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magnetic Schroedinger operators, asymptotics of Eigenvalues, density 
of states
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\begin{document}
\begin{center}
{\LARGE \bf Strong-magnetic-field asymptotics of the
integrated density of states  for a random 3D Schr\"odinger operator}
\end{center}
\
\vspace{0.5cm}
\begin{center}
{\sc W.Kirsch}\\
Department of Mathematics\\ 
Ruhr University\\
Universit\"atsstrasse, 150\\
44780 Bochum, Germany\\
E-mail: werner.kirsch@mathphys.ruhr-uni-bochum.de\\
\end{center}
\
\vspace{0.5cm}
\begin{center}
{\sc G.D.Raikov}\\ 
Institute of Mathematics and Informatics\\
Bulgarian Academy of Sciences\\
Acad.G.Bonchev Str., bl. 8\\ 
1113 Sofia, Bulgaria\\
E-mail: gdraikov@omega.bg\\
\end{center}
\
\vspace{1cm}

{\bf Abstract.} We consider the three-dimensional Schr\"odinger operator with constant magnetic field 
and bounded random electric potential. We investigate the asymptotic behaviour of the integrated density 
of states for this operator as the norm of the magnetic field tends to
infinity.\\

\section{Introduction}
\setcounter{equation}{0}
 Let ${\bf b}:= (0,0,b)$, $b > 0$, ${\bf x} = (x,y,z)
\in \rt$. 
 Introduce the unperturbed selfadjoint Schr\"odinger operator 
$H_0(b): = \left(i\nabla + \frac{{\bf b} \wedge {\bf x}}{2}\right)^2$ defined
originally  
 on $C_0^{\infty}(\rt)$ and then closed in $L^2(\rt)$. It is
well-known that for each $b>0$ we have 
 \be{1-1}
 \sigma(H_0(b)) =
\sigma_{\rm ess}(H_0(b)) = [b, +\infty)
 \ee
where $\sigma(H_0(b))$ denotes the spectrum of the operator $H_0(b)$, 
and  $\sigma_{\rm ess}(H_0(b))$ - its essential spectrum (see e.g. \cite{ahs}).\\ 
Let $(\Omega, {\cal F}, {\mathbb P})$ be a probability space. Let $\vo({\bf
x})$, $\omega \in \Omega$, ${\bf x} \in \rt$, be a real random field. We assume
that $\vo$ is $\gt$-ergodic with $\gge = \ze$ or $\gge = \re$ (see
\cite[Section 3.1]{k} or \cite[Section 1B]{pfi}). In other
words, there exists an ergodic group of automorphisms  ${\cal T}_{\bf
k}: \Omega \to \Omega$, ${\bf k} \in \gt$,   such that 
\be{s-1} 
\vo ({\bf x}+ {\bf
k}) = V_{{\cal T}_{\bf k} \omega}({\bf x}), \; {\bf x}  \in \rt, \; \omega \in
\Omega , \; {\bf k} \in \gt.   
\ee
We recall that ergodicity of a group $G$ of measure preserving automorphisms
of 
$\Omega$ means that the invariance of a given set ${\cal A} \in {\cal F}$
under the action of $G$ (i.e. $g{\cal A} = {\cal A}$ for each $g \in G$)
implies either ${\mathbb P}({\cal A}) = 1$ or ${\mathbb P}({\cal A}) = 0$. \\
Let ${\bf x} \in \rt$. We shall write occasionally ${\bf x} = (\xp,z)$ 
with $\xp \in \rd$ and $z \in \re$. We suppose that $\vo$ is $\gge$-ergodic 
with $\gge = \ze$ or $\gge = \re$ in
the direction of the magnetic field (or, in brief, in the $z$-direction), i.e.
that the subgroup $\left\{{\cal T}_{\bf k}| {\bf k} = (0,0,k), \; k \in
\gge\right\}$ is ergodic.\\
 Further, we assume that the realizations of $\vo$ are almost surely uniformly
bounded, i.e. we have 
\be{10} 
c_0: = {\rm ess} - \sup_{\omega \in \Omega} \; 
\sup_{{\bf x} \in \rt}|\vo({\bf 
x})| < \infty.  
\ee 
Finally, for simplicity we suppose that the realizations of $\vo$ are almost
surely continuous. \\

{\it Example}: Let $\alpha_{\bj}: \Omega
\to {\re}$, $\bj \in \zt$, be independent identically  distributed
almost surely bounded random variables.  Assume that  $q: \rt \to \re$ is a
continuous function
 satisfying 
$$
|q({\bf x})| \leq c\langle{\bf x}\rangle^{-\beta}, \quad c>0, \quad \beta > 3.
$$ 
Set $$ \vo({\bf x}):= \sum_{\bj \in \zt} \alpha_{\bj}(\omega)\;
q({\bf x} - \bj), \;  \omega \in \Omega, \; {\bf x} \in \rt. 
$$ 
Then the random field $\vo$ is $\zt$-ergodic (see  
\cite[Model II, Section 3.3]{k} or \cite[Example
1.15a, p.23]{pfi}), $\ze$-ergodic in the direction of the magnetic field (as a
matter of fact, in all directions; see \cite[Example 2, p.615]{ekss}).
Moreover, it is obvious that almost surely the realizations of $\vo$ are 
uniformly bounded and continuous. \\

On $D(H_0(b))$ define the perturbed Schr\"odinger operator 
$$
H(b,\omega): = H_0(b) + \vo, \;b>0, \; \omega \in \Omega. 
$$
It follows from (\ref{1-1}) and (\ref{10}) that almost surely we have 
\be{11/2}
\sigma(H(b,\omega)) \subseteq [b-c_0,+\infty).
\ee
The aim of this paper is to study the asymptotic behaviour as $b \to \infty$ 
of the integrated density of states (IDOS) for the operator $H(b, \omega)$. 
In order to recall the definition of the IDOS we need several auxiliary concepts. \\
Let $\varphi_r$, $r \in {\re}_+$, and $\varphi$ be non-decreasing functions defined on 
a common domain $I \subseteq \re$. We shall write 
$$
v-\lim_{r \to \infty}\,\varphi_r = \varphi 
$$ 
if we have $\lim_{r \to \infty} \varphi_r(t) = \varphi(t)$ at all continuity points $t$ of the 
function $\varphi$. In this case the function $\varphi$ is called the vague
limit as $r \to \infty$ of the family $\varphi_r$ (cf \cite[p.313]{k}). \\
 
Further, let $T=T^*$ be a selfadjoint operator in a Hilbert space. Denote
by $P_{{\cal I}}(T)$ its spectral projection corresponding to the interval
${\cal I} \subset {\re}$. Set
$$
N(\lambda;T):=   {\rm rank}\, P_{(-\infty,\lambda)}(T), \; \lambda \in \re,
$$
$$
n_{\pm}(s;T):= {\rm rank}\, P_{(s, +\infty)}(\pm T), s>0. 
$$
On the Sobolev space ${\rm H}^2\left(\qrt\right)$, $R>0$, with Dirichlet boundary conditions define the operator 
$H_{0,R}(b):= \left(i\nabla + \frac{{\bf b} \wedge {\bf x}}{2}\right)^2$. Put
\be{11}
{\cal D}_b({\bf .}):= v-\lim_{R \to \infty} \, R^{-3} \, N(\, {\bf .} \,;H_{0,R}(b) + \vo).
\ee
The function ${\cal D}_b(\mu)$, $\mu \in \re$, is called the IDOS for the operator 
$H(b, \omega)$. It is well-known that almost surely the vague limit (\ref{11}) exists and
the quantity ${\cal D}_b(\mu)$ is non-random (see e.g. 
\cite{bhl}, \cite{ma}, \cite{u}, and the references cited there). \\
Note that (\ref{11/2}) implies that almost surely 
$\inf \sigma(H_{0,R}(b) + \vo) \geq b - c_0$  
for all $R > 0$. Therefore, 
\be{gen}
{\cal D}_b(\mu) = 0, \quad  \mu < b - c_0.
\ee
For $\mu \in \re$ set
$$
D_b(\mu):= \frac{b}{2\pi^2} \sum_{k=1}^{\infty} (\mu - (2k - 1) b)_+^{1/2}.
$$
By \cite[Theorem 3.1]{cdv} the estimates 
$$
(R-R_0)^3 D_b(\mu - C R^{-2}_0 - c_0) \leq N(\mu ;H_{0,R}(b) + \vo) \leq 
$$
$$
R^3  D_b(\mu + c_0), \; \mu \in \re, \; R>0, \; R_0 \in (0,R),
$$
hold with $C$ independent of $\mu$, $R$, and $R_0$. Then (\ref{11}) easily
implies 
 \be{cdv}
D_b(\mu - c_0) \leq {\cal D}_b(\mu) \leq 
D_b(\mu + c_0), \; \mu \in \re.
\ee
 In this paper we study the asymptotic 
behaviour as $b \to \infty$ of ${\cal D}_b(\lambda + b)$, the 
parameter $\lambda \in \re$ being fixed. 


\section{Statement of the main result}
\setcounter{equation}{0}
 
On ${\rm H}^2\left(\qre\right)$ with Dirichlet boundary conditions define the operator 
$\chi_{0,R}:= -\frac{d^2}{dz^2}$. 
 
\begin{pr} \label{p21} 
Let ${\gge } = \ze$ or $\gge = \re$. 
Let $f_{\omega}(z)$, $\omega \in \Omega$, $z \in \re$, be a real $\gge$-ergodic
random field whose realizations are almost surely uniformly bounded and
continuous.  
Then for each $\lambda \in \re$ the  limit 
$$
\varrho (\lambda ; f): = 
\lim_{R \to \infty} R^{-1} \; N(\lambda;\chi_{0,R} +
f_{\omega}) 
$$
exists almost surely. Moreover, the function $\varrho (\lambda ; f)$   
is non-random, and  continuous with respect to $\lambda \in \re$. 
\end{pr}
The proof of the existence and the non-randomness of $\varrho(\lambda; f)$ for
much more general ergodic fields $f_{\omega}$ can be found in  \cite[Chapter
7]{k}. The continuity of $\varrho(\lambda; f)$ which is guaranteed by the 
fact that $\chi_{0,R} + f_{\omega}$ is an {\em ordinary} differential
operator,  is discussed in \cite[Chapter III]{pfi}. \\


\begin{lemma} \label{l21} Assume that the hypotheses of Proposition {\rm
\ref{p21}} 
 hold. 
Let ${\cal T}: \Omega \to \Omega$ be a measure preserving automorphism. Then
we have 
\be{1}
\lim_{R \to \infty} R^{-1} \; N(\lambda;\chi_{0,R} + f_{{\cal T} \omega}) = 
\lim_{R \to \infty} R^{-1} \; N(\lambda;\chi_{0,R} + f_{\omega}).
\ee
\end{lemma}

{\it Proof}. By \cite[Chapter 6, Theorem 7]{k} we have
$$
\lim_{R \to \infty} R^{-1} \; N(\lambda;\chi_{0,R} + f_{\omega}) = 
\sup_{R > 0} R^{-1} {\mathbb E}\left(N(\lambda;\chi_{0,R} + f_{\omega})\right)
\equiv 
$$
$$ 
\sup_{R > 0} R^{-1} \int_{\Omega}N(\lambda;\chi_{0,R} + f_{\omega})\;
d{\mathbb P}(\omega) 
$$
where ${\mathbb E}$ is used as the symbol of the mathematical expectation. 
Analogously, 
$$
\lim_{R \to \infty} R^{-1} \; N(\lambda;\chi_{0,R} + f_{{\cal T}\omega}) = 
\sup_{R > 0} R^{-1} \int_{\Omega}N(\lambda;\chi_{0,R} + f_{{\cal T}\omega})\;
d{\mathbb P}(\omega). 
$$
Since ${\cal T}$ is a measure preserving automorphism, we get 
$$
\sup_{R > 0} R^{-1} \int_{\Omega}N(\lambda;\chi_{0,R} + f_{{\cal T}\omega})\;
d{\mathbb P}(\omega) = 
\sup_{R > 0} R^{-1} \int_{\Omega}N(\lambda;\chi_{0,R} + f_{\omega})\;
d{\mathbb P}(\omega)
$$ 
which entails (\ref{1}). $\diamondsuit$

Our assumptions concerning  $\vo$ imply that the random 
field $f_{\omega} = \vo(X,.)$ depending on the parameter $X \in \rd$, 
satisfies  the hypotheses of Proposition \ref{p21}. 
Moreover, if 
$\gge = \ze$, then the function  $\varrho (\lambda ; V(\xp,.))$ is 
$\zd$-periodic with respect to $X \in \rd$. In order to see this, one 
may apply Lemma \ref{l21} with ${\cal T} = {\cal T}_{{\bf k}_{0}}$ 
(see (\ref{s-1})) with ${\bf k}_{0} = (K,0)$, $K \in \zd$, and conclude that 
$$
\varrho (\lambda ; V(\xp + K,.)) = \varrho (\lambda ; V(\xp,.)), \; \lambda
\in \re, \; X \in \rd. 
$$
Note that the continuity of $\vo({\bf x})$ with respect to ${\bf x} \in \rt$, 
and the continuity of $\varrho (\lambda ; f)$ with respect to $\lambda \in \re$, 
imply the continuity of
$\varrho (\lambda ; V(\xp,.))$ with respect to $X \in \rd$. Taking into
account also its periodicity, we find that $\varrho (\lambda ; V(\xp,.))$, $X
\in
 \rd$, is uniquely determined by its values for $X \in \qd$.\\
Similarly, if 
$\gge = \re$, the quantity  $\varrho (\lambda ; V(\xp,.))$ is 
independent of $X \in \rd$. In order to see this, one 
may apply Lemma \ref{l21} with ${\cal T} = {\cal T}_{{\bf x}_{0}}$ 
(see (\ref{s-1})) with ${\bf x}_{0} = (X,0)$, $X \in \rd$, and conclude that 
$$
\varrho (\lambda ; V(\xp,.)) = \varrho (\lambda ; V(0,.)), \; \lambda
\in \re, \; X \in \rd. 
$$
Finally, the elementary  estimate
$$
N(\lambda;\chi_{0,R} + \vo(\xp,.)) \leq 
\frac{R}{\pi}\left(\lambda + c_0\right)_+^{1/2}, \;   \xp \in \qd, \;   R>0, 
$$
implies
$$
\varrho (\lambda ; v(\xp,.)) \leq \frac{1}{\pi}\left(\lambda + c_0\right)_+^{1/2}, \;   \xp \in \qd.
$$
For $\lambda \in \re$ set 
$$ 
\delta(\lambda): = \left\{
\begin{array} {l}
\int_{{\qd}} \varrho (\lambda, V(\xp,.)) \; d\xp \quad {\rm if} \quad \gge =
\ze,\\
\varrho (\lambda, V(0,.))  \quad {\rm if} \quad \gge = \re. 
\end{array}
\right.
$$
Obviously, $\delta(\lambda)$ is continuous with respect to $\lambda$.  

\begin{theorem} \label{t21}
Let ${\gge } = \ze$ or $\gge = \re$. Let $\vo$ be a real $\gt$-ergodic random
field whose realizations are almost surely uniformly bounded and continuous. 
Assume in addition that $\vo$ is $\gge$-ergodic in the direction of the
magnetic field. Then we have
\be{21}
\lim_{b \to \infty} b^{-1} {\cal D}_b(\lambda + b) = \frac{1}{2\pi} \delta(\lambda), \; \lambda \in \re.
\ee
\end{theorem}

{\it Remark}: For definiteness, we shall prove Theorem \ref{t21} in the case
$\gge = \ze$. The proof in the case $\gge = \re$ is quite similar and only
simpler. \\

The asymptotics as $b \to \infty$ of the IDOS for the two-dimensional Schr\"odinger 
operator with constant magnetic field has been extensively investigated 
during the last two decades (see e.g. \cite{bgi}, \cite{bhl}, 
\cite{mpu}, \cite{pusc}, \cite{w}).  
As far as the authors are informed, no results concerning the strong-magnetic-field 
asymptotics of the IDOS for the three-dimensional Schr\"odinger operator considered 
in this paper, are known.\\
Beside the quantity ${\cal D}_b(\lambda + b)$ whose main asymptotic term is
obtained in (\ref{21}), 
 we could consider more general quantities 
${\cal D}_b(\lambda_2 + \epsilon b)
- {\cal D}_b(\lambda_1 + \epsilon b)$ 
with $\lambda_j\in \re$, $j=1,2$,
$\lambda_1 <\lambda_2$, and $\epsilon \in \re$. 
Recall that the numbers 
$\left\{(2k-1)b)\right\}_{k \geq 1}$ are called
Landau levels. For this reason we
 shall refer to the asymptotics as $b \to
\infty$ of 
 ${\cal D}_b(\lambda_2 + (2k-1) b)
- {\cal D}_b(\lambda_1 + (2k -1) b)$, $k \in \ze$, $k \geq 1$,
as the
asymptotics of the IDOS near the $k^{th}$ Landau level. Analogously, if
$\epsilon > 1$ is not an odd integer, we shall refer to the asymptotics
as $b \to \infty$ of  ${\cal D}_b(\lambda_2 + \epsilon b)
- {\cal D}_b(\lambda_1 + \epsilon b)$  as the
asymptotics of the IDOS far from the  Landau levels. Note that the case
$\epsilon < 1$ is trivial since (\ref{gen}) implies  
$
{\cal D}_b (\lambda + \epsilon b) = 0$, $\lambda \in \re$, $\epsilon <1$,  
provided that $b$ is big enough. \\ 
Since (\ref{21}) entails 
$$
\lim_{b \to \infty} b^{-1} \left({\cal D}_b(\lambda_2 + b) - {\cal
D}_b(\lambda_1 + b)\right) 
= \frac{1}{2\pi} \left(\delta(\lambda_2) - \delta(\lambda_1)\right),
$$
we can say that Theorem \ref{t21} contains the asymptotics of the IDOS 
near the first Landau Level. The problems of obtaining the first asymptotic
term  of the
 IDOS near the higher Landau levels and far from the Landau levels
remain open
 as far as the methods used in this paper are not directly
applicable to them. We hope to solve
 these problems in a future work. Here we
would like to note that we have 
 
\be{h1}
\lim_{b \to \infty} b^{-1} \left(D_b(\lambda_2 + \epsilon b) - D_b(\lambda_1
+ \epsilon b)\right)  = \frac{1}{2\pi^2} \left((\lambda_2)_+^{1/2} -
(\lambda_1)_+^{1/2}\right), 
\ee
if $\epsilon > 1$ is an odd integer, and 
\be{h2}
\lim_{b \to \infty} b^{-1/2} 
\left(D_b(\lambda_2 + \epsilon b) - D_b(\lambda_1
+ \epsilon b)\right)  = 
\frac{1}{4\pi^2} \left(\lambda_2 -
\lambda_1\right) \sum_{1 \leq k < (\epsilon + 1)/2}
(\epsilon - (2k - 1))^{-1/2},
\ee
if $\epsilon >1$ is not an odd integer. Combining (\ref{h1}) (respectively,
(\ref{h2})) with (\ref{cdv}) we obtain generically the 
correct asymptotic order of the IDOS near the higher Landau
 levels
(respectively, far from the Landau levels).\\  

Finally, note that 
(\ref{cdv}) implies immediately 
$$
\lim_{b \to \infty} b^{-3/2} {\cal D}_b(\lambda + \epsilon b) = 
D_1(\epsilon), \; \lambda \in \re, \; \epsilon > 1, 
$$
which however yields only the rough estimate 
$$
{\cal D}_b(\lambda_2 + \epsilon b) - {\cal D}_b(\lambda_1
+ \epsilon b) = o(b^{3/2}), \quad b \to \infty.
$$ 




The methods we apply are relatively simple. First of all, we give an equivalent 
representation of ${\cal D}_b(\lambda + b)$ which is more convenient for our purposes.
Namely, on $D(H_0(b))$ we define the operator 
\be{13}
\tilde{H}(b,\omega, \lambda, R):= H_0(b) - b + (V_{\omega} - \lambda - 1) 
{\bf 1}_{\qrt}, \; b>0, \; \lambda \in \re, \; \omega \in \Omega,
\ee 
which has purely discrete negative spectrum, and show that we probability one we have 
\be{14}
{\cal D}_b(\lambda + b) = \lim_{R \to \infty} R^{-3} N(-1; \tilde{H}(b,\omega, \lambda, R)) 
\ee
provided that $\lambda + b$ is a continuity point of ${\cal D}_b$ (see Proposition 4.1 below). \\
Moreover, we apply the Birman-Schwinger principle (see \cite{b}), and similarly to \cite{r1} -- \cite{r3} we 
employ the Kac-Murdock-Szeg\"o theorem in order to reduce the study of the asymptotics as $b \to \infty$ 
of ${\cal D}_b(\lambda +b)$ to the asymptotic analysis as $R \to \infty$ and $b \to \infty$ of the 
traces of the powers of certain trace-class operators $t_{b,R,\omega}$ 
(see (\ref{30}) below). The Birkhoff-Khintchine 
ergodic theorem plays a crucial r\^ole in this analysis.\\
The paper is organized as follows. In Section 3 we investigate the asymptotics of $R \to \infty$ and 
$b \to \infty$ of ${\rm Tr}\;t_{b,R,\omega}^l$, $l \geq 1$, and some related traces. 
Section 4 contains auxiliary results.
In particular, we demonstrate (\ref{14}) as well as an analogous formula
concerning $\varrho(\lambda, V(\xp,.))$,  $\lambda \in \re$, $\xp \in \qd$. 
Finally, the proof of Theorem \ref{t21} can be found in Section 5. 
 
\section{Trace asymptotics}
\setcounter{equation}{0}
Define the operator $r: = \left(-\frac{\partial^2}{\partial z^2} + 1\right)^{-1/2}$, bounded and 
selfadjoint in $L^2(\rt)$. Evidently, 
$$
(r^2 u)(x,y,z) = \frac{1}{2} \int_{\re} e^{-|z-z'|} u(x,y,z') \, dz',
\quad u \in L^2(\rt). $$
Introduce the orthogonal projection $p_b: L^2(\rt) \to L^2(\rt)$ by 
$$
(p_b u)(x,y,z) = \int_{\rd} {\cal P}_b(x,y;x',y') u(x',y',z) \, dx' dy'
$$
where
\be{boc}
{\cal P}_b(x,y;x',y'):=
\frac{b}{2\pi } \exp{\left\{-\frac{b}{4}\left[(x-x')^2
+ (y-y')^2 + 2i(xy'-yx')\right]\right\}}.
\ee 
Obviously, $p_b$ commutes with $H_0(b)$ and $\frac{\partial}{\partial z}$ 
(and, hence, with $r$). Moreover, we have
$$
H_0(b) p_b u = \left( - \frac{\partial^2}{\partial z^2} + b\right) p_b u, \quad u \in D(H_0(b)).
$$

Fix $\lambda \in \re$ and for brevity set
$$
\qo ({\bf x}) \equiv \qo ({\bf x}; \lambda)= \vo({\bf x}) - \lambda -1, \quad
{\bf x} \in \rt, \quad \omega \in \Omega.
$$
Evidently, almost surely we have  
\be{s*}
|\qo({\bf x})| \leq c_1, \;   {\bf x} \in {\rt}, 
\ee
with $c_1:  = c_0 + |\lambda + 1|$. Put 
$$
{\tilde{\qo}}_{,R}({\bf x}): = \qo({\bf x}) {\bf 1}_{\qrt}({\bf x}), \quad {\bf x} \in {\rt}.
$$
Define the operator
\be{30}
t_{b,R} (\qo):= p_b r {\tilde{\qo}}_{,R} r p_b,
\ee
compact and selfadjoint in $L^2(\rt)$. It is easy to check that we have
\be{est}
\|p_b r |{\tilde{Q}}_{\omega,R}|^{1/2}\|_2^2 = \frac{b}{4\pi} \int_{\qrt}
|\qo({\bf x})| \, d{\bf x} \leq  \frac{bR^3}{4\pi} c_1 
\ee
where $\|.\|_2$ denotes the Hilbert-Schmidt norm. Therefore, the operator $t_{b,R}(\qo)$ is in the 
trace class. Set
$$
M_1(b):= \frac{b}{4\pi}{\mathbb E}\left(\int_{\qt} \qo (\xp,z) \, d\xp dz
\right). $$

Let $l \geq 2$. Put  
$$
M_l(b): = 
$$
$$
\frac{b}{2\pi}{\mathbb E}\left(\int_{\qt}\qo (X_1,z_1)
\int_{{\re}^{3(l-1)}}  
\Pi_{s=2}^l \qo (X_1+b^{-1/2}X_s, z_1 + z_s)
 \right. 
$$
$$
\left. \psi_l(z_2,\ldots,z_l) \Psi_l(X_2, \ldots, X_l)
dX_2 \ldots dX_l \,dz_2 \ldots dz_l  \;  dX_1 dz_1 \right), 
$$  
where
$$
\psi_l(z_2,\ldots,z_l): = \frac{1}{2^l} \exp{\left\{-|z_2| -|z_l| - \sum_{s=2}^{l-1}|z_{s+1}-z_s|\right\}}, 
$$
$$
\Psi_l(X_2, \ldots, X_l) \equiv \Psi_l(x_2, y_2,\ldots, x_l, y_l):=
\frac{1}{(2\pi)^{l-1}}\exp{\left\{-\Phi_l(x_2, y_2, \ldots, x_l,
y_l)\right\}},
 $$
and
$$
\Phi_l(x_2, y_2, \ldots, x_l, y_l): =
$$
$$
\frac{1}{4}\left\{{x_2}^2 + {y_2}^2 + {x_l}^2 + {y_l}^2 +
\sum_{s=2}^{l-1} \left((x_{s+1}-x_s)^2 + (y_{s+1}-y_s)^2 +
2i(x_{s+1} y_s - y_{s+1} x_s)\right)\right\}.
$$
Note that $\psi_l \in L^1(\re^{l-1})$, $\Psi_l \in L^1(\re^{2(l-1)})$. Hence, 
(\ref{s*}) implies 
that the integral defining $M_l(b)$ is absolutely convergent. \\

\begin{pr} \label{p31}
Almost surely we have
\be{31}
\lim_{R \to \infty} R^{-3} {\rm Tr} \; t_{b,R}(\qo)^l = M_l(b), \quad l \geq 1, 
\ee
where the operator $t_{b,R}(\qo)$ is defined in {\rm (\ref{30})}. 
\end{pr}
{\it Proof}. We shall demonstrate (\ref{31}) in the generic case $l \geq 2$. \\
It is easy to verify that 
${\rm Tr} \; t_{b,R}(\qo)^l$ = ${\rm Tr} \left(p_b r^2 \tilde{\qo}_{,R}\right)^l$, and 
that ${\rm Tr} \left(p_b r^2 \tilde{\qo}_{,R}\right)^l$ can be written 
in a standard way as an integral over $\re^{3l}$ of the diagonal value of the integral kernel 
of the operator $\left(p_b r^2 {\qo}_{,R}\right)^l$. Hence,  we have
$$
{\rm Tr} \; t_{b,R}(\qo)^l = 
$$
$$
\int_{{\re}^{3l}}  
{{\Pi}_{s=1}^{l}}' \left(\frac{1}{2}{\cal P}_b(X_{s+1}, X_s)
e^{-|z_{s+1}-z_s|} \right) \Pi_{s=1}^{l} \tilde{\qo}_{,R}(X_{s},z_s)
dX_2 \ldots dX_l \,dz_2 \ldots dz_l 
$$
where the notation
${{\Pi}_{s=1}^{l}}'$ means that in the product of $l$ factors the
variables $X_{l+1}$ and $z_{l+1}$, should be set
equal respectively to  $X_{1}$, and $z_{1}$.
Changing the variables 
$$
X_1 = X'_{1}, \; 
X_s = X'_{1} + b^{-1/2} X'_{s}, \; s=2,\ldots, l,
$$
$$
z_1 = z'_1, \; 
z_s = z'_1 + z'_s, \; s=2,\ldots, l,
$$
we get
$$
{\rm Tr} \; t_{b,R}(\qo)^l=
$$
$$
\frac{b}{2\pi}\int_{\qrt} \qo(X'_{1}, z'_1)
\int_{{\re}^{3(l-1)}} \Pi^l_{s=2} 
\tilde{\qo}_{,R}(X'_{1} + b^{-1/2} X'_{s},  z'_1 + z'_s) 
$$
\be{32}
\psi_l(z'_2,\ldots, z'_l) \Psi_l(X'_{2},\ldots, X'_{l})  
 dX'_2 \ldots dX'_l \,dz'_2 \ldots dz'_l \; dX'_{1} dz'_1.
\ee 
Our next step is to show that if we replace in (\ref{32}) all the functions $\tilde{\qo}_{,R}$ 
by $\qo$, the error will be of order $o(R^3)$ as $R \to \infty$. More precisely, we set
$$
\tilde{M}_{l,R}(b):= 
$$
$$
\frac{b}{2\pi}\int_{\qrt} \qo(X_1, z_1)
\int_{{\re}^{3(l-1)}} \Pi^l_{s=2} 
\qo(X_1 + b^{-1/2} X_s, z_1 + z_s) 
$$
$$
\psi_l(z_2,\ldots, z_l) \Psi_l(X_2,\ldots, X_l) 
dX_2 \ldots dX_l \,dz_2 \ldots dz_l \; dX_1 dz_1, \quad l \geq 2,  
$$ 
write
\be{33}
{\rm Tr} \left(t_{b,R}(\qo)\right)^l = \tilde{M}_{l,R}(b)  + 
{\cal E}_l(R,b,\omega), \; l \geq 2, 
\ee
and shall demonstrate that almost surely we have 
\be{34}
\lim_{R \to \infty}R^{-3} {\cal E}_l(R,b,\omega) = 0.
\ee
Evidently, ${\cal E}_l(R,b,\omega)$ admits the estimate 
$$
|{\cal E}_l(R,b,\omega)| \leq 
$$
$$
\frac{b}{2\pi}c_1^l R^3 \int_{\qt} 
\int_{{\re}^{3(l-1)}} \Pi^l_{s=2} 
E_{R,b}(X_1, \ldots, X_l, z_1, \ldots, z_l)
$$
$$
\psi_l(z_2,\ldots, z_l) |\Psi_l(X_2,\ldots, X_l)|   
dX_2 \ldots dX_l \,dz_2 \ldots dz_l \; dX_1 dz_1  
$$ 
where
$$
E_{R,b}(X_1, \ldots, X_l, z_1, \ldots, z_l): = 
$$
$$
\left(1 - \Pi^l_{s=2} {\bf 1}_{\qrt} 
(R X_1 + b^{-1/2} X_{s}, R z_1 + z_s)\right).
$$
Since 
$\psi_l \Psi_l \in L^1 (\re^{3(l-1)})$, 
$\| E_{R,b}\|_{L^{\infty}\left(\qt \times \re^{3(l-1)}\right)} = 1$ for every $b > 0$ and $R>0$,
and 
$$
\lim_{R \to \infty}E_{R,b}(X_1, \ldots, X_l, z_1, \ldots, z_l) = 0
$$
for almost every  
$(X_1, \ldots, X_l, z_1, \ldots, z_l)$ $\in$ $\qt \times \re^{3(l-1)}$, 
the dominated convergence theorem yields (\ref{34}). \\
Set $L = L(R) = {\rm ent}\left(\frac{R}{2}\right)$ where ${\rm ent} (x)$
denotes the  integer part of $x \in \re$. Obviously, 
\be{35}
R^{-3} \tilde{M}_{l,R}(b) = (2L+1)^{-3} \tilde{M}_{l,2L+1}(b) + o(1), \; R \to \infty.
\ee
Introduce the random variables
$$
\gamma_{\bf g}(\omega, b): = \frac{b}{2\pi}\int_{\qt} \qo(X_1 +{\bf
g}, z_1) \int_{{\re}^{3(l-1)}} \Pi^l_{s=2} 
\qo(X_1 + {\bf g} + b^{-1/2} X_s, z_1 + z_s) 
$$
$$
\psi_l(z_2,\ldots, z_l) \Psi_l(X_2,\ldots, X_l) 
dX_2 \ldots dX_l \,dz_2 \ldots dz_l \; dX_1 dz_1, \quad l \geq 2,  
\quad {\bf g} \in \zt.
$$
Set
$$
\Gamma_L = \left\{{\bf g} = (g_1, g_2, g_3) \in \zt| |g_j| \leq L, \, j=1,2,3\right\}, \; L \in \ze, 
L \geq 1.
$$
It is easy to verify that 
\be{35a}
\tilde{M}_{l,2L+1} = \sum_{{\bf g} \in \Gamma_L} \gamma_{\bf g}(\omega, b).
\ee
On the other hand, it is obvious that for each ${\bf k} \in \zt$  we have
$$
\gamma_{{\bf g}+{\bf k}}(\omega , b) = 
\gamma_{{\bf g}}({\cal T}_{\bf k}\omega , b), \; {\bf g} \in \zt, \; \omega
\in \Omega,   $$ 
(see (\ref{s-1})). Hence, the sequence $\left\{\gamma_{\bf g}\right\}_{{\bf g}
\in \zt}$ is a $\zt$-ergodic random field. Therefore, we can apply the
Birkhoff-Khintchine theorem (see e.g. \cite[Theorem 2, Section 3.1]{k} or
\cite[Proposition 1.7]{pfi}). As a result we get almost surely  \be{36}
\lim_{L \to \infty} (2L+1)^{-3}\sum_{{\bf g} \in \Gamma_L} \gamma_{\bf
g}(\omega, b) = {\mathbb E}(\gamma_{\bf 0}) \equiv  M_l(b), \quad l \geq 2.  \ee
Now the combination of (\ref{33})-(\ref{36}) yields (\ref{31}) with $l \geq
2$.\\
 The proof in the case $l=1$ is similar but much simpler.
$\diamondsuit$\\
 
Fix $\xp \in \qd$. Introduce the operator
\be{dt}
\tau_R(\qo(\xp)):= 
\left(-\frac{d^2}{dz^2} + 1\right)^{-1/2} \qo(\xp,z) {\bf 1}_{\qre}(z) \left(-\frac{d^2}{dz^2} + 1\right)^{-1/2}
\ee
which is compact and selfadjoint in $L^2(\re_z)$, and depends on the parameters $\xp \in \qd$ and 
$\omega \in \Omega$. It is easy to check that the operator $\tau_R(\qo(\xp))$ is in the trace class. \\
For $\xp \in \qd$ set
$$
\mu_1(\xp):= \frac{1}{2} {\mathbb E}\left(\int_{\qe} \qo (\xp, z) \, dz
\right), $$
$$
\mu_l(\xp):= 
$$
$$
{\mathbb E} \left(\int_{\qe} \qo(\xp, z_1) \int_{{\re}^{l-1}} 
\Pi_{s=2}^l \qo(\xp, z_1 + z_s)  \psi_l(z_2,\ldots,z_l) dz_2 \ldots
z_l\right), \quad l \geq 2.  
$$

By analogy with Proposition \ref{p31} we can demonstrate the following 

\begin{pr} \label{p32}
Almost surely  we have
$$
\lim_{R \to \infty} R^{-1} {\rm Tr}\,\tau_R(\qo(\xp))^l = \mu_l(\xp), \; l \geq
1, \; \xp \in \qd. $$
\end{pr}

Set $m_l: = \int_{\qd} \mu_l(\xp) d\xp, \; l \geq 1$.
\begin{pr} \label{p33} We have
\be{38}
\lim_{b \to \infty} b^{-1} M_l(b) = \frac{1}{2\pi} m_l, \; l \geq 1.
\ee
\end{pr} 
{\it Proof}. Obviously $M_1(b) = \frac{b}{2\pi} m_1$ which entails immediately (\ref{38}) with $l=1$.\\ 
Let $l \geq 2$. Using the fact that $p_1$ is an orthogonal projection, and
taking into account the explicit form of its kernel ${\cal P}_1$ (see
(\ref{boc})), we get
 $$
 \int_{\re^{2(l-1)}} \Psi_l(X_2, \ldots, X_l) dX_2
\ldots dX_l = 
 $$
$$
2\pi \int_{\re^{2(l-1)}} {\cal P}_1(0;X_2) {\cal P}_1(X_2;X_3) \ldots {\cal
P}_1(X_{l-1};X_l) {\cal P}_1(X_l;0)   dX_2 \ldots dX_l  = 
$$
$$
2\pi {\cal P}_1(0;0) = 1.
$$
(cf. \cite[pp.16-17]{r2}). Hence, we have 
$$
m_l = 
\frac{1}{2\pi}{\mathbb E}\left(\int_{\qt}\qo (X_1,z_1)
\int_{{\re}^{3(l-1)}} \psi_l(z_2,\ldots,z_l) \Psi_l(X_2, \ldots,X_l) 
\right. 
$$
$$
\left. \Pi_{s=2}^l \qo (X_1, z_1 + z_s) dX_2 \ldots dX_l \, dz_2 \ldots dz_l  \; 
dX_1 dz_1 \right). 
$$
Then, obviously, 
$$
b^{-1}M_l(b) - m_l = 
\frac{1}{2\pi}{\mathbb E}\left(\int_{\qt}\qo (X_1,z_1)
\int_{{\re}^{3(l-1)}} \psi_l(z_2,\ldots,z_l) \Psi_l(X_2, \ldots,X_l) 
\right. 
$$
$$
\left( \Pi_{s=2}^l \qo (X_1 + b^{-1/2}X_s, z_1 + z_s) -
\Pi_{s=2}^l \qo (X_1, z_1 + z_s)\right) 
$$
\be{ebd}
\left. dX_2 \ldots dX_l \, dz_2 \ldots dz_l  \;  dX_1 dz_1 \right). 
\ee
Since $\qo$ is almost surely uniformly bounded and continuous, while 
$\psi_{l} \in L^1(\re^{l-1})$ and $\Psi_{l} \in L^1(\re^{2(l-1)})$,
we find that it follows from the dominated convergence
theorem that (\ref{ebd}) entail  (\ref{38}). 
$\diamondsuit$.\\


Combining Propositions \ref{p31}, \ref{p32}, and \ref{p33}, we get the following
\begin{follow} \label{f31}
For each $l \geq 1$  almost surely the limits
$$
\lim_{b \to \infty} \lim_{R \to \infty} b^{-1} R^{-3} {\rm Tr} \; t_{b,R}(\qo)^l
$$
and
$$
\frac{1}{2\pi}\lim_{R \to \infty}R^{-1} \int_{\qd} {\rm Tr} \; \tau_R(\qo(\xp))^l d\xp
$$
exist, coincide, and are non-random. 
\end{follow}
 
 \section{Auxiliary results} 
\setcounter{equation}{0}   
\begin{pr} \label{p41}
Let $\tilde{H}(b,\omega,\lambda,R)$ be the operator defined in {\rm
(\ref{13})}. Let $\lambda + b$ with $\lambda \in \re$ and $b>0$, be a
continuity point of ${\cal D}_b$. Then {\rm (\ref{14})} is valid. 
\end{pr}
{\it Proof}. First, note that we have
$$
N(\lambda + b; H_{0,R}(b) + \vo) = N(-1; H_{0,R}(b) - b + \vo - \lambda -1).
$$
Hence, the minimax principle implies
$$
N(-1; H_{0,R}(b) - b + \vo - \lambda -1) \leq N(-1;\tilde{H}(b,\omega, \lambda, R)).
$$
Therefore, 
$$
\liminf_{R \to \infty}R^{-3} N(-1;\tilde{H}(b,\omega, \lambda, R)) \geq 
\liminf_{R \to \infty}R^{-3} N(\lambda + b; H_{0,R}(b) + \vo) =
$$
\be{41}
\lim_{R \to \infty}R^{-3} N(\lambda + b; H_{0,R}(b) + \vo) = {\cal D}_b(\lambda + b).
\ee
Further, fix $R>0$, $R_0 \in (0,R)$, put 
$$
{\cal O}_1  = {\cal O}_{1,R} = \qrt, 
\quad {\cal O}_2 = {\cal O}_{2,R,R_0} = \rt \setminus \left[- \frac{R-R_0}{2},\frac{R-R_0}{2}\right]^3,
$$ 
and pick two functions $\varphi_1$ and $\varphi_2$ satisfying the following properties:\\

i) $\varphi_j \in C^{\infty}(\rt)$,  $j=1,2$;\\
ii) ${\rm supp} \; \varphi_j \subseteq {\cal O}_j$,  $j=1,2$;\\
iii) $\varphi_1^2({\bf x}) + \varphi_2^2({\bf x}) = 1$ for every ${\bf x} \in {\rt}$;\\
iv) $|\nabla \varphi_j({\bf x})| \leq c_2 R_0^{-1}$ for every ${\bf x} \in \rt$ with $c_2$ 
which is independent of ${\bf x}$, $R$, and $R_0$.\\

Introduce the selfadjoint operator
$$
\tilde{H}_D(b,\omega , \lambda , R,R_0):= \left(i\nabla + \frac{{\bf b} \wedge {\bf x}}{2}\right)^2 
- b  + (\vo - \lambda - 1) {\bf 1}_{\qrt}
$$
whose quadratic form is defined originally on 
$C_0^{\infty}({\cal O}_2)$, and then is
closed in $L^2({\cal O}_2)$ (cf. (\ref{13})). Then the ``magnetic'' version of
the so-called ISM localization formula  (see \cite[Section 3.1]{cfks}) yields
$$
N(-1;\tilde{H}(b,\omega, \lambda, R)) \leq 
$$
\be{42}
N(-1; H_{0,R}(b) - b + \vo - \lambda -1 - \sum_{j=1,2}|\nabla \varphi_j|^2) + 
N(-1;\tilde{H}_D(b,\omega , \lambda , R,R_0) - \sum_{j=1,2}|\nabla
\varphi_j|^2). \ee
Obviously, 
\be{43}
N(-1; H_{0,R}(b) - b + \vo - \lambda -1 - \sum_{j=1,2}|\nabla \varphi_j|^2)
\leq  N(\lambda + b +  2 c_2^2 R_0^{-2}; H_{0,R}(b) + \vo).
\ee
Choose a sequence $\left\{\varepsilon_r\right\}_{r\geq 1}$ such that $\varepsilon_r > 0$, $ r \geq 1$, 
$\lim_{r \to \infty} \varepsilon_r = 0$, and  $\lambda + b +\varepsilon_r$, $r \geq 1$, are continuity 
points of ${\cal D}_b$. Fix $r \geq 1$ and set $R_0 =
\sqrt{2} c_2/\sqrt{\varepsilon_r}$. Then we have $$
\limsup_{R \to \infty}R^{-3}N(\lambda + b + 2 c^2_2 R_0^{-2}; H_{0,R}(b) +
\vo) =  $$
\be{44}
\lim_{R \to \infty}R^{-3}N(\lambda + b + \varepsilon_r ; H_{0,R}(b) + \vo) = 
{\cal D}_b(\lambda + b + \varepsilon_r), \; r \geq 1.
\ee
The combination of (\ref{43}) and (\ref{44}) yields
\be{44a}
\limsup_{R \to \infty}R^{-3} N(-1; H_{0,R}(b) - b + \vo - \lambda -1 -
\sum_{j=1,2}|\nabla \varphi_j|^2)  \leq {\cal D}_b(\lambda + b +
\varepsilon_r), \; r \geq 1. \ee
On the other hand, the minimax principle entails
\be{45}
N(-1;\tilde{H}_D(b,\omega , \lambda , R,R_0) - \sum_{j=1,2}|\nabla
\varphi_j|^2) \leq  N(-1;H_0(b) - b  + W)
\ee
where
$$
W = W_{\omega , \lambda , R, R_0} = 
\left((\vo - \lambda - 1) {\bf 1}_{\qrt} - \sum_{j=1,2}|\nabla
\varphi_j|^2\right) {\bf 1}_{{\cal O}_2}. $$
Arguing as in \cite[Section 5]{r1}, we deduce the estimate 
\be{46}
N(-1;H_0(b) - b  + W) \leq c_3 \int_{\rt} |W|^{3/2} d{\bf x} \leq c_4
\left(R^3 - (R-R_0)^3\right)
 \ee
where the quantities $c_3$ and $c_4$ may depend on $b$, $\lambda$, and $R_0$,
but are independent of $R$. 
 The combination of (\ref{45}) and (\ref{46})
yields 
 \be{47}
\lim_{R \to \infty}R^{-3} N(-1;\tilde{H}_D(b,\omega , \lambda , R,R_0) -
\sum_{j=1,2}|\nabla \varphi_j|^2) = 0. \ee
Putting together (\ref{42}), (\ref{44a}), and (\ref{47}), we obtain
$$
\limsup_{R \to \infty}R^{-3} N(-1;\tilde{H}(b,\omega, \lambda, R))
\leq {\cal D}_b(\lambda + b + \varepsilon_r), \;   r \geq 1.
$$
Letting $r \to \infty$ (hence, $\varepsilon_r \to 0$), and bearing in mind that $\lambda + b$ is a continuity 
point of ${\cal D}_b$, we get
\be{48}
\limsup_{R \to \infty}R^{-3} N(-1;\tilde{H}(b,\omega, \lambda, R))
\leq {\cal D}_b(\lambda + b).
\ee
Now, the combination of (\ref{41}) and (\ref{48}) yields (\ref{14}). $\diamondsuit$\\

For $b > 0$, $\omega \in \Omega$, $\lambda \in \re$, and $R > 0$, introduce the operator 
\be{del}
T_{b,\omega,\lambda,R}: = (H_0(b) - b + 1)^{-1/2} (\vo - \lambda - 1) {\bf 1}_{\qrt} (H_0(b) - b + 1)^{-1/2},  
\ee
compact and selfadjoint in $L^2(\rt)$.\\ 
\begin{follow} \label{f41}
Let $\lambda + b$ be a continuity point of ${\cal D}_b$. Then we have
$$
{\cal D}_b(\lambda + b) = \lim_{R \to \infty}R^{-3} n_-(1;T_{b,\omega,\lambda,R})
$$ 
almost surely. 
\end{follow}
{\it Proof}. It suffices to note that the Birman-Schwinger principle (see \cite[Lemma 1.1]{b}) entails 
$N(-1;\tilde{H}(b,\omega,\lambda, R)) = n_-(1;T_{b,\omega,\lambda,R})$, 
and then to apply Proposition \ref{p41}. $\diamondsuit$\\

Fix $\xp \in \qt$ and $\lambda \in \re$. For $s \in \re$, $s \neq 0$, set 
$$
\trl(s; \xp): = - {\rm sign} (s) \; \varrho \left(-\frac{\lambda +1}{s} - 1;
-\frac{1}{s}V(\xp,.)\right) $$
Since $\varrho(\lambda,V(\xp,.))$ is a continuous function
with respect to $\lambda \in \re$  (see Proposition \ref{p21}), and 
$\vo$ is uniformly bounded (see (\ref{10})), 
we find that $\trl(s;\xp)$ is a continuous  function with respect to $s
\in \re \setminus \{0\}$ for any fixed $\lambda \in \re$. Moreover,  
$$
\trl(-1,\xp) = \varrho(\lambda, V(\xp,.)), \;   \xp \in \qd.
$$
For $R>0$, $\omega \in \Omega$, $\lambda \in \re$, $\xp \in \qd$, and $s \in \re \setminus \{0\}$ set 
$$
\nu_{R,\omega, \xp, \lambda}(s) = \left\{
\begin{array} {l}
n_-(-s; \tau_R(\vo(\xp,.) - \lambda -1))
\quad {\rm if} \quad s<0, \\
-n_+(s; \tau_R(\vo(\xp,.) - \lambda -1)) \quad {\rm if} \quad s>0,  
\end{array}
\right.
$$
where the operators $\tau_R$ are defined in (\ref{dt}). 
\begin{pr} \label{p42} 
For every $\lambda \in \re$, $\xp \in \qd$, and $s \in \re \setminus \{0\}$, we have
\be{49}
\trl(s;\xp) = \lim_{R \to \infty} R^{-1} \nu_{R,\omega, \xp, \lambda}(s)
\ee
almost surely. 
\end{pr}

{\it Proof}. We shall prove (\ref{49}) in the case $s<0$. In this case Proposition \ref{p21} implies
$$
\trl(s,\xp) = \lim_{R \to \infty} R^{-1} N(-\frac{\lambda +1}{s} -1; \chi_{0,R} - \frac{1}{s} \vo(\xp,.)) = 
$$
\be{410}
\lim_{R \to \infty} R^{-1} N(-1; \chi_{0,R} - \frac{1}{s} (\vo(\xp,.)-\lambda -1)).
\ee
On ${\rm H}^2(\re)$ define the operator
$$
\tilde{\chi}_{s,\omega,\xp,\lambda,R}:= -\frac{d^2}{dz^2} - \frac{1}{s} (\vo(\xp,.)-\lambda -1) {\bf 1}_{\qre}, \; R>0.
$$
Applying the minimax principle, and bearing in mind that $\chi_{0,R} - \frac{1}{s} (\vo(\xp,.)-\lambda -1)$ and 
$\tilde{\chi}_{s,\omega,\xp,\lambda,R}$ are second-order ordinary
differential operators, we get 
$$
0 \leq N(-1;\tilde{\chi}_{s,\omega,\xp,\lambda,R}) - N(-1; \chi_{0,R} - \frac{1}{s} (\vo(\xp,.)-\lambda -1)) \leq 2.
$$
Therefore, (\ref{410}) implies  
\be{411}
\trl(s,\xp) = \lim_{R \to \infty} R^{-1}
N(-1;\tilde{\chi}_{s,\omega,\xp,\lambda,R}). 
\ee 
On the other hand, the Birman-Schwinger principle entails  
\be{412}
N(-1;\tilde{\chi}_{s,\omega,\xp,\lambda,R}) = 
n_-(-s; \tau_R(\vo(\xp,.) - \lambda -1)) \equiv \nu_{R,\omega, \xp, \lambda}(s).  
\ee 
Putting together (\ref{411}) and (\ref{412}), we obtain  (\ref{49}) with $s<0$.
The proof in the case $s>0$ is  completely analogous. $\diamondsuit$ \\

For $\lambda \in \re$ and $s \in \re \setminus \{0\}$ set 
\be{412a}
\tdl(s):=\int_{\qd} \trl(s;\xp) d\xp.
\ee
Obviously, $\tdl(s)$ is continuous with respect to $s$. Moreover, 
\be{412b}
\tdl(-1) = \delta(\lambda), \; \lambda \in \re.
\ee

{\it Remark}. The function $\nu_{R,\omega, \xp, \lambda}(s)$ of the variable $s \in \re \setminus \{0\}$ 
is non-negative on $(-\infty,0)$, 
non-positive on $(0,\infty)$, and non-decreasing on $(-\infty,0)$ and $(0,\infty)$. By (\ref{49}) and 
(\ref{412a}), the functions $\trl(s;\xp)$ and $\tdl(s)$ have the same properties.\\

\begin{follow} \label{f42} 
For each $\lambda \in \re$ almost surely we have
\be{413} 
\lim_{b \to \infty} \lim_{R \to \infty} b^{-1} R^{-3} {\rm Tr} \; t_{b,R}(\vo - \lambda -1)^l 
= \frac{1}{2\pi} \int_{\re} s^l d\tdl(s), \; l \geq 1.
\ee
\end{follow}
{\it Proof}. By Corollary \ref{f31}, we have
$$
\lim_{b \to \infty} \lim_{R \to \infty} b^{-1} R^{-3} {\rm Tr} \, t_{b,R}(\vo
- \lambda -1)^l = 
 \frac{1}{2\pi} \lim_{R \to \infty} R^{-1} \int_{\qd} {\rm
Tr}\, \tau_R(\vo(\xp,.) - \lambda -1)^l d\xp =
 $$
\be{414}
\frac{1}{2\pi} \lim_{R \to \infty} R^{-1} \int_{\qd} \int_{\re} s^l d\nu_{R,\omega,\xp, \lambda}(s)\, d\xp.
\ee
Proposition \ref{p42} easily implies that
\be{415} 
\lim_{R \to \infty} R^{-1} \int_{\qd} \int_{\re} s^l d\nu_{R,\omega,\xp, \lambda}(s)\, d\xp =
\int_{\qd} \int_{\re} s^l d\trl(s;\xp)\, d\xp = \int_{\re} s^l d\tdl(s), 
\ee 
and the combination of (\ref{414}) and (\ref{415}) yields (\ref{413}). $\diamondsuit$\\

\begin{follow} \label{f43}
For each $\lambda \in R$ and $s < 0$ we have
\be{416}
\lim_{b \to \infty} \lim_{R \to \infty} b^{-1} R^{-3} n_-(-s;t_{b,R}(\vo - \lambda -1)) = 
\frac{1}{2\pi} \tdl(s).
\ee
\end{follow}

{\it Proof}. We have $\|t_{R,b}(\vo - \lambda - 1)\| \leq c_1$ (see
(\ref{s*})).  Moreover, $\tdl(s) = 0$ if $|s| > c_1$. \\
Hence we can apply the Kac-Murdock-Szeg\"o theorem (see \cite[Section 3]{r1}) and, taking into account 
the continuity of $\tdl(.)$, to conclude that (\ref{413}) implies (\ref{416}). $\diamondsuit$ \\

\section{Proof of Theorem 2.1}
\setcounter{equation}{0}
In order to prove (\ref{21}), it suffices to show that for each sequence $\{b_j\}_{j \geq 1}$ such that 
$b_j \to \infty$ as $j \to \infty$, we have 
\be{516a}
\lim_{j \to \infty} b_j^{-1} {\cal D}_{b_j}(\lambda + b_j) = \frac{1}{2\pi}
\delta (\lambda), \; \lambda \in \re. \ee
Fix two sequences $\{\lambda^{\pm}_m\}_{m \geq 1}$ such that $\lambda_m^- <
\lambda < \lambda_m^+$, $m \geq 1$,  $\lim_{m \to \infty}\lambda_m^{\pm} =
\lambda$, and $\lambda_m^{\pm} + b_j$ are continuity points of  ${\cal
D}_{b_j}$ for all $m \geq 1$ and $j \geq 1$. Then by Corollary \ref{f41} we
have  \be{517} \limsup_{j \to \infty} b_j^{-1} {\cal D}_{b_j}(\lambda + b_j)
\leq  \limsup_{j \to \infty} \lim_{R \to \infty} b_j^{-1} R^{-3} n_-(1;\tbop),
\ee
\be{518}
\liminf_{j \to \infty} b_j^{-1} {\cal D}_{b_j}(\lambda + b_j) \geq 
\liminf_{j \to \infty} \lim_{R \to \infty} b_j^{-1} R^{-3} n_-(1;\tbom),
\ee
where the operators $T_{b,\omega,\lambda,R}$ are defined in (\ref{del}). 
The minimax principle entails 
\be{519}
\liminf_{j \to \infty} \lim_{R \to \infty} b_j^{-1} R^{-3} n_-(1;\tbom) \geq
\liminf_{j \to \infty} \; \liminf_{R \to \infty} b_j^{-1} R^{-3} n_-(1; \pbj \tbom \; \pbj).
\ee
Note that the operator $p_b T_{b,\omega,\lambda,R} \; p_b$ coincides with the 
operator $t_{b,R}(\vo - \lambda -1)$ (see (\ref{30})). Hence Corollary
\ref{f43} and (\ref{412b}) imply $$
\liminf_{j \to \infty} \liminf_{R \to \infty} b_j^{-1} R^{-3} n_-(1; \pbj \tbom \pbj) \geq 
$$
\be{520}
\lim_{b \to \infty} \; \lim_{R \to \infty} b^{-1} R^{-3} n_-(1; t_{R,b}(\vo - \lambda^-_m -1)) = 
\frac{1}{2\pi}\tilde{\delta}_{\lambda^-_m}(-1) = \frac{1}{2\pi}\delta(\lambda^-_m).
\ee
The combination of (\ref{518})-(\ref{520}) yields
\be{520a}
\liminf_{j \to \infty} b_j^{-1} {\cal D}_{b_j}(\lambda + b_j) \geq \frac{1}{2\pi}\delta(\lambda^-_m).
\ee
On the other hand, we have
\be{521}
\tbop = \pbj \tbop \; \pbj + ({\rm Id}-\pbj) \tbop ({\rm Id} - \pbj) + 
2{\rm Re} \; \pbj \tbop ({\rm Id} - \pbj).
\ee
Set 
$$
\tilde{T}_{b,\omega,\lambda,R}:= 
$$
$$
({\rm Id} - p_b) (H_0(b) -b +1)^{-1/2} |\vo - \lambda -1| {\bf 1}_{\qrt} (H_0(b) -b +1)^{-1/2} ({\rm Id} -p_b).
$$
Applying the elementary operator inequalities
$$
({\rm Id}-\pbj) \tbop ({\rm Id} - \pbj) \geq - \tilde{T}_{b_j,\omega,\lambda_m^+,R},
$$
$$
2{\rm Re} \; \pbj \tbop ({\rm Id} - \pbj) \geq -\varepsilon^2 t_{b_j, R}(|\vo - \lambda_m^+ -1|) 
-\varepsilon^{-2} \tilde{T}_{b_j,\omega,\lambda_m^+,R}, \;   \varepsilon > 0,
$$
we find that (\ref{521}) entails 
$$
n_-(1;\tbop) \leq n_-(1; t_{b_j,R}(\vo - \lambda_m^+ -1) - \varepsilon^2t_{b_j,R}(|\vo - \lambda^+_m -1|)) \; + 
$$
\be{522}
n_+(1; (1+ \varepsilon^{-2}) \tilde{T}_{b_j,\omega,\lambda_m^+,R}), \;   \varepsilon > 0.
\ee
It easy to verify the estimate
\be{523}
\|\tilde{T}_{b_j,\omega,\lambda_m^+,R}\| \leq (b_j + 1)^{-1} (c_0 +
|\lambda_m^+ + 1|), \;   R>0. \ee
Fix $\varepsilon > 0$ and assume that $b_j$ is so big that we have
$$
(1+ \varepsilon^{-2}) (b_j + 1)^{-1} (c_0 + |\lambda_m^+| + 1) < 1.
$$
Then (\ref{523}) entails
\be{524}
n_+(1; (1+ \varepsilon^{-2}) \tilde{T}_{b_j,\omega,\lambda_m^+,R}) = 0.
\ee
The Weyl inequalities for the eigenvalues of compact selfadjoint operators imply 
$$
n_-(1; t_{b_j,R}(\vo - \lambda_m^+ -1) - \varepsilon^2t_{b_j,R}(|\vo - \lambda^+_m -1|)) \leq  
$$
\be{525}
n_-(1-\varepsilon; t_{b_j,R}(\vo - \lambda_m^+ -1)) + n_+(1; \varepsilon t_{b_j,R}(|\vo - \lambda^+_m -1|)), 
  \varepsilon \in (0,1).
\ee
The estimate (\ref{est}) implies
\be{526} 
n_+(1; \varepsilon t_{b_j,R}(|\vo - \lambda^+_m -1|))  \leq c_5 \varepsilon b_j R^3 
\ee
with $c_5:= (c_0 + |\lambda_m^+ + 1|)/4\pi$. Now, the combination of
(\ref{522}), (\ref{524}), (\ref{525}),  and (\ref{526}) yields 
$$
\limsup_{j \to \infty} \lim_{R \to \infty} b_j^{-1} R^{-3} n_-(1;\tbop) \leq 
$$
\be{527} 
\limsup_{j \to \infty} \limsup_{R \to \infty} 
b_j^{-1} R^{-3}n_-(1-\varepsilon; t_{b_j,R}(\vo - \lambda_m^+ -1)) 
+ c_5 \varepsilon, \;   \varepsilon \in (0,1).
\ee
Corollary \ref{f43} implies
$$
\limsup_{j \to \infty} \limsup_{R \to \infty} 
b_j^{-1} R^{-3}n_-(1-\varepsilon; t_{b_j,R}(\vo - \lambda_m^+ -1)) = 
$$
\be{528}
\lim_{b \to \infty} \lim_{R \to \infty} 
b^{-1} R^{-3}n_-(1-\varepsilon; t_{b,R}(\vo - \lambda_m^+ -1)) = 
\frac{1}{2\pi}\tilde{\delta}_{\lambda_m^+}(-1+\varepsilon), 
\; \varepsilon \in (0,1).
\ee
The combination of (\ref{517}), (\ref{527}), and (\ref{528}) yields
$$
\limsup_{j \to \infty} b_j^{-1} {\cal D}_{b_j}(\lambda + b_j) \leq 
\frac{1}{2\pi}\tilde{\delta}_{\lambda_m^+}(-1+\varepsilon) + c_5 \varepsilon, \;   \varepsilon \in (0,1).
$$
Letting $\varepsilon \downarrow 0$, we get 
\be{529}
\limsup_{j \to \infty} b_j^{-1} {\cal D}_{b_j}(\lambda + b_j) \leq 
\frac{1}{2\pi}\tilde{\delta}_{\lambda_m^+}(-1) = \frac{1}{2\pi}\delta(\lambda_m^+).
\ee
Letting $m \to \infty$ (hence, $\lambda^{\pm}_m \to \lambda$) in (\ref{520a}) and (\ref{529}), we 
arrive at (\ref{516a}).\\

{\Large \bf Acknowledgements} \\
This work was done during the second author's visits
 to the Ruhr University in 1998 and 1999. Financial support of 
{\em Sonderforschungsbereich 237 ``Unordnung und grosse Fluktuationen''},
and of Grant MM 612/96 of the Bulgarian Science Foundation,
is gratefully acknowledged. The authors thank Prof. H.Leschke and Dr
E.Giere for several stimulating discussions.


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