|_{1/p} \; < \infty ,
\eeq 
for some $p \geq 1$. Then, there exists a finite constant 
$C_p > 0$, so that for all $\lambda > 0$,
the SSF $\xi ( \lambda; H , H_0 )$ satisfies the bound
\beq
| \xi ( \lambda  ; H , H_0 ) | \; \leq \; C_p \| |V|^{1/2} ( H_0
- \lambda  - i 0 )^{-1} |V|^{1/2} \|_{1/p}^{1/p}.
\eeq
}

%\vspace{.1in}  

For one-dimensional \Schr\ operators, Kostrykin and Schrader \cite{[KS1]}
proved the following pointwise bound on the SSF.

\vspace{.1in}

\noindent
{\bf Theorem 2.5. \cite{[KS1]}}
{\it Let $H_0 = - d^2 / dx^2$ be the self-adjoint 
Laplacian on $L^2 ( \R )$, and
let $H = H_0 + V$, with the potential $V$ satisfying
\beq
\int_{\R} ( 1 + |x|^2 ) |V(x)| ~dx < \infty.
\eeq
Then, there exists a constant $0 \leq C_V < \infty$, depending on
$V$ and independent of $\lambda$, so that for all
$\lambda \in \R$, the SSF $\xi ( \lambda; H , H_0 )$ satisfies
\beq
| \xi ( \lambda; H , H_0 ) | \; \leq \; C_V.
\eeq
Moreover, there is a constant $0 \leq C_0 < \infty$,
independent of $V$ and $\lambda > 0$,
so that for all for $\lambda > 0$, the SSF $\xi ( \lambda; H ,
H_0 )$ satisfies
\beq
| \xi ( \lambda; H , H_0 ) | \leq   
C_0 \left\{ \frac{1}{2 \sqrt{\lambda} } \int_{\R} |V(x)| ~dx +
\frac{1}{ 4 \lambda } \left[ \int_{\R} |V(x)| ~dx \right]^2
\right\} .
\eeq
}

\subsection{Various Identities for the SSF}

In this subsection, we study various identities for the SSF. In this
setting, we consider a one-parameter family of self-adjoint operators
$H_\lambda, \lambda \in J \equiv [\lambda^- , \lambda^+ ] \subset
\R$. 

\begin{enumerate}
\item The family $\lambda \in J \rightarrow H_\lambda$ is self-adjoint
on the same domain $D_0$. The family is weakly differentiable on $J$ with
the derivative $\dot{H}_\lambda \equiv ( d H_\lambda / d \lambda
) \in {\cal I}_1$.
\item The map $\lambda \in J \rightarrow \| \dot{H}_\lambda \|_1 $ is
continuous.
\end{enumerate}

\vspace{.1in}
\noindent
The first fundamental result is the Birman-Krein trace formula
(cf.\ \cite{[BY],[Yafaev]}).

\vspace{.1in}
\noindent
{\bf Proposition 2.5.} {\it For any $f \in C_0^\infty ( \R^d)$, we have
\beq
Tr \{ f(H_{\lambda^+}) - f( H_{\lambda^-} ) \} =  \int_{\R}  
\; f'(E) \xi ( E ; H_{\lambda^+} , H_{\lambda^-} ) ~dE .
\eeq
}


%\vspace{.1in}

\noindent
We also have a form of the spectral averaging theorem
\cite{[CHM],[Simon1]}.

\vspace{.1in}
\noindent
{\bf Proposition 2.6.} {\it
Under the conditions stated above, we have 
\beq
\int_J Tr \{ E_\lambda ( I ) \dot{H}_\lambda \} ~d \lambda = 
\int_I \; \xi ( E ; H_{\lambda^+} , H_{\lambda^-} ) ~dE .
\eeq
}

\vspace{.1in}
\noindent
{\bf Sketch of the Proof.} 
We will sketch the proof of this identity by working formally.
First, for any $f \in C_0^\infty (\R)$,  we note the basic identity
\beq
\frac{d}{ds} Tr \{ f(H(s)) \} = Tr \{ f'(H(s)) \; \dot{H} (s) \}
.
\eeq
We now integrate this equation over the interval $J$,
\bea
\int_J \frac{d}{ds} Tr \{ f(H(s)) \} ~ds & =  & Tr \{ f(H(\lambda^+))
- f(H(\lambda^-)) \} \nonumber \\
 & = & \int_J Tr \{ f'(H(s)) \; \dot{H} (s) \} ~ds \nonumber \\
 & = & \int_{\R} f'(E) \xi (E ; H(\lambda^+) , H( \lambda^-) ) ~dE .
\eea
We used the Birman-Krein trace formula (2.20). We now use the
spectral theorem for $H(s)$ to write the integrand on the second
line of (2.23) as
\beq
Tr \{ f'(H(s)) \; \dot{H} (s) \} = \int_{\R} f'(E) d \mu_s (E) ,
\eeq
where $d \mu_s (E)$ is the measure on $\R$ with the formal density
given by $Tr \{ E_s(E) \dot{H} (s) \}$, with $E_s ( \cdot )$ 
the spectral family of $H(s)$. We integrate the identity (2.24)
over $J$ to obtain
\beq
\int_J Tr \{ f'(H(s)) \; \dot{H} (s) \} = \int_{\R} f'(E) \;
\int_J ~ds \; d \mu_s (E) .
\eeq
Comparing the formula on the right in (2.25) with the one on the
third line of (2.23), we obtain
\beq
\xi (E ; H(\lambda^+) , H( \lambda^-) ) ~dE =
\int_J \; ds \;  d \mu_s (E) .
\eeq
Integrating this identity over an interval $I \subset \R$, we
obtain
\beq
\int_I \xi (E ; H(\lambda^+) , H( \lambda^-) ) ~dE =  
\int_J Tr \{ E_s (I) \dot{H } ( s) \}  ~ds ,
\eeq
proving the proposition. $\Box$

Let us note that if we formally take $f$ so that $f' (x) = \chi_I (x)$,
then the result (2.27) follows from the second and third lines of (2.23).

\subsection{The Integrated Density of States}

The integrated density of states (IDS) is defined as follows. 
We consider the Hamiltonian $H_\omega$ restricted to
a cube $\Lambda$ with Dirichlet boundary conditions on $\partial
\Lambda$, the boundary of the cube. 
This operator, denoted by $H_\Lambda^{D}$, has discrete
spectrum. Let $N_\Lambda^{D} ( \lambda )$ be the number of
eigenvalues of $H_\Lambda^{D}$, including multiplicity, less than
or equal to $\lambda$. If the following limit exists,
\beq
\lim_{ | \Lambda | \rightarrow \infty } \frac { N_\Lambda^{D} (
\lambda ) }{ | \Lambda | } \equiv N ( \lambda ),
\eeq
it is called the IDS. It is known for the models discussed
here that $N( \lambda )$ exists, is nonrandom, and a monotone
increasing function of $\lambda$.
We refer to \cite{[Kr1]} for a proof of this result.

There is an interesting connection between
the IDS $N(\lambda)$
and the SSF for the pair $( H_\Lambda , H_0)$, with $H_\Lambda =
H_0 + V_\Lambda$, that involves the {\it spectral shift density} introduced by
Kostrykin and Schrader \cite{[KS1],[KS2]}. 
For any $g \in C_0^1 ( \R)$, they prove that the following limit
\beq
\lim_{ | \Lambda | \rightarrow \infty } \int g(\lambda) \;
\frac{ \xi ( \lambda ; H_0 + V_\Lambda , H_0 ) }{ | \Lambda | } ~d \lambda 
\eeq
exists and is nonrandom. 

\vspace{.1in}

\noindent
{\bf Theorem 2.5. \cite{[KS2]}} {\it For the models discussed
here, the integrated density of states
$N(E)$ exists, and belongs to
$L_{loc}^q ( \R)$, for any $q > 1$. Furthermore, if $N_0 ( \lambda
)$ is the IDS for $H_0$, we have the following identity,
for any $g \in C_0^1 ( \R)$,
\beq
\lim_{ | \Lambda | \rightarrow \infty } \int g(\lambda) \;
\frac{ \xi ( \lambda ; H_0 + V_\Lambda , H_0 ) }{ | \Lambda | }
~d \lambda = \int g(\lambda) ( N_0 ( \lambda ) - N( \lambda )) ~d
\lambda .
\eeq
}

We remark that the proof of $N(\lambda) \in L_{loc}^q ( \R)$, for
any $q > 1$, uses the estimate (2.9).



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

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\setcounter{chapter}{3}
\setcounter{equation}{0}

\section{Proof of Wegner's Estimate}\label{S.3} 

We first formulate Wegner's estimate in general terms for a
family of random operators satisfying some assumptions. We then
show that these assumptions are verified for some Anderson-type
models.


\subsection{An Abstract Wegner's Estimate}

We give a rather general proof of a Wegner estimate under the
following assumptions. 

\begin{enumerate}
\item[(A1)] The operator $H_\Lambda$ depends on $N = {\cal O} ( |
\Lambda | ) $ random variables $\{ \lambda_1 , \ldots , \lambda_N
\} $, distributed according to the distribution $h_0 ( \lambda ) d
\lambda $ with $h_0 \in L^\infty (( \lambda^- , \lambda^+ ))$, for
finite $\lambda^- , \lambda^+ \in \R$.
\item[(A2)] For a bounded interval $I \subset \R$,
the following identity holds for some finite $C_0 >
0$:
\beq   
Tr \{ E_\Lambda ( I_\eta ) \} \;  \leq \;  C_0  Tr
\left\{  \Sum_{j =1}^N  \left( \frac{ \partial
H_\Lambda }{ \partial \lambda_j } \right) E_\Lambda ( I )
\right\} .
\eeq  
\item[(A3)] Let $\omega_j^\pm \equiv \{ ( \lambda_1, \ldots,
\lambda_j = \lambda^\pm , \ldots , \lambda_n ) \}$ be
the set of all configurations for which the random variable
$\lambda_j$ is fixed at the minimum, respectively, maximum,
value. For any $p > 1$,
let $g(x) = (x + M_0)^{-k}$, for some $k > (pd/2) +2$.
Then, there is a finite constant $C = C(p, d, M_0 ) > 0$ so that
\beq
\sup_{j = 1 , \ldots , N} \left( \sup_{k \neq j } \;
\| g(H_{\omega_j^+} ) - g( H_{\omega_j^-} ) \|_{1/p} \right) \; \leq
C < \infty .
\eeq
\end{enumerate}

These assumptions can be modified for the multiplicatively
perturbed model (1.1), but we don't do this here, and concentrate
on the additively perturbed model (1.2).
 
\vspace{.1in}
\noindent
{\bf Theorem 3.1.} {\it Assume that the random family of
Hamiltonians satisfy assumptions (A1)--(A3). Then, for any $q >
1$, there exists a finite constant $C_W = C_W ( q,d,C_0, C_1, k,
\mbox{dist} \; ( I , M_0))
> 0$, so that
\beq
\E \{ Tr ( E_\Lambda ( I ) )  \}  \; \leq \; C_W \| h_0 \|_\infty
|I|^{1/q} | \Lambda | .
\eeq
}

\vspace{.1in}

\noindent
{\bf Proof.} \\  
\noindent
1. Due to hypothesis (A2), we have
\bea
\E \{ Tr ( E_\Lambda ( I ) ) \} & \leq & C_0 \E \left[ Tr
\left\{ \Sum_{j =1}^N \left( \frac{ \partial
H_\Lambda }{ \partial \lambda_j } \right) E_\Lambda ( I )
\right\} \right] \nonumber \\ 
 & \leq & C_0 \Sum_{j = 1 }^N  \; \E \left\{ Tr
\left( \frac{ \partial
H_\Lambda }{ \partial \lambda_j } \right) E_\Lambda ( I )
\right\} .
\eea
As usual, we select one random variable, say $\lambda_j$, and
integrate with respect to it, using positivity,  
\bea
\lefteqn{\E \{ Tr ( E_\Lambda ( I ) ) \} } \nonumber \\
 & \leq & C_0 \Sum_{j=1}^N  \int_I \Pi_{k \neq
  j} h_0 (\lambda_k) d \lambda_k \int_{[\lambda_j^- , \lambda_j^+
]}  h_0 ( \lambda_j) d \lambda_j 
  \; Tr \left\{ \left( 
  \frac{ \partial H_\Lambda }{ \partial \lambda_j } \right) 
  E_\Lambda ( I ) \right\} \nonumber \\
& \leq & C_0 \| h_0 \|_\infty \E' \left\{ \int_{[\lambda_j^- ,
\lambda_j^+ ]}  ~d
\lambda_j Tr \left[ \left( \frac{ \partial H_\Lambda }{ \partial
\lambda_j } \right) E_\Lambda ( I ) \right] \right\} ,
\eea
where $\E'$ denotes the expectation with respect to the other
random variables $\lambda_k$, for $k \neq j = \{ 1, \ldots, N \}$.


\noindent
2. We use the spectral averaging formula, Proposition 2.6, to
evaluate the integral on the right side of (3.5). This gives
\beq
\int_{[\lambda_j^- , \lambda_j^+ ]} ~d
\lambda_j Tr \left[ \left( \frac{ \partial H_\Lambda }{ \partial
\lambda_j } \right) E_\Lambda ( I ) \right] = \int_{I}  ~dE \; \xi ( E ;
H_{\lambda_j^+ } , H_{\lambda_j^- }) .
\eeq
At this stage, we use the $L^p$-estimate on the SSF and
H\"older's inequality. Let $\chi_I$ be the characteristic
function on the energy interval $I$. For any $q>1$, let
$p>1$ be the conjugate index so that $(1/p) + (1/q) = 1$. We then
have
\beq
\int \chi_I (E) \xi ( E ;
H_{\lambda_j^+ } , H_{\lambda_j^- }) ~dE \; \leq \; | I |^{1/q}
\| \xi ( \cdot; H_{\lambda_j^+ } , H_{\lambda_j^- }) \|_{L^p (I)} .
\eeq
 
\noindent
3. The SSF appearing in (3.7) is defined through the invariance
principle due to the fact that the Hamiltonians are unbounded.
Let $g(E) = (E + M_0)^{-k}$, for some $M_0 >> 0$, the existence
of which is guaranteed by (H1), and for some $k > (2d /p) + 2$,
where $p > 1$. Note that $\mbox{sgn} g' = -1$, for $E > - M_0$.
We recall, as in (2.12), that the SSF is defined by
\beq
\xi ( E; H_{\lambda_j^+ } , H_{\lambda_j^- }) = 
- \xi( E ; g(H_{\lambda_j^+ }) , g(H_{\lambda_j^- })) .
\eeq
Using Theorem 2.1, after changing variables in the integral, we
find
\beq
\| \xi ( \cdot; H_{\lambda_j^+ } , H_{\lambda_j^- }) \|_{L^p (I)} \;
\leq \; C_1 ( E_0 - |I| / 2 + M_0 )^{-(k+1)/pk} \|
g(H_{\lambda_j^+ }) - g(H_{\lambda_j^- }) \|_{1/p}^{1/p} .
\eeq 
By Proposition 3.2 ahead, the trace ideal functional is bounded
independently of $| \Lambda |$.
Hence, from (3.5)--(3.7), we obtain
\beq
\E \{ Tr ( E_\Lambda ( I ) ) \} \; \leq \; C_2 |I|^{1/q} \|
h_0 \|_\infty | \Lambda | ,
\eeq
proving the theorem. $\Box$ 

The trace estimate used above is the following.
We let $H_0$ be the Schr\"odinger 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 proof of the following proposition is given in \cite{[CHN]}.

\vspace{.1in}
\noindent
{\bf Proposition 3.2.} {\it Let $H_0$ be as above, 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 Kato-class function
supported in $B(R)$, 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 )^{-k} - (H_1 + M )^{-k} \in {\cal
I}_{1/p} ,
\eeq
provided $k > dp / 2 + 2$. Under these conditions, there exists a
constant $C_0$, depending on $p, k, H_0, M_1 , \| V \|_{K_d}$,
and $R$, so that
\beq
\| V_{eff} \|_{1/p} \; \leq \; C_0 .
\eeq
}

We remark that for the case of a locally perturbed
\Schr\ operator $H_0$, with $H = H_0 +
V_\Lambda$, Kostrykin and Schrader \cite{[KS3]} showed that the constant $C_0$
in (3.13) is bounded above by $C_1 | \Lambda |^p$, for a constant
$C_1$ independent of $| \Lambda|$.

 
\subsection{Application to Anderson-type Models}

We indicate how to verify the assumptions (A1)--(A3)
for the Anderson-type
additive models described in
section one. This
provides a simpler proof
than the one presented in \cite{[CHN]} provided we add hypothesis
(H3b).
{\it To simplify the notation, we will drop the notation $H_0^A$
and $H_\Lambda^A$, and write $H_0$ and $H_\Lambda$, respectively, for
the additive case.}


As with the proof in \cite{[CHN]},
we note that this proof of the Wegner estimate does not require
spectral averaging \cite{[CHM]}.
It does, however, rely upon some monotonicity
of the eigenvalues with respect to the random variables (for
comparison, see the work 
\cite{[BuschmannStolz]} in one dimension).
Furthermore, the comparison theorem of Kirsch, Stollmann, and
Stolz \cite{[KSS]}, used in \cite{[CHN]}
is not needed for this version of the proof.
In the next section, we present a technique that removes the
positivity assumption.

\vspace{.1in}

\noindent
{\bf Proposition 3.3.} {\it Let us suppose that $H_\Lambda$
satisfies hypotheses (H1) or (H1a), (H2), (H3b), and
(H4). Then, the additively-perturbed Anderson model 
satisfies assumptions (A1)--(A3).}

\vspace{.1in}  
\noindent
{\bf Proof.} Assumption (A1) is obviously satisfied by the
Anderson-type potentials with $N = | {\tilde \Lambda}|$, the
number
of lattice points in $\Lambda$. We turn to the proof of (A2).
Hypothesis (H3b) implies that the single-site potentials
satisfy $u_i u_j = \delta_{ij} u_j$. 
Let $E_0 \in G$, where $G \subset \rho (H_0)$ is a subset
of an unperturbed spectral gap for $H_0$, and choose $\eta > 0$
so that the interval $I$ of (A2)
is $I = I_\eta \equiv [ E_0 - \eta , E_0 + \eta ] \subset G$.
Since the perturbation
$V_\Lambda$ is relatively $H_0$ compact, we know that $\sigma
(H_0) \cap G$ is discrete. 
Let $\phi \in E_\Lambda ( I_\eta) L^2 ( \R^d)$ be a normalized 
eigenfunction of $H_\Lambda$ with eigenvalue $E \in I_\eta$.
Using the eigenvalue equation, we
easily verify that
\beq
\| ( H_0 - E ) \phi \| \; = \; \| V_\Lambda \phi \| .
\eeq
Furthermore, we can expand the right side as
\bea
\| V_\Lambda \phi \|^2 & = & \langle \phi , V_\Lambda^2 \phi
\rangle \nonumber \\
  & = & \sum_{j \in {\tilde \Lambda}} \; \lambda_j^2 
  \langle \phi, u_j^2 \phi \rangle \nonumber \\
 & = & \| ( H_0 - E ) \phi \|^2 \nonumber \\
 & \geq & [ \mbox{dist} \; ( \sigma(H_0) , I_\eta ) ]^2 ,
\eea  
using hypothesis (H3b). Now, we know that
\beq
\frac{ \partial H_\Lambda }{ \partial \lambda_j } \; = \; 
\frac{ \partial V_\Lambda }{ \partial \lambda_j } \; = 
 \; u_j ( \cdot - j ) ,
\eeq
so that, as $ u_j ( \cdot - j ) \geq C_0 u_j ( \cdot - j )^2$, we have
\bea
\sum_{j \in {\tilde \Lambda} } \langle \phi , 
  \left( \frac{ \partial H_\Lambda }{ \partial \lambda_j } \right)
  \phi \rangle & \geq & \sum_{j \in {\tilde \Lambda}
   } C_0 \langle \phi , u_j ( \cdot - j )^2 \phi  \rangle \nonumber \\
 & \geq &  C_0 ( \lambda^+)^{-2} \langle \phi , V_\Lambda^2 \phi \rangle 
    \nonumber \\
  & \geq & C_0 ( \lambda^+)^{-2} [ \mbox{dist} \; ( \sigma(H_0) ,
I_\eta ) ]^2 \| \phi \|^2,
\eea
where we assume, without loss of generality, that $| \lambda^+ | \geq
| \lambda^- |$.
This inequality immediately implies (A2) since
\bea 
\lefteqn{ Tr \left\{  \sum_{j \in {\tilde \Lambda}}
\left( \frac{ \partial H_\Lambda }{ \partial \lambda_j } \right) 
E_\Lambda ( I ) \right\} } \nonumber \\
 & = & \sum_{k} \; \sum_{j \in {\tilde \Lambda}} \langle \phi_k , 
  \left( \frac{ \partial H_\Lambda }{ \partial \lambda_j } \right)
  \phi_k  \rangle \nonumber \\
 & \geq &  C_0 ( \lambda^+)^{-2} [ \mbox{dist} \; ( \sigma(H_0) ,
I_\eta ) ]^2 \; \sum_{k} \| \phi_k \|^2  \nonumber \\
 & \geq & C_0 (\lambda^+)^{-2} [ \mbox{dist} \; ( \sigma(H_0) ,
I_\eta ) ]^2 \; Tr E_\Lambda ( I_\eta ) .
\eea
Finally, we verify (A3). We note that $ ( H_{\omega_j^+}^\Lambda -
H_{\omega_j^-}^\Lambda) = ( \lambda^+ - \lambda^- ) u_j ( \cdot -
j )$. It is proved in \cite{[CHN]} that given any $p > 1$, for
any $k > (pd/2) + 2$, if we set $g(E) = (E + M_0)^{-k}$, then
\beq
\| g(H_{\omega_j^+}^\Lambda) - g(H_{\omega_j^-}^\Lambda) \|_{1/p} 
\; \leq \; C_k < \infty ,
\eeq
where the constant $C_k$ is independent of the index $j$, and
it is independent of $|\Lambda|$ and
depends only on $| \mbox{supp} \; u_j |$.  $\Box$  

\vspace{.1in}
The verification of assumption (A2) is more difficult in the
general case. 
In the absence of hypothesis (H3b), there are two possibilities:
1) the $u_j$ satisfy hypothesis (H3a), i.\ e.\
$u_j \geq C_0 \chi_{\Lambda_1
(0)}$, or 2) the $u_j$ satisfy hypothesis (H3), i.\ e.\
$u_j$ is nonnegative and the support of $u_j$ is compact.
In the first case, assumption (H3a), the single-site potentials
satisfy 
\beq
\sum_{j \in {\tilde \Lambda}} u_j ( x - j) \; \geq \; C_0
\chi_\Lambda ,
\eeq
which is a strong monotonicity condition. Under this condition,
we have the following global result.

\vspace{.1in}

\noindent
{\bf Proposition 3.3.} {\it We define the local Hamiltonian
$H_\Lambda$ by $H_\Lambda = ( H_0 + V_\omega ) |
\Lambda$, with Dirichlet boundary conditions on $\partial
\Lambda$. Suppose that the local Hamiltonian $H_\Lambda$
satisfies (H1), (H2), (H3a), and (H4). Then, for any
$E_0 \in \R$, and any interval $I_\eta = [ E_0 - \eta, E_0 + \eta
] \subset \R$, there exists a finite constant $C_W > 0$,
depending on $(d, \eta , E_0 )$, so that we have
\beq
\E \{ Tr ( E_\Lambda ( I_\eta ) ) \} \; \leq \; C_W \eta | \Lambda | .
\eeq
Consequently, the IDS is Lipschitz continuous at all energies.}

\vspace{.1in}
\noindent
{\bf Sketch of the Proof.} The proof of this proposition follows
the lines of the proof given in \cite{[CH1]}. 
We suppose that $\Lambda$ is a cube, and
work on the Hilbert space $L^2 ( \Lambda )$. Since $V_\omega$
is bounded, there is a finite, positive constant $V_0$ so that $- V_0 \leq
V_\Lambda$, so that $H_\Lambda \geq H_0^\Lambda - V_0$, where
$H_0^\Lambda \equiv H_0 | \Lambda$, with Dirichlet boundary
conditions. This lower bound and Jensen's inequality lead to the
bound
\bea
Tr E_\Lambda ( I_\eta) & \leq & e^{(E_0 + \eta)} \; Tr \{
e^{-H_\Lambda} E_\Lambda ( I_\eta) \} \nonumber \\
 & \leq & e^{(E_0 + \eta + V_0 )} \; Tr \{ e^{-H_0^\Lambda }
E_\Lambda ( I_\eta) \} .
\eea
We decompose the cube $\Lambda$ into unit cubes $\Lambda_j$, so
that $\Lambda = \mbox{Int} \; \overline{ \cup_{j} \Lambda_j }$.
Dirichlet-Neumann bracketing and the diamagnetic inequality imply
that
\beq
e^{-H_0^\Lambda } \; \leq e^{\oplus_j \Delta_{\Lambda_j}^N} =
\sum_{j \in {\tilde \Lambda}} \; \chi_j e^{\Delta_{\Lambda_j}^N}
\chi_j ,
\eeq
where $\chi_j$ is the characteristic function on $\Lambda_j$ and
$- \Delta_j^N$ is the nonnegative Neumann Laplacian on $\Lambda_j$.
Substituting this into (3.22), we obtain
\beq
Tr E_\Lambda ( I_\eta) \; \leq \; e^{(E_0 + \eta + V_0 )} \;
\sum_{j \in {\tilde \Lambda}} \; Tr \{ e^{\Delta_{\Lambda_j}^N}
\chi_j E_\Lambda ( I_\eta) \chi_j \} .
\eeq
We now expand the trace in the eigenfunctions of
$\Delta_{\Lambda_j}^N$ and use spectral averaging. The result
follows by noting that $Tr \{ e^{\Delta_{\Lambda_j}^N} \} $ is
bounded. $\Box$ 

\vspace{.1in}

The general case of hypothesis (H3),
was treated in \cite{[CHN]} using
a result of Kirsch, Stollmann, and Stolz \cite{[KSS]} on 
the localization of the eigenfunctions of the local Hamiltonian
$H_\Lambda$. This theorem provides precise information about the
eigenfunctions in the region $\Lambda$.
The proof of this theorem is simple and we refer the reader to
\cite{[KSS],[CHN]}.

\vspace{.1in}

\noindent
{\bf Proposition 3.4.} {\it Let $H_0$ and $V_\Lambda$ be as above and
$H_\Lambda \phi = E \phi$ with $E \in G$ and $\phi \in L^2 ( \R^d
)$. Suppose that the following two conditions are satisfied:
\begin{enumerate}
\item There exists a potential $V_0$ such that, with $H^0_\Lambda
\equiv H_0 + V_0$, we have $E \in \rho ( H^0_\Lambda)$;
\item There exists a subset $F \subset \Lambda$ and a constant
$\theta > 0$ so
that $\mbox{dist} \; ( F \cup \Lambda^c , \{ x \; | \; V_\Lambda
( x
) \neq V_0 ( x ) \} ) > \theta > 0$.
\end{enumerate}
We then have,
\beq
\| \phi \| \leq ( 1 + \| ( H^0_\Lambda - E)^{-1} W_1 \| )
\| ( 1 - \chi_F ) \phi \| ,
\eeq
where $W_1 \equiv [ H_0 , \chi_1 ]$, with $\chi_1$ is defined in
the
proof, and $\chi_F$ is the characteristic function of $F$.}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\section{The Nonsign-definite Case}\label{S.4}  

Although the proof presented in section 3 is elementary, it does
require that the single-site potentials $u_j$ have a definite sign. The
case of nonsign definite single-site potentials is more delicate since the
eigenvalues are no longer monotonic functions of the random
variables. We have two main results in the nonsign-definite case.
The first, and more general, result applies to energies below the bottom
of the spectrum of the background operator $H_0$. The second
result concerns the Wegner estimate at energies in an internal
gap of the spectrum of $H_0$. This requires that the disorder be
small. The basic idea of the proofs is to combine the vector field
method of Klopp \cite{[Klopp]} with the techniques of section 3.

The single-site potentials $u_j$ must satisfy hypothesis (H3c),
that is weaker than the other hypotheses (H3), (H3a), or (H3b).
Basically, we need that $u_j$ is continuous and nonvanishing on
some bounded, open set. As we mention in the proof below, we need
a slightly stronger hypothesis on the common distribution $h_0$ of the
random variables. This is given in hypothesis (H4a). 

\subsection{Below the Infimum of the Spectrum of $H_0^A$}

For energies $E <  \; \mbox{inf} ~ \sigma (H_0^A) \equiv \Sigma_0^A$, 
the operator
$(H_0 - E ) $ is strictly positive. This allows us to reformulate the
Wegner estimate as a statement concerning a Birman-Schwinger-type operator.
The main result, under hypotheses (H1a), (H2), (H3c), and (H4a)
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 4.1}. {\em Assume (H1a), (H2), (H3c), and (H4a).  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 \P \left\{ \dist ( E_0 ,
\sigma(H_\Lambda^A) ) \leq \eta \right\} \leq C_{E_0}
\eta^{1/q} \, |\Lambda|\ .  
\eeq 
}

As an immediate corollary of Theorem 4.1, and of the definition
of the density of states, we obtain

\vspace{.1in}
\noindent
{\bf Corollary 4.2}. {\em
Assume (H1a), (H2), (H3c), and (H4a), 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}

Following \cite{[Klopp]}, we formulate the Wegner estimate in
terms of the resolvent of $H_\Lambda^A$
using the fact that if $E_0 < \mbox{inf} \; \sigma (
H^A_0 )$, we have that $( H^A_0 - E_0 ) > 0$. 
So, for an energy $E_0$ in the resolvent
set of $H^A_\Lambda$, we have
\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 Birman-Schwinger-type
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 (4.2) that
\bea
\| R_\Lambda ( E_0 ) \| & \leq & \{ \mbox{dist} \; ( \sigma
(H^A_0), E_0 ) \}^{-1} \; \| ( 1 + \Gamma_\Lambda ( E_0 ; \omega ) )^{-1}  \|
 \nonumber \\
 & \leq  & \delta^{-1} \| ( 1 + \Gamma_\Lambda ( E_0 ; \omega ) )^{-1}
   \|.
\eea
It follows from (4.4) that
\beq
\P \{ \| R_\Lambda ( E_0 ) \| \leq 1 / \eta \} \geq
\P \{ \| ( 1 + \Gamma_\Lambda ( E_0 ; \omega ) )^{-1} \|  \leq
\delta / \eta
\} .
\eeq
Consequently, Wegner's estimate can be reformulated as
\bea
\P \{ \mbox{dist} \; ( \sigma ( H^A_{\Lambda} ) , E_0 ) < \eta
\}
    & = & \P \{ \; \| R_\Lambda ( E_0 ) \| > 1 / \eta  \}
\nonumber \\
    & \leq & \P \{ \; \| ( 1 + \Gamma_\Lambda  ( E_0 ; \omega
) )^{-1}  \| \;
          > \; \delta / \eta  \}     \nonumber \\
 & = & \P \{ \; \mbox{dist} \; ( \sigma ( \Gamma_\Lambda  (
E_0 ; \omega ) )
        , -1 ) < \;  \eta / \delta \} .
\eea
Hence, it suffices to compute
\beq
\P \{ \; \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]}, Proposition 3.4, 
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.

\vspace{.1in}
\noindent
{\bf Sketch of the Proof of Theorem 4.1.} \\ 
\noindent
1.  It follows from the reduction given above that we need to
estimate the probability in (4.7). 
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 (4.7) 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
\P \{ \; \mbox{dist} \; ( \sigma ( \Gamma_\Lambda ( E_0  ) ) ,
-1 ) < \kappa
\}
 & = &  \P \{ 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 in (4.10),
following the original argument of Wegner \cite{[Wegner]} as modified by
Kirsch \cite{[Kr2]}.
Let $\rho$ be a nonnegative, smooth function 
such that $\rho ( x ) = 1 $, for $
- M_1 < x < - \kappa /2 $, and $\rho ( x) = 0$, for $ x \geq \kappa /2 $
and for $x \leq -M_1$, for some $M_1 > 0$.
We can assume that $M_1 < \infty$, so that $\rho$ has
compact support, since $\Gamma_\Lambda (E_0 )$ is lower
semibounded,
independent of $\Lambda$.
We further assume that $\rho$ is monotone decreasing
for $ x > - M_1$.
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 (4.9). 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 (4.11) 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
As in the proof of Theorem 3.1,
we select one random variable, say $\lambda_k$, with $k \in
{\tilde \Lambda}$, and first integrate with respect to this variable
using hypothesis (H4a). The local absolute continuity property is
necessary here because a single term in the sum of (4.14) 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 (4.15)
is obtained. We remark that each term in
(4.16) 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 (4.16) 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. A result similar to Proposition
3.2 holds in this case, and 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
the invariance principle (2.12).
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 (4.16). 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 (4.20) 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 $| \Lambda |$.
As in section 3, one can prove 
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 (4.23) is bounded above by $C_0
\kappa^{(1-q)/q}$, independent of $| \Lambda |$. This estimate,
equations (4.16)
and (4.20), lead us to the result
\beq
\P \{ \mbox{dist} \; ( - 1 , \sigma( \Gamma_\Lambda (E_0) ) )
< \kappa \} \leq
C_W \kappa^{ 1 / q} \|g \|_\infty  | \Lambda |,
\eeq
for any $q > 1$. $\Box$

\subsection{The Case of a General Band Edge and Small Disorder}

Suppose now that the background operator $H_0$ has an open, internal
spectral gap, as in hypothesis (H1). In the case of nonsign-definite
single-site potentials, the behavior of the eigenvalues created by
$V_\Lambda$, as a coupling constant $\lambda_j (\omega)$ varies, may be very
complicated. In order to compensate for this, we must work in
the weak disorder regime. The main result is the following.   

\vspace{.1in}
\noindent
{\bf Theorem 4.3}. {\it We assume that $H_0^X$ and $V_\omega$
satisfy (H1), (H2), (H3c), and (H4a), 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
  \P \{
  \: \mbox{dist} ( \sigma ( H_\Lambda^X (\lambda) ) , E_0 ) \leq
\eta \}
  \leq C_{E_0} \eta^{1/q} | \Lambda | .
  \eeq 
Consequently, for ergodic models, the IDS is H\"older continuous
in a neighborhood of $E_0$. }

\vspace{.1in}
We give some ideas concerning the proof.
Formula (4.2) is no longer valid so we replace 
it with the Feshbach projection formula.
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 $P_+ P_- = 0$.
The Feshbach method permits us 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 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
$H_\pm ( \lambda )$.
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

We notice that if $E_0 \in G$, then $(H_+ - E_0 ) > 0$, so that
we can use the same ideas as in the previous subsection to treat
this operator. For example, let us suppose that $E_0$ is close to
the upper gap edge $B_+$. We then apply formula 
(4.27) with $Q = P_+$ and $P = P_-$. With
this choice, we see that $R_P ( E_0 ) = ( P_-H_0 + \lambda P_-
V_\Lambda P_- - zP_- )^{-1}$ is bounded provided $| \lambda |$ is
small enough. This implies that the singularity
of the resolvent comes from the operator ${\cal G }(z)$, for $z$
near $E_0$. Following the general proof of Theorem 4.1, we reduce
the statement of the Wegner estimate to a statement concerning
the norm of the operator ${\cal G }(E_0)$:
\bea
\P \{ \mbox{dist} ( \sigma ( H_\Lambda ) , E_0 ) < \eta \}
 &  = & \P  \{ \; \| R_\Lambda ( E_0 ) \| > 1 / \eta \}
\nonumber \\
 & \leq &  \P \{ \; \| {\cal G } ( E_0 ) \| \; > 1 / ( 8 \eta
) \}.
\eea

Looking closely at the operator ${\cal G } ( E_0 )$ in (4.28), we
see that it can be written 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
This operator ${\tilde \Gamma}_+ ( E_0 )$ is the analog of the
operator $\Gamma_\Lambda (E_0 ; \omega)$ appearing in (4.3).
Equations (4.29) and (4.30) show that we can, as in subsection 4.1,
reduce the Wegner estimate as follows:
\bea
\P \{ \; \mbox{dist} ( \sigma ( H_\Lambda , E_0 ) < \eta \}
 & = & \P \{ \| R_\Lambda ( E_0 ) \| \; > 1/ \eta \} \nonumber
\\
 & \leq & \P \{ \; \| ( 1 + {\tilde \Gamma}_+ ( E_0 ))^{-1} \|
\; >
   \delta_+ (E_0) / ( 8 \eta ) \} \nonumber \\
 & = & \P \{ \: \mbox{dist} ( \sigma ( {\tilde \Gamma}_+ (E_0
) ) , -1 )
    < 8 \eta / \delta_+ (E_0) \} . \nonumber \\
  &  &
\eea

We can now proceed as in subsection 4.1. The final difficulty is
that the operator ${\tilde \Gamma}_+ ( E_0 )$ is no longer an
eigenvector of the operator $A_\Lambda$ defined in (4.8).
Instead, a calculation yields the relation
\beq
A_\Lambda {\tilde \Gamma}_+ (E_0) = {\tilde \Gamma}_+ (E_0) +
\lambda^2 W(E_0).
\eeq
The second constraint of $| \lambda |$ originates with this
expression. We want $| \lambda |$ small enough so that the
leading term in (4.33) dominates. With this, the proof continues
as in the proof of Theorem 4.1.

We conclude by mentioning the model studied by I.\ Veseli\'c
\cite{[Veselic]}. Let $\Gamma \subset \Z^d$, be a finite subset
containing the origin $k = 0$,
and consider a finite set of real numbers $\alpha \equiv \{
\alpha_k \; | \; k \in \Gamma \}$. We assume that $\alpha_0 = 1$,
and that the remaining terms $\alpha_k , k \neq 0$, 
satisfy $\sum_{k \neq 0 } | \alpha_k
| < 1 $. In the simplest case, let $\chi_0$ be the
characterisitc function on the unit cell centered at the origin
in $\Z^d$. We define a compactly-supported, single-site potential $u$ by
\beq
u(x) \equiv \Sum_{k \in \Gamma } \alpha_k \chi_0 ( x - k ) .
\eeq
This potential has no fixed sign if some of the terms $\alpha_k, k \neq
0$, are negative. Veseli\'c considers the Anderson-type
potentials (1.5) constructed with this single-site potential
\beq
V_\omega (x) = \Sum_{i \in \Z^d } \lambda_i ( \omega )
u ( x - i ) ,
\eeq
with the coupling constants $\lambda_i ( \omega)$ being
independent and identically distributed with common density $h_0$, 
as considered in this paper.
Veseli\'c observes that the potential 
$V_\omega$
can be written as
\bea
V_\omega (x) & = & \Sum_{i \in \Z^d} \lambda_i ( \omega) 
  \left[ \Sum_{k \in \Gamma }
   \alpha_k \chi_0 ( x - k - i ) \right] \nonumber \\
 & = & \Sum_{m \in \Z^d} \nu_m ( \omega ) \chi_0 ( x - m ) .
\eea
where the new family of random variables $\{ \nu_m ( \omega ) \; |
\; m \in \Z^d \}$ is defined by
\beq
\nu_m ( \omega ) = \Sum_{k \in \Gamma} \lambda_{m-k} ( \omega)
\alpha_k .
\eeq

The Anderson-type potential $V_\omega$ in (4.36) is constructed from
a sign-definite, single site potential $\chi_0$, but the coupling
constants $ \nu_m ( \omega)$ are not necessarily independent and have a
different distribution that no longer has a product form. 
Note that if $\| k - m \| $ is sufficiently large,
depending upon $\Gamma$, then the random variables
$\nu_m ( \omega )$ and $\nu_k ( \omega )$ are independent. That
is, the correlation is finite range. 
The distribution of the family $\{ \nu_m ( \omega ) \; | \; m \in
\Z^d \}$ can be easily calculated. Let $A$ be the infinite
Toeplitz matrix with entries $A_{ij} = \alpha_{i - j}$. It
follows from (4.37) that $\nu = A \lambda$. Formally, the
probability distribution for the family $\nu$ is given by
\beq
\P \{ \nu \in B \} = \int_B \: | det (A^{-1})| \: \Pi_{k \in \Z^d }
f ( ( A^{-1} \nu ) ) \: d \nu_k ,
\eeq
for any measurable subset $B \subset A [ \mbox{supp} \: h_0
]^{\Z^d}$.
These comments can be restricted to a finite cube. It follows
that the conditional probability distribution of one random
variable $\nu_k$, conditioned on the others in a cube, is
absolutely continuous. Consequently, the results of \cite{[CHM2]}
apply, and, because hypothesis (H3a) is satisfied,
one can prove a Wegner estimate at any energy (cf.
Proposition 3.3).



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

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