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Random Schroedinger Operators, Anderson Model, Multiparticle System
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\begin{document}
\title{An interacting multielectron Anderson model}
\author{
Heribert Zenk\\
Mathematisches Institut, Ludwig-Maximilians-Universit\"at\\
Theresienstra{\ss}e 39, 80333 M\"unchen \\
{\em email:} Heribert.Zenk@mathematik.uni-muenchen.de}
\maketitle
\begin{abstract}
This article is a first tiny step towards a rigorous description of an 
interacting multielectron system in a random potential of Anderson
type. Deterministic
spectrum and the Wegner estimate for this model are proven.
\end{abstract}
%
\noindent
{\bf Keywords:}
Random Schr\"odinger Operators, Anderson Model, Multiparticle System \\
{\bf Mathematics Subject Classification 2000:} 
81Q10, 35J10,47A10
\section{Introduction}
\setcounter{equation}{0}
This paper combines some techniques known from multiparticle
Schr\"odinger operators with the random operator techniques for a
single electron. We consider $n$ interacting electrons moving under
the influence of an additional random perturbation of Anderson type in
the $d$-dimensional configuration space $\R^d$. 
\subsection{The interacting multielectron model}
The Hamiltonian
is of the form
\be \label{eq1.1}
H(\omega):=-\lap+V_I-V_n(\omega),
\ee
where the Laplacian $-\lap$ on $\R^{nd}$
describes the free kinetic energy of
the $n$ electrons.
\be \label{eq1.2a}
V_I(x_1,...,x_n):=\sum_{1 \leq k < l \leq n} v(x_k-x_l)
\ee
is a sum of repulsive two body interaction potentials $v(x)=v(-x) \geq 0$.
We assume that $V_I$ is
infinitesimally $-\lap$-bounded in $L^2(\R^{nd})$, like in the case
$V_I=V_C$, where
\be \label{eq1.2b}
V_C(x_1,...,x_n):=\sum_{1 \leq k < l \leq n} \frac{1}{|x_k-x_l|}
\ee
is the repulsive Coulomb potential generated by the electrons. The
random potential $V_n(\omega)$ should describe the interaction of $n$
electrons with a potential created by a randomly chosen configuration
(denoted by $\omega$) of atoms or ions, hence it is written as 
\be \label{eq1.3}
V_n(\omega)=\sum_{k=1}^n \one_{L^2(\R^d)}^{\otimes (k-1)}
\otimes V_1(\omega) \otimes
\one_{L^2(\R^d)}^{\otimes (n-k)},
\ee
or $\displaystyle V_n(\omega,x_1,...,x_n)=\sum_{k=1}^n V_1(\omega,x_k)$,
where the one particle random potential $V_1(\omega)$ models the
interaction of a single electron with the potential created by the
configuration $\omega$ of atoms. 
Before we comment on the special form  $V_1(\omega)$ takes in the
Anderson model, we construct the probability space $\Omega$,
collecting all configurations $\omega$ of atoms under consideration:
\begin{enumerate}
\item Take a probability measure $\P_0$ on the Borel sets $\B(\R)$ of
$\R$, such that $\P_0$ is supported in a finite interval. 
\item Take a probability measure $\P_1$ on $\B(\R^d)$ and $R \in
]0,\frac{1}{2}[$, such that
\be \label{eq1.5}
\P_1(\{x \in \R^d:|x|<R\})=1
\ee
\item Define the probability space
\be \label{eq1.6}
\Omega:=((\R \times \R^d)^{\Z^d}, \B(\Omega), (\P_0 \otimes
\P_1)^{\otimes \Z^d})
\ee
as the product of the probability spaces. $\B(\Omega)$ is taken as the
completion with respect to $\P=(\P_0 \otimes \P_1)^{\otimes \Z^d}$-zero-sets of
the product $\sigma$-algebra. 
\end{enumerate}
Denote by $\omega'$ and $\omega''$ the images of $\omega$ under the
canonical projections \linebreak
$pr':\Omega \longrightarrow \Omega':=(\R^{\Z^d},\P_0^{\otimes \Z^d})$ and
$pr'':\Omega \longrightarrow \Omega'':=((\R^d)^{\Z^d},\P_1^{\otimes
  \Z^d})$ 
and by $\omega_j'$ and $\omega_j''$, $j \in \Z^d$ the image of
$\omega$ under the projections
$pr_j':\Omega \longrightarrow \R$ and
$pr_j'':\Omega \longrightarrow \R^d$, we write 
$\omega=(\omega',\omega'')=(\omega_j',\omega_j'')_{j \in \Z^d}$.
Thus the random potential
\be \label{eq1.8}
V_1(\omega,x):=\sum_{j \in \Z^d} \omega_j' \wp(x-j-\omega_j'')
\ee
of Anderson type, acting in $L^2(\R^d)$ describes the interaction of a 
single electron with the atoms of a non crystalline solid. 
The random variables $pr_j'$ and $pr_j''$ determine the sort of atom
and the displacement of the atom at lattice site $j \in \Z^d$. For a
given realization $\omega \in \Omega$ the kind of atom at site $j \in
\Z^d$ is given by $\omega_j'$ and the displacement at site $j$ by
$\omega_j''$. Note, that due to assumption $|\omega_j''| \leq
\frac{1}{2}$, we can loosely speak of an atom at site $j \in \Z^d$
even if $\omega_j'' \not =0$, because the is no doubt, to which
lattice site $j \in Z^d$ an atom at $j+\omega_j''$ belongs to.
We remark, that the product structure of $\Omega$, we have assumed, is
an equivalent way of getting independent, identical distributed families of
random variables for sort and displacement of atoms - but this
formulation is more convenient in the following proofs.
As a consequence of the 
identical distribution of $(pr_j')_{j \in \Z^d}$, the single
site potential $\wp$ must satisfy a decay estimate at $\infty$,
otherwise the series defining $V_1(\omega,x)$ does not converge. 
To ensure convergence and to exclude $V_1=0$, we will assume:
\begin{enumerate}
\item $\wp \geq 0$
\item There is a $c_0 >0$, such that $c_0 1^0_{1+2R} \leq \wp$ holds,
where $1^x_l$ denotes the characteristic function of a box
$\Lambda(x,l)$ of sidelength $l$ centered at $x$.
\item There are $c_1 < \infty$ and $\varsigma >d$, such that
\be \label{eq1.8a}
\wp(x) \leq c_1 \lkl x \rkl^{-\varsigma} := c_1
(1+|x|^2)^{-\frac{\varsigma}{2}} 
\ee
\end{enumerate}
Under these assumptions $V_1(\omega)$ is a bounded potential
$\P$-almost everywhere (the displacements are bounded, so the series
converges, see \cite{Ze} Lemma 1.1), hence
$\displaystyle |\cV_1|:=\P-\mathrm{ess}\sup_{\omega \in \Omega}
\|V_1(\omega)\|_{L^{\infty}},
|\cV_n|:=\P-\mathrm{ess}\sup_{\omega \in \Omega}
\|V_n(\omega)\|_{L^{\infty}} <  \infty$.


\subsection{Which phenomena in random media are due to interacting electrons?}
A motivation for the study of this interacting $n$-electron model in
an Anderson background comes from the discovery of ``universality
classes'' in the measurement of the conductivity $\sigma$ of doted
semiconductors at low temperatures. Most of the experimental results can be
fitted very well to a function of the type
\[\sigma(n)=\left\{ \begin{array}{rl}
0 & {\mathrm{if }}\, n \leq n_c \\
\sigma_c (\frac{n}{n_c}-1)^{\nu} & {\mathrm{if }}\, n \geq n_c
\end{array} \right. \]
where $n$ is the donator concentration and $\sigma_c,n_c$ are
fit parameters. For
experimental data see e.g. \cite{RATB}
on $P$ doted $Si$ or \cite{KKSK} on $Si$ doted
$Al_{0,3}Ga_{0,7}As$. The astonishing fact about the exponent $\nu$
is, that in all these experiments it could be chosen to be roughly
$1$ or $\frac{1}{2}$. A heuristic explanation of the first class
$\nu \sim 1$ argues, that one can neglect electron-electron
interactions and approximate this situation in the well known Anderson
model for one electron. In contrast
the $\nu \sim \frac{1}{2}$ case 
should be an effect of the electron-electron interaction.\\
The experimental results mentioned above are interpreted as a
metal-insulator transition for electrons in three dimensional random
media. But for electrons confined to a two dimensional
space it is still an open question if there is such a transition and
if this is an effect of interaction.\\

\subsection{Rigorous results}
A rigorous proof of this behavior of the conductivity in disordered
systems in the framework of the Anderson model is far away from the
present techniques. The detection of absolutely continuous spectrum in
the Anderson model as a sign for nonzero conductivity remains an open 
problem, which has been mentioned in Barry Simon's list
of problems in mathematical physics \cite{Si} in the twenty-first
century. In the one electron Anderson model deterministic spectrum and
localization for low energies have been proven, see \cite{St} for a
summary of recent results.
The crucial ingredient of current localization proofs in
the continuous Anderson model is an estimate of the norm of localized 
resolvents arising from a multiscale analysis procedure. 
The starting point for
multiscale analysis is Fr\"ohlich and Spencer's paper \cite{FrSp} 
for discrete Anderson models in $l^2(\Z^d)$, which is made
applicable for continuous Anderson models in $L^2(\R^d)$ by Combes,
Hislop and Kirsch, Stollmann, Stolz \cite{CoHi}, \cite{KSS1} and
\cite{KSS2}. An important ingredient in the multiscale
analysis is the Wegner estimate, serving as an a priori bound.\\
In contrast to the one electron models, there are at least to my
knowledge no rigorous results for the multielectron Anderson model. In
the next section we check some basic properties of the multielectron
Anderson model, which enable us to apply the general theory of ergodic
operator families, implying deterministic spectrum. Chapter
\ref{kap3} lists the differences between one and many electron model
in a proof of Wegner's estimate.
%
%
\section{Selfadjointness, ergodicity and deterministic spectrum of H}
\setcounter{equation}{0}
%
%
In this section we will prove the existence of nonrandom sets
$\Sigma$, $\Sigma_c$, $\Sigma_{ac}$,... such that
$\Sigma=\sigma(H(\omega))$, $\Sigma_c=\sigma_c(H(\omega))$,... for
$\P$-almost every $\omega \in \Omega$. This goal is achieved by use of
the theory of ergodic operator families. A first step is the proof of
selfadjointness for the operators $H(\omega)$ $\P$-almost
everywhere.
%
%
\begin{theorem} \label{t2.1}
For $\P$-almost every realization $\omega \in \Omega$ the Hamiltonian
\be \label{eq2.0}
H(\omega)=-\lap+V_C-V_n(\omega)
\ee
is bounded from below, selfadjoint on $\D(-\lap)$ and 
$C_0^{\infty}(\R^{nd})$ is a core for $H(\omega)$.
\end{theorem}
\begin{proof}
In multiparticle Schr\"odinger theory, see e. g. \cite{HuSi},
 it is well known, that the
Coulomb potential
$\displaystyle V_C=\sum_{1 \leq k < l \leq n} \frac{1}{|x_k-x_l|}$ 
is a Kato potential, i.e. for each $\alpha >0$ there is a $\beta(\alpha) <
\infty$, such that
$\|V_C\Psi\| \leq \alpha \| -\lap \Psi\| + \beta(\alpha) \|\Psi\|$
for each $\Psi \in C_0^{\infty}(\R^{nd})$, but the 
closure $H_0^2(\R^{nd})$ of $C_0^{\infty}(\R^{nd})$ with respect
to the $H^2$-norm equals $H^2(\R^{nd})$, see \cite{Wl} chap. 3.2, so
this extends to
\be \label{eq2.2}
\|V_C\Psi\| \leq \alpha \| -\lap \Psi\| + \beta(\alpha) \|\Psi\|
\ee
for all $\Psi \in \D(-\lap)=H^2(\R^{nd})$. From (\ref{eq2.2}) as an
application of the Kato-Rellich theorem, we see,
that $\displaystyle H_0=-\lap+V_C \geq  -\min_{\alpha >0} 
\max \{ \frac{\beta(\alpha)}{1-\alpha},\beta(\alpha)\}$ 
is selfadjoint on $\D(-\lap)$ and
$C_0^{\infty}(\R^{nd})$ is a core for $H_0$. The assumptions on the
support of $\P_0$ and $\P_1$ and the decay assumptions on $\wp$ imply,
that  there is a $|\cV_n| < \infty$, such that
$\|V_n(\omega,\cdot)\|_{L^{\infty}} \leq |\cV_n|< \infty$,  
$\P$-almost everywhere. 
Hence $H(\omega)=H_0-V_n(\omega)$ is bounded from below, selfadjoint
on $\D(-\lap)$ and $C_0^{\infty}(\R^{nd})$ is a common core for $\P$
almost every $\omega \in \Omega$.
\end{proof}
%
The same theorem and proof applies for the potentials $V_I$ from
(\ref{eq1.2a}), when we assume that $V_I$ is infinitesimally $-\lap$-bounded.
%
\begin{theorem} \label{t2.2}
There are closed sets $\Sigma=\Sigma_{\mathrm{ess}}$, $\Sigma_{\mathrm{ac}}$,
$\Sigma_{\mathrm{sc}}$, $\Sigma_{\mathrm{pp}}$, $\Sigma_{\mathrm{c}}$, $\Sigma_{\mathrm{s}} \subseteq \R$,
such that
$\Sigma=\sigma(H(\omega))=\sigma_{\mathrm{ess}}(H(\omega))$,
$\Sigma_{\mathrm{ac}}=\sigma_{\mathrm{ac}}(H(\omega))$,
$\Sigma_{\mathrm{sc}}=\sigma_{\mathrm{sc}}(H(\omega))$,
$\Sigma_{\mathrm{pp}}=\sigma_{\mathrm{pp}}(H(\omega))$,
$\Sigma_{\mathrm{c}}=\sigma_{\mathrm{c}}(H(\omega))$ and
$\Sigma_{\mathrm{s}}=\sigma_{\mathrm{s}}(H(\omega))$
for $\P$-almost every $\omega \in \Omega$.
\end{theorem}
%
\begin{proof}
The result of Theorem \ref{t2.1} establishes measurability of the
operator $H$ by \cite{CL} Proposition V.3.1 or \cite{St}
Example 1.2.7. The next ingredient is a family of ergodic
transformations on $\Omega$, which we choose in the following way:
The product structure of $\Omega$ immediately shows
\be \label{eq2.4}
\lim_{k \to \infty} \P(\theta_j^{-k} A \cap B)=\P(A) \P(B)
\ee
for all $j \in \Z^{d} \backslash \{0\}$ and all cylinder sets $A,B$ in
$\B(\Omega)$, where $\theta_j:\Omega \longrightarrow \Omega$ is the
measure preserving transformation defined by 
\be \label{eq2.3}
\theta_j((\omega_k)_{k \in \Z^d})=(\omega_{j+k})_{k \in \Z^d}.
\ee
Equality (\ref{eq2.4}) extends to all $A,B \in \B(\Omega)$, cf. \cite{Wa}
Theorem 1.17, hence it is the mixing property, which implies 
ergodicity of the system $(\Omega,\P,(\theta_j)_{j \in \Z^d})$.
Apart from the ergodic transformations $(\theta_j)_{j \in \Z^d}$, we
have to find a family $(\cT_j)_{j \in \Z^d}$ of unitary operators in
$L^2(\R^{nd})$, such that the ``covariance condition''
\be \label{eq2.5}
H(\theta_j \omega) f= \cT_j^* H(\omega) \cT_j f
\ee
holds for each element $f$ in the common core $C_0^{\infty}(\R^{nd})$
of the operators $H(\omega)$, $\omega \in \Omega$, see \cite{CL}
Remark V.2.6 and Proposition V.2.7. We choose $\cT_j$ as the unitary
operators on $L^2(\R^{nd})$, which are given by
\be \label{eq2.6}
(\cT_j f)(x_1,...,x_n):=f(x_1-j,...,x_n-j)
\ee
for each $f \in C_0^{\infty}(\R^{nd})$. Thus $(\cT_j)_{j \in \Z^d}$ is
a unitary representation of $\Z^d$ in $L^2(\R^{nd})$ which is total,
see \cite{CL} page 247. The operators $\cT_j$ are tensor products of
the translation operators $T_j$ in $L^2(\R^d)$ given by 
$(T_j g)(x):=g(x-j)$ for $x \in \R^d$ and $g \in
C_0^{\infty}(\R^d)$. So
\begin{eqnarray}
\cT_j^* V_n(\omega) \cT_j&=&
(T_j^*)^{\otimes n}
\left[ \sum_{k=1}^n \one_{L^2(\R^d)}^{ \otimes (k-1)}
\otimes V_1(\omega) \otimes
\one_{L^2(\R^d)}^{ \otimes (n-k)} \right]
T_j ^{\otimes n} \nonumber\\
&=&\sum_{k=1}^n 
\one_{L^2(\R^d)}^{ \otimes (k-1)}
\otimes T_j^* V_1(\omega)T_j \otimes
\one_{L^2(\R^d)} ^{\otimes (n-k)} \label{eq2.7}
\end{eqnarray}
and the standard calculation
\begin{eqnarray*}
\lefteqn{(V_1(\theta_j\omega)g)(x)= V_1(\theta_j\omega,x)g(x)=
\sum_{k \in \Z^d} \omega_{k+j}' \wp(x-k-\omega_{k+j}'') g(x)=}\\
&=&\sum_{k \in \Z^d} \omega_{k+j}' \wp((x+j)-(k+j)-\omega_{k+j}'') g(x)=
V_1(\omega,x+j) g(x)=\\
&=&(V_1(\omega)T_jg)(x+j)=(T_j^* V_1(\omega)T_jg)(x)
\end{eqnarray*}
for each $x \in \R^d$ and $g \in C_0^{\infty}(\R^d)$ in the one
electron case proves
\be \label{eq2.8}
\cT_j^* V_n(\omega) \cT_j f =V_n(\theta_j \omega) f
\ee 
for $f \in C_0^{\infty}(\R^{nd})$. From the invariance of the pair interaction
potential with respect to simultaneous translations, i.e.
$V_I(x_1,...,x_n)=V_I(x_1-x,...,x_n-x)$ for each $x \in \R^d$ we
conclude
\begin{eqnarray*}
\lefteqn{ (\cT_j^* V_I \cT_jf)(x_1,...,x_n)=V_I(x_1+j,...,x_n+j)(\cT_j
f)(x_1+j,...,x_n+j)=}\\
&=&V_I(x_1,...,x_n)f(x_1,...,x_n). \hspace{7cm}
\end{eqnarray*}
Translation invariance of the Laplacian now finally implies the
covariance condition (\ref{eq2.5}), so we may apply ergodic operator
family theory; this finishes the proof, see \cite{CL} chap. V.2 or \cite{St}
chap. 1.2.
\end{proof}
%
\noindent Theorem \ref{t2.2} allows us to speak of $\Sigma$ as the
deterministic spectrum of $H$. In the same way the essential, continuous,
absolute continuous, singular continuous, singular and pure point spectrum
are deterministic. Note that there are no isolated
eigenvalues of finite multiplicity and that the eigenvalues of $H$ depend on
the realizations $\omega$, but the pure point spectrum of $H$, which is the
closure of the set of eigenvalues, is deterministic.
%
\section{The Wegner estimate} \label{kap3}
\setcounter{equation}{0}
Let $\Lambda \subseteq \R^{nd}$ be an open bounded cube, let
$H_{\Lambda}(\omega)$ denote the restriction of $H(\omega)$ to
$\Lambda$ with Dirichlet boundary conditions and
$(\pi_{\Lambda}(A,\omega))_{A \in \B(\R)}$ the associated spectral
measure. Then Wegner's estimate
\[\bbE ({\mathrm{Trace}}\, \pi_{\Lambda}(]E_0-\eta,E_0+\eta[)) \leq C_W
    \eta |\Lambda| \]
bounds the expectation value of the trace of the spectral projection
of $H_{\Lambda}$ onto the interval $]E_0-\eta,E_0+\eta[$ by the length
of the interval and the volume $|\Lambda|$ of the cube. The proof in
\cite{CoHi} in the one electron model we want to adapt uses
a spectral averaging procedure plus Dirichlet Neumann bracketing. 
So we have to assume, that there is a density $g \in L^{\infty}(\R)$ for
the probability distribution of $\P_0$, i.e.
\be \label{eq1.4}
\P_0(]-\infty,t[)=\int\limits_{-\infty}^t g(x) dx.
\ee
This assumption for $\P_0$, which seems rather strange
from physical point of view is the price one has to pay
- even in the single electron Anderson model- 
for a proof of Wegner's estimate with a volume dependent bound proportional to
$|\Lambda|$. This type of spectral
averaging works with a family of operators
$H(\mu)=H_0-\mu V, \mu \in \C $, with a selfadjoint operator $H_0$ and
a bounded $V \geq 0$. We use the spectral averaging from section
\ref{kap3.2} by putting kinetic and electron-electron interaction
terms of $H_{\Lambda}(\omega)$ into the operator $H_0$ of spectral
averaging plus a $\omega$-dependent decomposition of the Anderson
random potential
$V_n(\omega)$ into  $-\mu V$ and $H_0$ parts. To be sure that we are
dealing with well defined operators, we look first at the restrictions
of $-\lap+W$ to an open bounded cube $\Lambda$ with Dirichlet and
Neumann boundary conditions, where $W$ is an infinitesimally $-\lap$
bounded perturbation. 
\subsection{Restricting the nonrandom Hamiltonian \protect \boldmath $H_0$}
%
The potential
\be \label{eq1}
V_C=\sum_{1 \leq k < l \leq n} \frac{1}{|x_k-x_l|}
\ee
acting in $L^2(\R^{nd})$ describes the repulsive Coulomb interaction 
of $n$ electrons. In Theorem \ref{t2.1} we have defined $H_0=-\lap+V_C$ 
as a selfadjoint
operator on $\D(-\lap)=H^2(\R^{nd})$, then we remarked, that the same
can be done for all infinitesimally $-\lap$ bounded potentials $W$.
Now we want to restrict the
configuration space from $\R^{nd}$ to a bounded open cube 
$\Lambda \subseteq \R^{nd}$ and define $H_{0,\Lambda}$ and
$H_{0,\Lambda,N}$, the restrictions of $H_0$ to $\Lambda$ with
Dirichlet and Neumann boundary conditions. 

\begin{lemma} \label{l2}
Let $W$ be a real valued function on $\R^{nd}$, such that the
corresponding multiplication operator is infinitesimally
$-\lap$-bounded in $L^2(\R^{nd})$. Then
the form domain $Q(W)$ of $W$ contains $H^1(\R^{nd})$ and for each
$\alpha >0$ there is a $\gamma(\alpha)<\infty$, such that
\be \label{eq5}
|\lkl \Psi, W\Psi \rkl| \leq \alpha \| \nabla \Psi \|^2 + \gamma(\alpha)
\|\Psi\|^2
\ee
for all $\Psi \in H^1(\R^{nd})$.
\end{lemma}
\begin{proof}
According to \cite{RS2} Theorem X.18 the infinitesimally $-\lap$-bounded
selfadjoint perturbation $W$ of the positive operator $-\lap$ is
infinitesimally form bounded, which is by definition, see again
\cite{RS2} p.~168, the claim of the lemma.
\end{proof}
%
\begin{theorem} \label{t2}
Let $W$ be a infinitesimally $-\lap$-bounded potential acting in
$L^2(\R^{nd})$ and let 
$\Lambda \subseteq \R^{nd}$ be an open bounded cube. The
restriction $H_{0,\Lambda}$ of $H_0=-\lap+W$ to $\Lambda$ with Dirichlet
boundary conditions, given by the closure of the quadratic form
\begin{eqnarray}
q_{\Lambda}: C_0^{\infty}(\Lambda) & \longrightarrow & \R \label{eq6}\\
\Psi & \longmapsto & \lkl \nabla \Psi, \nabla \Psi \rkl+ \lkl \Psi,
W\Psi \rkl \nonumber
\end{eqnarray}
is a selfadjoint operator, bounded from below with compact resolvent.
\end{theorem}
\begin{proof}
{From} the construction of the Dirichlet Laplacian $-\lap^{\Lambda}$
we know that 
$\displaystyle \begin{array}[t]{rcl}H_0^1(\Lambda) & \longrightarrow &
\R \\
\Psi & \longmapsto & \|\nabla \Psi\|^2
\end{array}$ 
is the closure of 
$\displaystyle \begin{array}[t]{rcl}C_0^{\infty}(\Lambda) & \longrightarrow &
\R \\
\Psi & \longmapsto & \|\nabla \Psi\|^2
\end{array}$. The form bound (\ref{eq5}) is true for all $\Psi \in
C_0^{\infty}(\Lambda) \subseteq H^1(\R^{nd})$, hence $q_{\Lambda}$ is
closable, see \cite{Ka} VI. Theorem 1.33. So the closure
$\overline{q_{\Lambda}}$ of $q_{\Lambda}$ is a closed form bounded
from below on
a dense domain of $L^2(\Lambda)$, so the representation theorems, see
\cite{Ka} VI. \S 2, apply and give a selfadjoint operator
$H_{0,\Lambda}$ bounded from below. The form domain 
$H_0^1(\Lambda)$ of $H_{0,\Lambda}$
is a subset of $H^1(\R^{nd})$, so due to (\ref{eq5}) 
\[\lkl \Psi,-\lap^{\Lambda} \Psi \rkl (1-\alpha) -\gamma(\alpha) \|\Psi\|^2 
\leq \lkl \Psi, H_{0,\Lambda}
\Psi \rkl \leq \lkl \Psi,-\lap^{\Lambda} \Psi \rkl (1+\alpha)+
\gamma(\alpha) \|\Psi\|^2\]
for all $\Psi \in H_0^1(\Lambda)$
hence the numbers
\begin{eqnarray*}
\mu_k(H_{0,\Lambda})&:=& \sup \{
\inf \{ \lkl \Psi, H_{0,\Lambda} \Psi \rkl: \Psi \in H_0^1(\Lambda),
\Psi \in \lin \{\varphi_1,...,\varphi_{k-1}\}^{\perp}\\
&& \hspace{3.9cm}\|\Psi\|=1\} :
\varphi_1,...,\varphi_{k-1} \in L^2({\Lambda})\}
\end{eqnarray*}
in the min-max principle, \cite{RS4} Theorem XIII.2 are bounded by
\be \label{eq7}
\mu_k(-\lap^{\Lambda})(1-\alpha)-\gamma(\alpha) \leq \mu_k(H_{0,\Lambda}) \leq 
\mu_k(-\lap^{\Lambda})(1+\alpha)+\gamma(\alpha).
\ee
Compact resolvent of $-\lap^{\Lambda}$ is equivalent to 
$\mu_k(-\lap^{\Lambda}) \stackrel{k \to \infty} \longrightarrow \infty$, see
\cite{RS4} Theorem XIII.64, so $\mu_k(H_{0,\Lambda})
\stackrel{k \to \infty}{\longrightarrow} \infty$ by choosing $\alpha <1$ 
in (\ref{eq7}), hence
$H_{0,\Lambda}$ has compact resolvent.
\end{proof}
%
%
%
For the definition of $H_{0,\Lambda,N}$, the restriction of $H_0$ with
Neumann boundary conditions the shape of $\Lambda$ comes into
play. 
%
%
\begin{lemma} \label{l3}
Fix a bounded open cube $\Lambda \subseteq \R^{nd}$ and an
infinitesimally $-\lap$-bounded potential $W\geq 0$ in $L^2(\R^{nd})$. 
Then for all $\alpha >0$ there is a $\eta(\alpha) <\infty$, such that
\be \label{eq8}
\lkl \Psi, W \Psi\rkl \leq \alpha \|\nabla \Psi\|^2 + \eta(\alpha)
\|\Psi\|^2
\ee
for all $\Psi \in H^1(\Lambda)$.
\end{lemma}
%
%
\begin{proof}
Regularity of the boundary of $\Lambda$ allows the construction of a
linear operator 
\[F_{{\Lambda}_{\varepsilon}}:H^1(\Lambda) \longrightarrow
H_0^1(\Lambda_{\varepsilon}) \]
from the Sobolev space $H^1(\Lambda)$ into the $H^1$ closure
$H_0^1(\Lambda_{\varepsilon})$ of $C^{\infty}$-functions with compact
support inside the $\varepsilon$-ball 
$\Lambda_{\varepsilon}:=\{x \in \R^{nd}:\mathrm{dist}(x,\Lambda) <
\varepsilon\}$ around $\Lambda$ with the following properties, see
\cite{Wl} Satz 5.6 and Folgerung 5.2:
\begin{enumerate}
\item For each $\Psi \in H^1(\Lambda)$ the restriction
$F_{{\Lambda}_{\varepsilon}} \Psi |_{\Lambda}$ of
$F_{{\Lambda}_{\varepsilon}} \Psi$ to $\Lambda$ coincides with $\Psi$:
$F_{{\Lambda}_{\varepsilon}} \Psi |_{\Lambda}=\Psi$.
\item $F_{{\Lambda}_{\varepsilon}}$ is
continuous with respect to the $H^1$-norms and the $L^2$-norms, i.~e.
there are $c_{H^1},c_{L^2}<\infty$, such that
$\|F_{{\Lambda}_{\varepsilon}} \Psi\|_{H^1(\Lambda_{\varepsilon})} \leq
c_{H^1} \|\Psi\|_{H^1(\Lambda)}$ and
$\|F_{{\Lambda}_{\varepsilon}} \Psi\|_{L^2(\Lambda_{\varepsilon})} \leq
c_{H^1} \|\Psi\|_{L^2(\Lambda)}$.
\end{enumerate}
From the orthogonal decomposition 
$F_{\Lambda_{\varepsilon}}\Psi=\Psi+(F_{\Lambda_{\varepsilon}}\Psi-\Psi)$ 
in $L^2(\R^{nd})$ and the fact, that the positive multiplication
operator $W$ in $L^2(\R^{nd})$ does not change the support, we
conclude
\be \label{eq9}
\lkl F_{\Lambda_{\varepsilon}}\Psi, W F_{\Lambda_{\varepsilon}}\Psi\rkl=
\lkl \Psi,W \Psi\rkl+
\lkl(F_{\Lambda_{\varepsilon}}\Psi-\Psi),W(F_{\Lambda_{\varepsilon}}\Psi-\Psi)\rkl
\geq \lkl \Psi,W\Psi\rkl.
\ee
For $F_{\Lambda_{\varepsilon}}\Psi \in H_0^1(\Lambda_{\varepsilon})
\subseteq H^1(\R^{nd})$ estimate (\ref{eq5}) applies, so for all
$\Psi \in H^1(\Lambda)$ and $\alpha >0$ we get:
\begin{eqnarray}
0 &\leq& \lkl \Psi, W \Psi \rkl \leq \lkl
F_{\Lambda_{\varepsilon}}\Psi, W F_{\Lambda_{\varepsilon}}\Psi\rkl
\leq \alpha \| \nabla F_{\Lambda_{\varepsilon}}\Psi \|^2_{L^2}+
\gamma(\alpha) \|F_{\Lambda_{\varepsilon}}\Psi\|^2_{L^2}\nonumber\\
&=& \alpha \|F_{\Lambda_{\varepsilon}}\Psi\|^2_{H^1}+
(\gamma(\alpha)-\alpha) \|F_{\Lambda_{\varepsilon}}\Psi\|^2_{L^2} \leq 
\nonumber\\
&\leq& c_{H^1}^2 \alpha \|\Psi\|^2_{H^1}+c_{L^2}^2
(\gamma(\alpha)-\alpha) \|\Psi\|^2_{L^2}\leq\nonumber\\
&\leq& \alpha c_{H^1}^2 \|\nabla \psi\|^2_{L^2}+
(c_{L^2}^2 (\gamma(\alpha)-\alpha)+\alpha c_{H^1}^2) \|\Psi\|^2_{L^2},
\label{eq10}
\end{eqnarray}
which is the desired bound.
\end{proof}
%
%
\begin{theorem} \label{t3}
Let $\Lambda \subseteq \R^{nd}$ be an open bounded cube and let $W\geq
0$ be an infinitesimally $-\lap$-bounded operator in $L^2(\R^{nd})$. The
restriction $H_{0,\Lambda,N}$ of $H_0=-\lap+W$ to $\Lambda$ with Neumann
boundary conditions, given by the quadratic form
\begin{eqnarray}
q_{\Lambda,N}: H^1(\Lambda) & \longrightarrow & \R \label{eq11}\\
\Psi & \longmapsto & \lkl \nabla \Psi, \nabla \Psi \rkl+ \lkl \Psi,
W\Psi \rkl \nonumber
\end{eqnarray}
is a positive selfadjoint operator with compact resolvent.
\end{theorem}
\begin{proof}
{From} Lemma \ref{l3} we conclude, that 
$\begin{array}[t]{rcl}
H^1(\Lambda) & \longrightarrow & \R \\
\Psi & \longmapsto & \lkl \Psi,W \Psi \rkl \end{array} $
is an infinitesimally form bounded perturbation of the Neumann
Laplacian $-\lap^{\Lambda}_N$ on $\Lambda$. Hence $q_{\Lambda,N}$
defines a closed positive form on a dense domain and the corresponding
operator $H_{0,\Lambda,N}$ is selfadjoint and positive. By (\ref{eq8})
\[\lkl \Psi,-\lap^{\Lambda}_N \Psi \rkl \leq
\lkl \Psi, H_{0,\Lambda,N} \Psi \rkl \leq
\lkl \Psi,-\lap^{\Lambda}_N \Psi \rkl (1+\alpha)+\eta(\alpha)
\|\Psi\|^2\]
for all $\Psi \in H^1(\Lambda)$, hence by the min-max principle
\be \label{eq12}
\mu_k(-\lap^{\Lambda}_N) \leq \mu_k(H_{0,\Lambda,N}) \leq
\mu_k(-\lap^{\Lambda}_N)(1+\alpha) +\eta(\alpha),
\ee
so $\mu_k(H_{0,\Lambda,N}) \stackrel{k \to \infty}{\longrightarrow}
\infty$, which is equivalent to $H_{0,\Lambda,N}$ being an operator
with compact resolvent.
\end{proof}
%
%
%
\subsection{Spectral Averaging} \label{kap3.2}
In this section we have a look at the following quite general
situation, which is already sketched in \cite{CoHi}, chapter 4:
$\hr$ is a separable complex Hilbert space and $H_0$ a selfadjoint
operator in $\hr$ with domain $\D(H_0)$. $V \geq 0$ is a bounded
Operator on $\hr$ and for $\mu \in \C$, we define the operator
\be \label{eq13}
H(\mu):=H_0-\mu V
\ee
on the domain $\D(H(\mu))=\D(H_0)$. The operators $H(\mu)$ are not
selfadjoint for $\mu \not \in \R$, so we first list some properties:
%
%
%
\begin{lemma} \label{l4}
$H(\mu)$ is closed, $H(\mu)^*=H(\bar{\mu})$ and
for $z=x+iy \in \C$, $y=\Im z \not=0$ and 
$\mu = \alpha + i \beta \in \C$ with $\beta y \geq 0$ 
\[W(\mu):=\{\lkl H(\mu)u,u\rkl: u \in \D(H_0),\|u\|=1\} \subseteq
\{w \in \C: \beta \Im w \leq 0\} \]
is the numerical range of $H(\mu)$ and $H(\mu)-z$ 
has a bounded inverse. 
\end{lemma}
%
%
\begin{proof}
Let $u,v \in \hr$ and $(u_n)_{n \in \N}$ be a sequence in $\D(H_0)$ 
such that $u_n \to u$ and $H(\mu) u_n \to v$. 
Continuity of $V$ implies
$Vu_n \to Vu$, hence $H_0 u_n=H(\mu) u_n + \mu Vu_n \to v + \mu Vu \in \hr$.
As $ H_0$ is closed, we get $u\in \D(H_0)$ and $H_0 u =v +\mu Vu$
or $H(\mu) u=H_0 u - \mu Vu =v$, so $H(\mu)$ is closed.
For $u,v \in \D(H_0)$ 
\[\lkl H(\mu) u,v \rkl = \lkl(H_0-\mu V)u,v\rkl=\lkl u,(H_0-\bar{\mu} V)v\rkl
=\lkl u,H(\bar{\mu})v\rkl=\lkl u,H(\mu)^*v\rkl,\]
so $H(\mu)^*=H(\bar{\mu})$.
In the rest of the proof we may assume $\beta \not =0$. 
For each $u \in \D(H_0)$
\[\lkl H(\mu) u,u \rkl=
\lkl H_0 u,u\rkl - \alpha \lkl Vu,u \rkl -i\beta \lkl Vu,u\rkl \in 
\{w \in \C: \beta \Im w \leq 0\}, \]
so the numerical range satisfies
$W(\mu) \subseteq \{w \in \C: \beta \Im w\leq 0\}$.
 The two scalarproducts
$\lkl \beta V u,u\rkl$ and $\lkl yu,u\rkl$ are of the same sign for
each $u \in \D(H_0)$, hence
\begin{eqnarray*}
|\lkl (H(\mu)-z)u,u\rkl|&=&
\sqrt{\lkl(H_0-\alpha V-x)u,u\rkl^2+\lkl(\beta V+y)u,u\rkl ^2}\geq\\
&\geq& |\lkl (\beta V+y)u,u\rkl |\geq |y|\,\|u\|^2,
\end{eqnarray*}
so Kern$(H(\mu)-z)=$ Kern$(H(\bar{\mu})-\bar{z})=\{0\}$
and 
Kern$((H(\mu)-z)^*)=$ Kern$(H(\bar{\mu})-\bar{z})=$ 
Ran$(H(\mu)-z)^{\perp}=\{0\}$.
This means that Ran$(H(\mu)-z)$ is dense in $\hr$, so according to 
\cite{HiSi} Proposition 19.7 the resolvent set $\rho (H(\mu))$ of
$H(\mu)$ satisfies
$\C \backslash \overline{W(\mu)}\subseteq \rho (H(\mu))$, which proves
$\{z \in \C : \beta \Im z >0 \} \subseteq \rho(H(\mu))$.
\end{proof}
%
%
\begin{proposition} \label{p1}
Let $B$ be a bounded selfadjoint operator on $\hr$, such that there is a
$c_2>0$ satisfying $0 \leq c_2 B^2\leq V$.
For $t>0$ and $z \in \C \backslash \R$ define
\be \label{eq14}
F_t(z):=\int\limits_{\R} \frac{1}{1+t\alpha^2} B(H(\alpha) -z)^{-1}B d\alpha ,
\ee
then
\be \|F_t(z)\| \leq \frac{\pi}{c_2} \ee
\end{proposition}
%
%
\begin{proof}
According to Lemma \ref{l4}  and \cite{HiSi} Proposition 19.7
the operator
$K(\mu,z):=B(H(\mu) -z)^{-1} B$ is bounded
for $z=x+iy \in \C$, $\Im z>0$ and
$\mu=\alpha+i\beta \in \C,\Im \mu \geq 0$  by 
\begin{eqnarray}
\lefteqn{
\|K(\mu,z)\|=\|B(H(\mu)-z)^{-1} B\| \leq \|B\|^2 \|(H(\mu)-z)^{-1}\| 
\leq \label{eq18}}\\
&\leq& \|B\|^2 \frac{1}{{\mathrm{dist}}(z,\overline{W(\mu)})} \leq 
\|B\|^2 \frac{1}{{\mathrm{dist}} (z, \{w \in \C: \Im w \leq 0\})} =
\|B\|^2 \frac{1}{|\Im z|}. \nonumber
\end{eqnarray}
Thus for
$\mu=\alpha+i\beta,\beta \in [0,\frac{1}{2\sqrt{t}}]$ 
the integral
\[F_t(\beta,z):=\int\limits_{-\infty}^{\infty} \frac{1}{1+t(\alpha+i\beta)^2}
K(\alpha+i\beta,z)d\alpha\]
exists, because of the estimate
\be \label{eq46} \left| \frac{1}{1+t\mu^2}\right| = \frac{1}
{\sqrt{\left(1+t\alpha^2-t\beta^2\right)^2+\left(2t\alpha \beta\right)^2}}\leq
\frac{1}{\frac{3}{4}+t\alpha^2}.
\ee
The strategy of the proof is a calculation of $F_t(\beta,z)$ for
$\beta >0$ plus limiting process $\beta \searrow 0$:
{For} $\mu,\nu \in \C$, $\Im \mu,\Im \nu \geq 0$ 
the second resolvent equation for $z \in \C$, $\Im z >0$ yields:
\be K(\mu,z)-K(\nu,z)=(\mu-\nu)B(H(\nu)-z)^{-1} V (H(\mu)-z)^{-1} B,
\label{eq19} \ee
so $\|(H(\mu)-z)^{-1}\|\leq \frac{1}{|\Im z|}$ implies the operator
norm bound
\be \label{eq20}
\|K(\mu,z)-K(\nu,z)\| \leq |\mu-\nu|\,\|V\| \frac{\|B\|^2}{(\Im z)^2}.
\ee
Equation (\ref{eq19}) shows the existence of
\[K'(\mu,z)=\lim_{\nu \to \mu} \frac{K(\nu,z)-K(\mu,z)}{\nu-\mu}=
B(H(\mu)-z)^{-1} V(H(\mu)-z)^{-1} B,\]
for all $\mu \in \C$ with $\Im \mu >0$. So given
$t>0$ and $z \in \C$, $\Im z >0$ 
the function
\begin{eqnarray*}
f_t:\{\mu \in \C:\Im \mu >0\}&\longrightarrow& L(\hr)\\
\mu &\longmapsto& \frac{1}{1+t\mu^2}K(\mu,z)
\end{eqnarray*}
has a pole of order 1 at $\frac{i}{\sqrt{t}}$ and 
is analytic else. The residue of $f_t$ at $\frac{i}{\sqrt{t}}$ is
\[{\mathrm{Res}}(f_t,\frac{i}{\sqrt{t}})=\lim_{\mu \to \frac{i}{\sqrt{t}}}
\frac{(\mu-\frac{i}{\sqrt{t}})K(\mu,z)}{(1+i\sqrt{t}\mu)(1-i\sqrt{t}\mu)}=
\frac{-i}{2\sqrt{t}}K(\frac{i}{\sqrt{t}},z).\]
Choose
$\beta \in ]0,\frac{1}{2\sqrt{t}}]$ and $r>\frac{1}{\sqrt{t}}$
and integrate $f_t$ along the complex paths
$\displaystyle \begin{array}[t]{rcl}        
C_1(\beta,r):[-r,r]&\longrightarrow& \C\\
s&\longmapsto& s+i\beta
\end{array}$ and 
$\displaystyle \begin{array}[t]{rcl}  
C_2(\beta,r):[0,\pi]&\longrightarrow& \C\\
s&\longmapsto& i\beta+r e^{is}
\end{array}$
where $f_t$ is analytic. Using the bound
$\|K(\mu,z)\|\leq \frac{\|B\|^2}{|\Im z|}$, which is independent of $r$,
\[\lim_{r \to \infty} \int\limits_{C_2(\beta,r)} \frac{1}{1+t\mu^2}
K(\mu,z) d\mu=0,\]
so for $\beta \in ]0,\frac{1}{2\sqrt{t}}]$ 
\begin{eqnarray*}
F_t(\beta,z)&=&
\lim_{r \to \infty} \left( \int\limits_{C_1(\beta,r)} f_t(\mu) d\mu+
\int\limits_{C_2(\beta,r)} f_t(\mu) d\mu \right)=\\
&=& 2\pi i 
{\mathrm{Res}} (f_t,\frac{i}{\sqrt{t}})= \frac{\pi}{\sqrt{t}} 
K(\frac{i}{\sqrt{t}},z).
\end{eqnarray*}
The estimate
$\|\frac{1}{1+t(\alpha+i\beta)^2}K(\mu,z)\|\leq 
|\frac{1}{\frac{3}{4}+t\alpha^2}| \frac{\|B\|^2}{|\Im z|}$ together with
convergence 
$K(\alpha+i\beta,z)\stackrel{\beta \searrow 0}{\longrightarrow}
K(\alpha,z)$ 
in operator norm resulting from (\ref{eq18}), (\ref{eq46}) respectively. (\ref{eq20}) 
allow us to apply Lebesgue dominated convergence
theorem, which yields
\be \label{eq47}
F_t(z)= \int\limits_{\R} \frac{1}{1+t\alpha^2} K(\alpha,z) d \alpha=
\lim_{\beta \searrow 0} F_t(\beta,z)=
\frac{\pi}{\sqrt{t}}K(\frac{i}{\sqrt{t}},z).
\ee
The last equality reduces the proof to a
clever estimate of $\|K(\mu,z)\|$:\\
Using selfadjointness of
$H_0$, we see
$\left[(H(\mu)-z)^{-1}\right]^* = ((H(\mu)-z)^*)^{-1} =
(H(\bar{\mu})-\bar{z})^{-1}$, so together with the resolvent equation, the
imaginary part of $K(\mu,z)$ is 
\begin{eqnarray}
\Im K(\mu,z)
&=& \frac{1}{2i}(K(\mu,z)-K(\mu,z)^*)=\label{eq15}\\ 
&=&\frac {1}{2i}
B\left[(H(\mu)-z)^{-1}-(H(\bar{\mu})-\bar{z})^{-1}\right]B= 
\nonumber\\
&=&\frac{1}{2i} B\left[(H(\mu) -z)^{-1} (2i \Im \mu V +2i \Im z)
(H(\bar{\mu}) -\bar{z})^{-1}\right]B= 
\nonumber\\
&=&B\left[(H(\mu) -z)^{-1} (y+\beta V)(H(\bar{\mu})
-\bar{z})^{-1}\right]B 
\nonumber
\end{eqnarray}
So for $u \in \hr$ we estimate:
\begin{eqnarray}
\lkl (\Im K(\mu,z))u,u \rkl &=&
\lkl B(H(\mu) -z)^{-1} (y+\beta V)(H(\bar{\mu}) -\bar{z})^{-1} B u,u
\rkl= 
\nonumber\\
&=& \lkl (y+\beta V)(H(\bar{\mu}) -\bar{z})^{-1} Bu, (H(\bar{\mu}) -
\bar{z})^{-1} Bu \rkl =\nonumber\\
&=& \beta \lkl V(H(\bar{\mu})-\bar{z})^{-1}Bu,(H(\bar{\mu})-\bar{z})^{-1}Bu
\rkl +\nonumber\\
&& +y \|(H(\bar{\mu})-\bar{z})^{-1} Bu \|^2 \nonumber\\
&\geq& \beta c_2 \lkl B^2 (H(\bar{\mu})-\bar{z})^{-1}Bu,
(H(\bar{\mu})-\bar{z})^{-1}Bu\rkl =\nonumber\\
&=& \beta c_2 \lkl B(H(\mu) -z)^{-1} BB(H(\bar{\mu})-\bar{z})^{-1}
Bu,u\rkl = \nonumber\\
&=& \beta c_2\lkl K(\mu,z)K(\bar{\mu},\bar{z})u,u \rkl. \label{eq16}
\end{eqnarray}
This estimate leads to
\begin{eqnarray*}
\|K(\mu,z)^* u \|^2 &=&
\lkl K(\mu,z)^* u,K(\mu,z)^*u\rkl=
\lkl K(\mu,z)K(\mu,z)^*u,u\rkl \leq \\
&\leq& \frac{1}{|\beta| c_2} \lkl \Im K(\mu,z)u,u\rkl =
\frac{1}{|\beta| c_2 2i} \lkl (K(\mu,z)-K(\mu,z)^*)u,u \rkl  \\
&\leq& \frac{\|u\|^2}{2 |\beta| c_2}(\|K(\mu,z)\|+\|K(\mu,z)^*\|)
=\frac{\|u\|^2}{|\beta| c_2} \|K(\mu,z)\|
\end{eqnarray*}
for each $u \in \hr$, hence
\[\|K(\mu,z)\|^2=\|K(\mu,z)^*\|^2=
\left(\sup_{u \not=0} \frac{\|K(\mu,z)^* u\|}{\|u\|}\right)^2
\leq \frac{\|K(\mu,z)\|}{|\beta| c_2}\]
or 
\be \label{eq17}
\|K(\mu,z)\| \leq \frac{1}{|\beta| c_2}. 
\ee
Combining (\ref{eq17}) with (\ref{eq47}) we conclude
\[ \|F_t(z)\|=\frac{\pi}{\sqrt{t}} \|K(\frac{i}{\sqrt{t}},z)\| \leq 
\frac{\pi}{\sqrt{t}} \frac{1}{c_2 \frac{1}{\sqrt{t}}}=\frac{\pi}{c_2},\]
which completes the proof.
\end{proof}
%
%
%
\begin{lemma}\label{l5}
For $\alpha \in \R$ let $\pi_{\alpha}$ denote the spectral measure
corresponding to the selfadjoint operator $H(\alpha)=H_0 +\alpha V$.
Let $g$ be a bounded nonnegative function on $\R$ with compact support
and  $M \in \B(\R)$ a Borel set of finite Lebesgue measure $\lambda(M)<\infty$.
Then
\be \label{eq24}
\| \int\limits_{\R} g(\alpha)B \pi_{\alpha}(M)B d\alpha \| 
\leq \frac{ \|g\|_{L^{\infty}} \lambda(M)}{c_2}. \ee
\end{lemma}
%
%
\begin{proof}
For $s,x \in \R$ and $y \in \R_+$,
$\frac{y}{(s-x)^2+y^2}=\Im (\frac{1}{s-x-iy}) \geq 0$,
hence functional calculus implies
$\Im((H(\alpha)-x-iy)^{-1}) \geq 0$.
Choosing $a,b \in \R$, $a<b$ 
Stone's formula, see \cite{RS1} Theorem VII.13
together with
$\pi_{\alpha} (]a,b[) \leq \pi_{\alpha}([a,b])$ 
implies:
\be\lkl u, \pi_{\alpha} (]a,b[) u \rkl \leq \frac {1}{\pi} 
\liminf\limits_{y\searrow 0} \int\limits_a^b \Im 
\lkl u,(H(\alpha)-x-iy)^{-1}u \rkl dx \label{eq21}\ee
Writing an open set $U \subseteq \R$ as a countably union of open
pairwise disjoint intervals,
$\displaystyle U=\bigcup_{k \in \N} ]a_k,b_k[$,
we conclude:
\begin{eqnarray*}
\lkl u, \pi_{\alpha}(U) u \rkl &\nwarrow& 
\lkl u, \pi_{\alpha}(\bigcup_{k=1}^m]a_k,b_k[) u\rkl =
\sum\limits_{k=1}^{m} \lkl u, \pi_{\alpha}(]a_k,b_k[)u\rkl \leq \\
&\leq&\frac{1}{\pi} \sum\limits_{k=1}^{m} \liminf\limits_{y \searrow 0} 
\int\limits_{a_k}^{b_k} \Im \lkl u,(H(\alpha)-x-iy)^{-1}u \rkl 
dx\leq\\
&\leq& \frac{1}{\pi} \liminf\limits_{y \searrow 0} 
\int\limits_U \Im \lkl u, (H(\alpha)-x-iy)^{-1}u \rkl dx,
\end{eqnarray*}
so for each open set $U \subseteq \R$ we obtain the estimate:
\be \label{eq22}
\lkl u,\pi_{\alpha}(U) u \rkl \leq 
\liminf\limits_{y \searrow 0} \frac{1}{\pi} \int\limits_{U} 
\Im \lkl u,(H(\alpha)-x-iy)^{-1}u\rkl dx \ee
Regularity of the Lebesgue measure allows us to choose for each $k \in \N$ 
an open set $U_k$, such that $M\subseteq U_k$
and $\lambda(U_k) \leq \lambda(M)+\frac{1}{k}$, hence
$\pi_{\alpha}(M) \leq \pi_{\alpha}(U_k)$, too. Due to
$\Im\left(B(H(\alpha)-x-iy)^{-1}B\right)$$=\Im K(\alpha,x+iy) \geq 0$
we may use Fatou's lemma and Fubini's theorem for nonnegative
functions, so together with Proposition \ref{p1} this yields:
\begin{eqnarray*}
&\,&\int\limits_{\R} \frac{1}{1+t\alpha^2} \lkl u,B\pi_{\alpha}(M)Bu\rkl 
d\alpha \leq
\int\limits_{\R} \frac{1}{1+t\alpha^2} \lkl u,B\pi_{\alpha}(U_k)Bu\rkl 
d\alpha\leq\\
&\leq& \frac{1}{\pi} \int\limits_{\R} \frac{1}{1+t\alpha^2} \liminf\limits_
{y\searrow 0}\left( \; \int\limits_{U_k} \Im \lkl B u,
(H(\alpha)-x-iy)^{-1}B u\rkl dx \right) d\alpha \leq \\
&\leq &\frac{1}{\pi} \liminf\limits_{y\searrow 0} \int\limits_{\R}
\frac{1}{1+t\alpha^2}\left( \; \int\limits_{U_k} \Im \lkl u,
K(\alpha,x+iy)u\rkl dx \right) d\alpha =\\
&=&\frac{1}{\pi} \liminf \limits_{y\searrow 0} \Im\left(\; 
\int\limits_{U_k}\int\limits_{\R} \frac{1}{1+t\alpha^2}
\lkl u,K(\alpha,x+iy)u\rkl d\alpha dx\right)=\\
&=&\frac{1}{\pi} \liminf\limits_{y\searrow 0} \Im \int\limits_{U_k}
\lkl u,F_t(x+iy)u\rkl dx\leq
\frac{\|u\|^2 \lambda(U_k)}{c_2} 
\leq \frac{\|u\|^2}{c_2}(\lambda(M)+\frac{1}{k}). 
\end{eqnarray*}
for each $k \in \N$, hence
\be \label{eq23}
\int\limits_{\R} \frac{1}{1+t\alpha^2} \lkl u,B\pi_{\alpha}(M) Bu \rkl 
d\alpha \leq \frac {\lambda(M) \|u\|^2}{c_2}. \ee
Now we use the real function
$h_t(\alpha):=g(\alpha) (1+t\alpha^2)$ to estimate the norm of  
the selfadjoint operator 
$\int\limits_{\R}g(\alpha) B\pi_{\alpha}(M) B d\alpha$  
by (\ref{eq23}) as follows:
\begin{eqnarray}
\lefteqn{ \|\int\limits_{\R}g(\alpha) B\pi_{\alpha}(M) B d\alpha\|= 
\sup\{ |\int\limits_{\R} g(\alpha) 
\lkl u,B\pi_{\alpha}(M) B u \rkl d\alpha |:
\|u\|=1 \}= \nonumber }\\
&=& \sup\{ |\int\limits_{\R} h_t(\alpha) \frac{1}{1+t \alpha^2}  
\lkl u,B\pi_{\alpha}(M) B u \rkl d\alpha |:
\|u\|=1 \}\leq \nonumber \\
&\leq& \sup\{\inf_{t>0} \|h_t\|_{L^{\infty}} 
\frac{\|u\|^2 \lambda(M)}{c_2}: \|u\| \leq 1\}=
\inf_{t>0} \|h_t\|_{L^{\infty}}\frac{\lambda(M)}{c_2}. \hspace{2cm}
\label{eq25}
\end{eqnarray}
Note that the compact support of $g$ implies 
$\displaystyle \|g\|_{L^{\infty}} = \inf_{t>0} \|h_t\|_{L^{\infty}}$, 
which finishes the proof.
\end{proof}
%
%
%
\subsection{Proof of the Wegner estimate}
%
%
\begin{proposition}\label{p2}
Let $\Lambda \subseteq \R^{nd}$ be a bounded open cube and let
\begin{eqnarray}
H_{\Lambda} :\Omega & \longrightarrow & SA(L^2(\Lambda)) \label{eq26}\\
\omega &\longmapsto& H_{\Lambda}(\omega)=H_{0,\Lambda}-V_n(\omega)|_{\Lambda}
\nonumber
\end{eqnarray}
be the sum of the nonrandom operator $H_{0,\Lambda}$ constructed in
Theorem \ref{t2} as $-\lap+V_I$ restricted to $\Lambda$
with Dirichlet boundary conditions and the
restriction $V_n(\omega)|_{\Lambda}$ of the random potential
$V_n(\omega)$ for $n$ electrons.
Let $\pi_{\Lambda}$ denote the spectral measure of
$H_{\Lambda}$ and $\bbE$ the expectation value on $\Omega$ 
with respect to $\P$ and suppose, that $\P_0$ is absolutely continuous
with respect to Lebesgue measure with a bounded density 
$g \in L^{\infty}(\R)$ of compact support. 
For each $j \in \Z^{nd}$ 
we write $1^j:=1^j_1 \cdot 1_{\Lambda}$, then for each $M \in \B(\R)$, 
$\lambda(M)<\infty$ the following estimate holds true:
\be \label{eq27}
\|\bbE(1^j \pi_{\Lambda}(M) 1^j)\| \leq 
\frac{ \lambda(M) \|g\|_{L^{\infty}}}{c_0}\ee
\end{proposition}
%
%
\begin{proof}
For $j=(j_1,...,j_n) \in \Z^{nd}$ we write the probability space
\[(\Omega,\P)=
(\R^{\Z^d} \times (\R^d)^{\Z^d}, \P_0^{\otimes \Z^d} \otimes
\P_1^{\otimes \Z^d})\]
as the product of the probability spaces
\[\hat{\Omega}_{j_1}:=
\left( \R^{\Z^d \backslash \{j_1\}} \times 
\left(\R^d\right)^{\Z^d\backslash \{j_1\}}, 
\hat{\P}=\P_0^{\otimes \Z^d \backslash \{j_1\}}
\otimes \P_1^{\otimes \Z^d \backslash \{j_1\}} \right),\]
$(\R^d,\P_1)$ and $(\R,\P_0)$ and according to this product decomposition we
write $\omega=(\hat{\omega},\omega_{j_1}'',\omega_{j_1}')$.
We find this product structure also for the random potential, which can be
written as
\begin{eqnarray}
V_n(\omega)&=&V_n(\hat{\omega},\omega_{j_1}'',\omega_{j_1}')= \label{eq28}\\
&=& \sum_{k=1}^n \left[ \underbrace{\one \otimes ... \otimes \one}_{k-1} 
\otimes
(\sum_{i \in \Z^d \atop i \not=j_1} \omega_i' T_{i+\omega_i''}\wp)
\otimes \underbrace{\one \otimes ... \otimes \one}_{n-k} \right]
\nonumber\\
&&+ \omega_{j_1}' 
\sum_{k=1}^n \underbrace{\one \otimes ... \otimes \one}_{k-1} 
\otimes T_{j_1+\omega_{j_1}''}\wp
\otimes \underbrace{\one \otimes ... \otimes \one}_{n-k} 
\nonumber\\
&=:&\widehat{V}^{j_1}(\hat{\omega})+\omega_{j_1}'\tilde{V}(\omega_{j_1}'')
\nonumber
\end{eqnarray}
The assumptions on $\wp$ and $\P_1$ imply 
$T_{j_1+\omega_{j_1}''} \wp \geq c_0 (1^{j_1})^2$ as operators in
$L^2(\R^d)$ for $\P_1$-almost every $\omega_{j_1}''$, hence
\be \label{eq29}
T_{j_1+\omega_{j_1}''} \wp \otimes \underbrace{\one \otimes
... \otimes \one}_{N-1} \geq c_0 (1^j)^2
\ee
as operators in $L^2(\R^{nd})$. From
$\displaystyle \sum_{k=2}^n \underbrace{\one \otimes ... \otimes \one}_{k-1} 
\otimes T_{j_1+\omega_{j_1}''} \wp 
\underbrace{\otimes \one \otimes ... \otimes \one}_{n-k}
\geq 0$
and (\ref{eq29}) we conclude $\tilde{V}(\omega_{j_1}'') \geq c_0
(1^j)^2$ in $L^2(\R^{nd})$ and
\be \label{eq30}
\tilde{V}(\omega_{j_1}'')|_{\Lambda} \geq c_0 (1^j)^2|_{\Lambda}
\ee
in $L^2(\Lambda)$. On the other hand 
\be \label{eq31}
H_{\Lambda,j_1}(\hat{\omega}):=H_{0,\Lambda}-\widehat{V}^{j_1}(\hat{\omega})|_{\Lambda}
\ee
is for $\hat{\P}$-almost every $\hat{\omega} \in \hat{\Omega}_{j_1}$ a
selfadjoint operator with
\be \label{eq32}
H_{\Lambda}(\omega)=H_{\Lambda,j_1}(\hat{\omega})-\omega_{j_1}'
\tilde{V}(\omega_{j_1}'').
\ee
So we are in the position to use Lemma \ref{l5} with
$\hr=L^2(\Lambda)$, $H_0=H_{\Lambda,j_1}(\hat{\omega})$ and 
$V=\tilde{V}^{j_1}(\omega_{j_1}'')$
for the estimate of
\begin{eqnarray}
\lefteqn{
\lkl \phi, \bbE (1^j\pi_{\Lambda} (M)1^j) \phi \rkl =\label{eq33}}\\
&=&\int\limits_{\hat{\Omega}_{j_1}} d\hat{\P} (\hat{\omega})
\int\limits_{\R^d} d\P_1(\omega_{j_1}'')
\int\limits_{\R} d\P_0(\alpha)
\lkl \phi, 1^j \pi_{\Lambda}(M,(\hat{\omega},\omega_{j_1}'',\alpha)) 
1^j \phi \rkl = \nonumber\\
&=&
\int\limits_{\hat{\Omega}_{j_1}} d\hat{\P} (\hat{\omega})
\int\limits_{\R^d} d\P_1(\omega_{j_1}'')
\int\limits_{\R} d\alpha g(\alpha)
\lkl \phi, 1^j \pi_{\Lambda}(M,(\hat{\omega},\omega_{j_1}'',\alpha)) 
1^j \phi \rkl \leq \nonumber\\
&\leq& \int\limits_{\hat{\Omega}_{j_1}} d\hat{\P} (\hat{\omega})
\int\limits_{\R^d} d\P_1(\omega_{j_1}'')
\frac{\lambda(M) \|g\|_{L^{\infty}}}{c_0} \|\phi\|^2=
\frac{\lambda(M) \|g\|_{L^{\infty}}}{c_0} \|\phi\|^2,
\nonumber
\end{eqnarray}
hence $\| \bbE (1^j\pi_{\Lambda} (M)1^j)\| \leq \frac{\lambda(M)
\|g\|_{L^{\infty}}}{c_0}$. 
\end{proof}
%
%
%
\begin{theorem} \label{t4}
Let $\P_0$ have a density $g \in L^{\infty}(\R)$ of compact support.
For each bounded open cube $\Lambda \subseteq \R^{nd}$ of sidelength
greater one, $H_{\Lambda}$, $\pi_{\Lambda}$ and $\bbE$ as in
Proposition \ref{p2} and each $E_0 \in \R$, there is a 
$C_W=C_W(E_0,\|g\|_{L^{\infty}})<\infty$, such that
\be \label{eq34}
\bbE(\mathrm{Trace} (\pi_{\Lambda}(]E_0-\eta,E_0+\eta[))) \leq C_W
\eta \lambda(\Lambda)
\ee
for $\eta >0$. Moreover the real function
$E_0 \mapsto C_W(E_0,\|g\|_{L^{\infty}})$ 
remains bounded on each interval, which is bounded from above.
\end{theorem}
%
%
\begin{proof} 
We start the proof with some remarks about $H_{\Lambda}(\omega)$.
Let $J=J(\Lambda):=\{j \in \Z^{nd}: \Lambda(j,1) \cap \Lambda \not= 
\emptyset\}$ and for $j \in J$ define $\Lambda_j:= \Lambda(j,1) \cap
\Lambda$.
According to \cite{RS4} XIII.15 Proposition 3 and Proposition 4 the
Dirichlet and Neumann Laplacians satisfy
\[-\lap^{\Lambda} \geq -\lap_N^{\Lambda} \geq 
-\lap_N^{\Lambda\backslash \Lambda'} = 
-\bigoplus_{j \in J} \lap_N^{\Lambda_j}.\]
$\displaystyle \Lambda':=
\Lambda \backslash \bigcup_{j \in J} \Lambda_j$ has Lebesgue measure
zero, so there is no restriction on the domain arising from the
multiplication operator with the Coulomb potential $V_C|_{\Lambda'}$
restricted to $\Lambda'$, so by the $\P$ almost sure boundedness
$\|V_n(\omega)\| \leq |\cV_n|$, the operator inequality
\be \label{eq38}
H_{\Lambda}(\omega) \geq H_{N,\Lambda}:=\bigoplus_{j \in J}
H_{0,N,\Lambda_j}-|\cV_n|
\ee 
holds true, where $H_{0,N,\Lambda_j}$ is the restriction of
$-\lap+V_I$ to $\Lambda_j$ with Neumann boundary conditions.
Fix $E_0 \in \R$ and set
$I_{\eta}:=]E_0-\eta,E_0+\eta[$.
Calculating the trace in an orthonormal basis
$(e_k)_{k \in \N}$ of $L^2(\Lambda)$, from
\begin{eqnarray*}
\lkl e_k,\pi_{\Lambda}(I_{\eta})e_k\rkl &=& 
\int\limits_{\R} 1_{I_{\eta}}(x) d\pi_{\Lambda,e_k,e_k}(x)=
\int\limits_{\R} 1_{I_{\eta}} (x) e^{-x} e^x d\pi_{\Lambda,e_k,e_k}(x) \leq\\
&\leq&e^{E_0+\eta} \int\limits_{\R} 1_{I_{\eta}} (x) e^{-x} 
d\pi_{\Lambda,e_k,e_k}(x)=
e^{E_0+\eta} \lkl e_k,\pi_{\Lambda}(I_{\eta}) e^{-H_{\Lambda}} e_k\rkl
\end{eqnarray*}
we see
\be \label{eq36}
\bbE({\mathrm{Trace}}(\pi_{\Lambda}(I_{\eta}))) 
\leq e^{E_0+\eta}
\bbE({\mathrm{Trace}}(\pi_{\Lambda}(I_{\eta})e^{-H_{\Lambda}})).
\ee
Boundedness $\|V_n(\omega)\|\leq |\cV_n|$ of the random potential 
$\P$-almost everywhere implies the estimate of $H_{\Lambda}(\omega)$
in terms of the restriction $H_{0,\Lambda}$ of $-\lap+V_I$ to
$\Lambda$ with Dirichlet boundary conditions:
\be \label{eq37}
\lkl \varphi,(H_{0,\Lambda}-|\cV_n|) \varphi \rkl 
\leq \lkl \varphi,H_{\Lambda}(\omega) \varphi\rkl 
\leq \lkl \varphi, (H_{0,\Lambda}+|\cV_n|) \varphi\rkl ,
\ee
so the min-max principle gives an estimate for the eigenvalues of 
$H_{\Lambda}(\omega)$ as 
\be \label{eq43}
\mu_k(H_{0,\Lambda}) -|\cV_n|  \leq \mu_k(H_{\Lambda}(\omega)) \leq 
\mu_k(H_{0,\Lambda}) +|\cV_n|
\ee
and proves, that $H_{\Lambda}(\omega)$ has compact resolvent. Hence
for almost every $\omega \in \Omega$
there is an orthonormal basis $(\varphi_k)_{k \in \N}$ of $L^2(\Lambda)$
consisting of eigenvalues of $H_{\Lambda}(\omega)$ - thus depending on
the realization $\omega$.
Let $M:=\{k \in \N:\mu_k \in I_{\eta}\}$, then the Trace calculated in the
orthonormal basis $(\varphi_k)_{k \in \N}$ is estimated by 
\begin{eqnarray}
\lefteqn{
{\mathrm{Trace}}(\pi_{\Lambda}(I_{\eta},\omega) e^{-H_{\Lambda}(\omega)} ) = 
\sum\limits_{k \in \N} \lkl \varphi_k, \pi_{\Lambda}(I_{\eta},\omega) 
e^{-H_{\Lambda}(\omega)} \varphi_k \rkl = \label{eq45}}\\
&=&\sum\limits_{k \in \N} \lkl \pi_{\Lambda}(I_{\eta},\omega)
\varphi_k, e^{-H_{\Lambda}(\omega)} \varphi_k \rkl 
= \sum \limits_{k \in M} \lkl \varphi_k, e^{-H_{\Lambda}(\omega)} \varphi_k \rkl=
\sum \limits_{ k \in M} e^{-\mu_k}=\nonumber\\
&=&\sum \limits_{k \in M} e^{\lkl \varphi_k, -H_{\Lambda}(\omega)
\varphi_k \rkl} \leq 
\sum \limits_{k \in M} e^{\lkl\varphi_k,- H_{N,\Lambda} \varphi_k\rkl} \leq
\sum\limits_{k \in M} \lkl \varphi_k, e^{-H_{N,\Lambda}} \varphi_k \rkl =
\nonumber \\
&=& \sum \limits_{k \in \N} \lkl \pi_{\Lambda}(I_{\eta},\omega) \varphi_k, 
e^{-H_{N,\Lambda}} \varphi_k \rkl =
\sum \limits_{k \in \N} \lkl \varphi_k, \pi_{\Lambda}(I_{\eta},\omega) 
e^{-H_{N,\Lambda}} \varphi_k \rkl= \nonumber\\
&=&{\mathrm{Trace}}(\pi_{\Lambda}(I_{\eta},\omega) e^{-H_{N,\Lambda}}).
\nonumber
\end{eqnarray}
%
The last inequality of the third line above uses $e^{\lkl \varphi,-H
\varphi\rkl} \leq \lkl \varphi,e^{-H} \varphi\rkl$ for each
selfadjoint operator $H$ bounded from below and for $\varphi \in \D(H)$, 
which is a result of spectral calculus and Jensen's inequality.
For the calculation of 
${\mathrm{Trace}}(\pi_{\Lambda}(I_{\eta},\omega) e^{-H_{N,\Lambda}})$, we 
use the orthonormal basis $(\phi_{k,j})_{k \in \N}$ of
$L^2(\Lambda_j)$ consisting out of eigenvectors for
$H_{0,N,\Lambda_j}$ with eigenvalues $E_{k,j}$. Then with the eigenvectors
\[ \psi_{k,j} :=(0,...,0,\phi_{k,j},0,...,0) \in 
\bigoplus_{j \in J} L^2(\Lambda_j)\]
of $H_{N,\Lambda}$ we obtain
\begin{eqnarray}
\lefteqn{
{\mathrm{Trace}}(\pi_{\Lambda}(I_{\eta},\omega) e^{-H_{N,\Lambda}} )=
\sum\limits_{k \in \N} \sum\limits_{j \in J} 
\lkl \psi_{k,j} ,\pi_{\Lambda}(I_{\eta},\omega)e^{-H_{N,\Lambda}} 
\psi_{k,j} \rkl =\label{eq41}}\\
&=&\sum\limits_{k \in \N} \sum\limits_{j \in J}
\lkl \psi_{k,j} ,\pi_{\Lambda}(I_{\eta},\omega) \psi_{k,j} \rkl 
e^{-E_{k,j}+|\cV_n|}= \nonumber\\
&=& \sum \limits_{k \in \N} \sum \limits_{j \in J}
\lkl 1^j \psi_{k,j} ,1^j \pi_{\Lambda}(I_{\eta},\omega) 1^j \psi_{k,j} \rkl
e^{-E_{k,j}+|\cV_n|}=\nonumber\\
&=&\sum \limits_{k \in \N} \sum \limits_{j \in J}
\lkl 1^j \phi_{k,j} ,1^j \pi_{\Lambda}(I_{\eta},\omega) 1^j \phi_{k,j} \rkl
e^{-E_{k,j}+|\cV_n|}\nonumber \hspace{4cm}
\end{eqnarray}
Taking expectation values on both sides of (\ref{eq41}), 
by Proposition \ref{p2} we obtain
\begin{eqnarray}
\lefteqn{
\bbE({\mathrm{Trace}}(\pi_{\Lambda}(I_{\eta}) e^{-H_{N,\Lambda}}))=
\bbE \left( \sum\limits_{j \in J} \sum\limits_{k \in \N}
\lkl \phi_{k,j}, 1^j\pi_{\Lambda}(I_{\eta}) 1^j \phi_{k,j} \rkl 
e^{-E_{k,j}+|\cV_n|} \right)=\nonumber }\\
&=&\sum\limits_{j \in J} \sum\limits_{k \in \N} \lkl \phi_{k,j}, \bbE
(1^j \pi_{\Lambda}(I_{\eta}) 1^j) \phi_{k,j} \rkl e^{-E_{k,j}+|\cV_n|}
\leq 
\nonumber\\
&\leq& \sum\limits_{j \in J} \sum \limits_{k \in \N} 
\|\bbE(1^j \pi_{\Lambda}(I_{\eta})1^j)\| e^{-E_{k,j}+|\cV_n|} \leq \nonumber\\
&\leq& \sum\limits_{j \in J} \sum \limits_{k \in \N} 
\frac{\lambda(I_{\eta}) \|g\|_{L^{\infty}}}{c_0} e^{-E_{k,j}+|\cV_n|}= 
\frac{2\eta \|g\|_{L^{\infty}}}{c_0} \sum\limits_{j \in J} 
\sum \limits_{k \in \N} e^{-E_{k,j}+|\cV_n|}, \hspace{1.5cm}
\label{eq42} 
\end{eqnarray}
the remaining problem is deterministic and we have to prove
convergence of $\sum\limits_{j \in J} 
\sum \limits_{k \in \N} e^{-E_{k,j}}$.
Suppose that for a given $j \in J$ the cube
$\Lambda_j$ has sidelength $0< l_1^j,...,l_{nd}^j \leq 1$, then
\[\pi^2 \leq a^j:=\min\{\left( \frac{\pi}{l_k^j} \right)^2:1 \leq k \leq nd \} 
< \infty.\]
Due to \cite{RS4} p. 266, the eigenvalues of the 
Neumann Laplacian are given by
\[\tilde{E}^j_{m_1,...,m_{nd}} = 
\sum\limits_{k=1}^{nd} \left( \frac{\pi}{l_k^j} \right)^2 m_k^2
\geq a^j \sum_{k=1}^{nd} m_k,\] 
indexed with $(m_1,...,m_{nd}) \in \N_0^{nd}$. The corresponding
eigenvalues $E^j_{m_1,...,m_{nd}}$ of $H_{0,N,\Lambda_j}$ are
estimated due to (\ref{eq12}) by
$\tilde{E}^j_{m_1,...,m_{nd}} \leq E^j_{m_1,...,m_{nd}}$, so
\begin{eqnarray}
\lefteqn{
\sum_{j \in J} \sum\limits_{(m_1,...,m_{nd}) \in \N_0^{nd}} 
\hspace{-0.5cm} e^{-E_{m_1,...,m_d}^j}
\leq \sum_{j \in J} \sum\limits_{(m_1,...,m_{nd}) \in \N_0^{nd}} 
\hspace{-0.5cm} e^{-\tilde{E}_{m_1,...,m_d}^j} \leq \label{eq44}}\\
&\leq& 
{\mathrm{Card}} J \sum_{(m_1,...m_{nd})\in \N_0^{nd}} \hspace{-0.5cm}
e^{-\pi^2 \sum\limits_{n=1}^{nd} m_n}=
{\mathrm{Card}}J \left( \sum\limits_{n \in \N_0} e^{-\pi^2 n}
\right)^{nd} \leq \nonumber\\
&\leq& 3^{nd} \lambda(\Lambda) 
\left( \frac{e^{\pi^2}}{e^{\pi^2}-1} \right)^{nd}. \nonumber
\end{eqnarray}
Pasting together (\ref{eq36}), (\ref{eq45}),
(\ref{eq42}) and (\ref{eq44}) we obtain:
\begin{eqnarray}
\lefteqn{
\bbE({\mathrm{Trace}}(\pi_{\Lambda}(I_{\eta}))) \leq 
e^{E_0+\eta}
\bbE({\mathrm{Trace}}(\pi_{\Lambda}(I_{\eta})e^{-H_{\Lambda}})) \leq
\label{eq48}}\\
&\leq&
e^{E_0+\eta}
\bbE({\mathrm{Trace}}(\pi_{\Lambda}(I_{\eta}) e^{-H_{N,\Lambda}}))
\leq e^{E_0+\eta+|\cV_n|}
\frac{2 \|g\|_{L^{\infty}}}{c_0} 
\left( \frac{3e^{\pi^2}}{e^{\pi^2}-1} \right)^{nd}\eta
\lambda(\Lambda)\nonumber\\
&:=& C_W(E_0,\|g\|_{L^{\infty}}) \eta \lambda(\Lambda). 
\nonumber
\end{eqnarray}
{From} the explicit form of $C_W(E_0,\|g\|_{L^{\infty}})$ given in
(\ref{eq48}), we see that the real function 
$E_0 \mapsto C_W(E_0,\|g\|_{L^{\infty}})$ is
bounded on intervals, which are bounded from above.
\end{proof}
%
\begin{corollary}
In the situation of Theorem \ref{t4}
\be \label{eq50}
\P\{\omega \in \Omega: {\mathrm{dist}}(\sigma(H_{\Lambda}(\omega)),E_0)<\eta \}
\leq C_W \eta \lambda(\Lambda). \ee
\end{corollary}
\begin{proof}
$\{\omega \in \Omega: {\mathrm{dist}}(\sigma(H_{\Lambda}(\omega),E_0)<\eta \}=
\{ \omega \in \Omega: {\mathrm{Trace}}(\pi_{\Lambda}(I_{\eta},\omega))
\geq 1\}$, 
so by Tschebyschev's inequality
\be \label{eq35}
\P \{\omega \in \Omega: 
{\mathrm{dist}}(\sigma(H_{\Lambda}(\omega),E_0)<\eta \} 
\leq \bbE({\mathrm{Trace}}(\pi_{\Lambda}(I_{\eta}))).
\ee
\end{proof}
%
\noindent 
{\bf Acknowledgement:}
I thank B. Baumgartner, F. Klopp, U. Krey and \linebreak
H. Narnhofer for helpful discussions and remarks.
%
%
%
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\end{document}
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