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Random Schroedinger operator, Poisson potential
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\begin{document}
%%%%%%% DOUBLE SPACED %%%%%%%%
\setlength{\baselineskip}{15pt}
%
\title{
The spectrum of Schr\"odinger operators with Poisson type random potential
}
\author{Kazunori Ando\thanks{Department of Information Sciences, Tokyo
Denki University, Hatoyama-machi, Hiki-gun, Saitama 350-0394, Japan},
Akira Iwatsuka\thanks{Kyoto Institute of Technology,
Matsugasaki, Sakyo-ku,
Kyoto, 606-8585, Japan},
Masahiro Kaminaga\thanks{Department of Electrical Engineering and
Information Technology,
Tohoku-Gakuin University, 1-13-1, Chuo, Tagajo, 985-8537, Japan}\\
and Fumihiko Nakano\thanks{Faculty of Science, 
Department of Mathematics and Information Science,
Kochi University,
2-5-1, Akebonomachi, Kochi, 780-8520, Japan. }}
%
\date{}
\maketitle
%%%%%%% ABSTRACT %%%%%%%%%%%%%
\begin{abstract}
We consider
the Schr\"odinger operator with Poisson type random potential, and derive
the spectrum which is deterministic almost surely.
%
Apart from some exceptional cases, the spectrum is equal to
$[0, \infty)$
if the single-site potential is non-negative, and is equal to
${\bf R}$
if the negative part of it does not vanish with positive probability, which
is consistent with the naive observation.
%
To prove that,
we use
the theory of admissible potential and the Weyl asymptotics.
\end{abstract}

Mathematics Subject Classification (2000): 81Q10, 82B44

%\tableofcontents
%%%%% INTRODUCTION %%%%%
\section{Introduction}
We consider
the Schr\"odinger operator on 
${\bf R}^d$ 
with a Poisson type random potential.
%
\[
H_{\omega} =
-\triangle + V_{\omega}(x),
\quad
V_{\omega}(x) =
\sum_{j=1}^{\infty} q_j (\omega)
f(x - X_j(\omega)).
\]
%
$\omega \in \Omega$
and
$(\Omega, {\cal F}, {\bf P})$
is a probability space.
%
We 
assume the following conditions throughout this paper. \\
%

\noindent
{\bf Assumption}\\
%
{\it
{\bf (H1) }
$f \ne 0$
is real-valued, measurable, and
$f \in l^1(L^p({\bf R}^d))$, i.e.,
$f \in L^p_{loc}({\bf R}^d)$
and
%
$\| f \|_{l^1(L^p({\bf R}^d))}
=
\sum_{n \in {\bf Z}^d} \| f \|_{L^p (C_n)} < \infty$\footnote{
$\| f \|_{L^p (C)}
=
\left(
\int_{C} | f(x) |^p dx
\right)^{\frac 1p}$
is the
$L^p (C)$-norm
%
($1 \le p  <\infty$).
}.
%
$C_n =  [-\frac 12, \frac 12)^d + n$, $n \in {\bf Z}^d$
and
%
$p > p(d)$
with
%
\[
p(d) = \cases{
2 & $(d \le 3)$ \cr
\frac d2 & $(d \ge 4)$ \cr}
\]
%
{\bf (H2) }
$\{ q_j (\omega) \}_{j=1}^{\infty}$
is an i.i.d.
such that
$q_j \ne 0$
and
${\bf E} [ | q_1 (\omega) |^r ] < \infty$
%
with
%
\[
r > \max \left\{1,
\frac {pd}{2 (p - p(d))}
\right\}.
\]
%
{\bf (H3) }
$\{ X_j (\omega) \}_{j=1}^{\infty}$
is the Poisson configuration
(support of the Poisson random measure)
with intensity measure
$\rho \, m (d x)$
($\rho >0$,
$m$
is the Lebesgue measure on
${\bf R}^d$)
and is independent of
$\{ q_j (\omega) \}_{j=1}^{\infty}$.
%
That is, 
%

\noindent
(i)
For any
$E_1, E_2, \cdots, E_n (\subset {\bf R}^d)$
disjoint Borel sets on
${\bf R}^d$,
the random variables
$\sharp \{ j : X_j (\omega) \in E_k \}$,
$k=1, 2, \cdots, n$
are mutually independent, \\
%
(ii)
If
$E \subset {\bf R}$
is a Borel set with finite Lebesgue measure,}
%
\[
{\bf P} \left(
\sharp \{ j : X_j (\omega) \in E \} = n
\right)
=
\frac {(\rho m (E))^n}{n!} e^{- \rho m (E)},
\quad
n=0, 1, 2, \cdots.
\]
%

$H_{\omega}$
describes electrons in amorphous materials where the atoms are distributed
randomly.
%
$H_{\omega}$
is essentially self-adjoint on
$C_0^{\infty}({\bf R}^d)$
almost surely\footnote{$C^{\infty}_0({\bf R}^d)$
is the space of infinitely differentiable functions on 
${\bf R}^d$ 
with compact support.}, 
and measurable on $\Omega$
\cite[Proposition V.3.2, Corollary V.3.4]{Ca-La}.
%
$\{ H_{\omega} \}_{\omega \in \Omega}$
is an ergodic family of self-adjoint operators on
$(\Omega, {\cal F}, {\bf P})$.
%
That is,
for any
$x \in {\bf R}^d$,
there exists a measure preserving transformation
$T^x : \Omega \to \Omega$
such that
%
\[
U_x^* H_{\omega} U_x = H_{T^x \omega},
\quad
x \in {\bf R}^d.
\]
%
where
$(U_y \varphi)(x) = \varphi(x-y)$, $\varphi \in L^2({\bf R}^d)$
is unitary on
$L^2({\bf R}^d)$
%
and
$\{ T^x \}_{x \in {\bf R}^d}$
is an ergodic family.
%
Then it is well-known that there exists a closed set
$\Sigma (\subset {\bf R})$
such that
$\sigma (H_{\omega}) = \Sigma$ a.s.
%
%
The aim
of this paper is to derive
$\Sigma$
explicitly.
%
A naive observation tells us that,
%
in the case of
$q_j (\omega) > 0$,
%
\[
(*)
\quad
\mbox{
{\it
If
$f \ge 0$
then
$\Sigma = [0, \infty)$,
otherwise
$\Sigma = {\bf R}$.}
}
\]
%
$f \ge 0$
means that
$f(x) \ge 0$ for a.e. $x$.
%
There is a discussion on the assertion $(*)$ in
\cite{Kirsch}, 
\cite[Theorem 5.34]{Pastur-Figotin}, 
under some assumptions on $f$,  
%
which does not, however, seem to be fully convincing  to us.
%
The assertion $(*)$ 
is also stated in \cite{Klopp, Klopp-Pastur} without proof. 
%
It is possible to prove 
$(*)$
under some conditions on the regularity of the distribution of 
$q_1 (\omega)$. 
%
Thus our aim
in this paper is to prove $(*)$ in wider generality,  without using the 
randomness of
$q_j (\omega)$.
%
This problem 
then becomes harder, 
in view of the representation of
$\Sigma$
in terms of the admissible potential
%
(Theorem \ref{admissible potential}).
%

An important topic, 
which is not discussed in this paper, 
is whether the Anderson localization holds in some region of
$\Sigma$,
in which the spectrum is dense pure point with exponentially decaying
eigenfunctions.
%
This problem
has been solved for
$d=1$
and for all energy by
Stolz\cite{Stolz} and
Buschmann-Stolz\cite{Buschmann-Stolz}.
%
Recently,
for any
$d$,
Kirsch-Veseli\'c
\cite{Kirsch-Veselic}
proved the Wegner estimate for negative energy under some regularity
assumptions on $f$ and $q_j$.
%

We state
the main results of this paper below.
%
Let
${\bf P}_q$
be the distribution of
%
$q_1 (\omega)$
%
:
${\bf P}_q (E) = {\bf P} (\{ \omega\in \Omega : q_1 (\omega) \in E \})$,
$E \in {\cal B}({\bf R})$
and let
$f_{\pm} (x) = \max \{ \pm f(x), 0 \}$.
%
\begin{theorem}
%
Assume 
(H1), (H2), (H3)
and let the space dimension 
$d \ne 2$.\\
%
(1) Assume supp ${\bf P}_q \subset [0, \infty)$. \\
%
(i)
If
$f_- = 0$, then $\Sigma = [0, \infty)$, 
%
(ii)
If 
$f_- \ne 0$, then $\Sigma = {\bf R}$. \\
%

\noindent
(2) Assume supp ${\bf P}_q \subset (-\infty, 0]$. \\
%
(i)
If
$f_+ = 0$, then $\Sigma = [0, \infty)$, 
%
(ii)
If 
$f_+ \ne 0$, then $\Sigma = {\bf R}$. \\
%

\noindent
(3)
%
Assume
supp ${\bf P}_q \cap (0, \infty) \ne \emptyset$
and
supp ${\bf P}_q \cap (-\infty,0) \ne \emptyset$,
then
$\Sigma = {\bf R}$.
%
\label{location of the spectrum-1}
\end{theorem}
%

We give 
here the outline of the proof of Theorem \ref{location of the  spectrum-1}.
%
Along the theory developed in
\cite{Kir-Mar},
we consider the family of admissible potentials
${\cal A}$
which is composed of all the superposition of a finite number of
translates of $f$
(Definition \ref{definition}).
%
We show that
$\Sigma$
is equal to the closure of the union of the spectrum of the Schr\"odinger
operator with the admissible potential
(Theorem \ref{admissible potential}).
%
Then,
by the definition of
${\cal A}$,
$\sigma_{ess}(-\triangle + W) = [0, \infty)$
for
$W \in {\cal A}$,
so that the statements
``$\Sigma = [0, \infty)$"
in Theorem \ref{location of the spectrum-1}
are easy to prove.
%
%
To show
``$\Sigma = {\bf R}$"
in the other case, 
we aim to deduce a contradiction, 
supposing that there exists
$b \in \Sigma^c$.
%
Let
${\cal A}_n$
be the set of elements in
${\cal A}$
which are superposition of
$n$-translates of $f$:
$W(x) = \sum_{j=1}^n c f(x - u_j) \in {\cal A}_n$, 
$c \in \mbox{supp }{\bf P}_q$. 
%
The number of eigenvalues of
$-\triangle + W$
less than
$b$
is independent of
$\{ u_j \}_{j=1}^n$
by continuity.
%
By taking each
$u_j$
far apart each other, this number should be proportional to
$n$.
%
On the other hand,
by taking
$u_j = 0$
for all $j$,
the Weyl asymptotics
(\ref{concentrate})
implies that this number should be proportional to
$n^{\frac d2}$
in leading order,
hence we arrive at a contradiction if
$d \ne 2$.
%

The above strategy
also works for
$d=2$
under some additional assumptions.
%
%
%%%%%
\begin{theorem}
Assume 
(H1), (H2), (H3)
%
and let
$d=2$.\\
%
(1)
Assume
$\mbox{supp }{\bf P}_q \subset [0, \infty)$.\\
%
(i)
If
$f _-  = 0$,
then
$\Sigma = [0, \infty)$, \\
%
(ii)
If
$f_- \ne 0$,
then
$\Sigma = {\bf R}$
if one of the following conditions hold.\\
%
(a)
$f_+ \ne 0$,
%
(b)
$c\int_{{\bf R}^2} f_- dx \notin 4 \pi {\bf N}$
for some
$c \in \mbox{supp }{\bf P}_q$,\\
%

\noindent
(2)
Assume
$\mbox{supp }{\bf P}_q \subset ( -\infty,0]$.\\
%
(i)
If
$f _+  = 0$,
then
$\Sigma = [0, \infty)$, \\
%
(ii)
If
$f_+ \ne 0$, then
$\Sigma = {\bf R}$
if one of the following conditions hold.\\
%
(a)
$f_- \ne 0$, 
(b)
$c\int_{{\bf R}^2} f_+ dx \notin 4 \pi {\bf N}$
for some
$c \in \mbox{supp }{\bf P}_q$,\\

\noindent
(3)
Assume
$\mbox{supp }{\bf P}_q \cap (0, \infty) \ne \emptyset$
and
$\mbox{supp }{\bf P}_q \cap (-\infty, 0) \ne \emptyset$,
then
$\Sigma = {\bf R}$.
%
\label{two dimension-1}
\end{theorem}
%
The statement of
Theorem \ref{two dimension-1}
implies that there are some cases in which we cannot prove the assertion 
$(*)$
(e.g.,
$\mbox{ supp }{\bf P}_q =\{1\}$,
$f \le 0$
and
$\int f_- dx \in 4 \pi{\bf N}$).
%
In this case, we can show 
(Theorem \ref{Sigma})
that 
$\Sigma$
has at most one gap below the origin.
%
Moreover
an example of
$f$
which gives a gap in
$\Sigma$
would have an unusual property
(Remark \ref{special property of the counterexample})
%
so that
we believe it is quite unlikely that there exists such
$f$.
%

In the following sections,
we prove the theorems given above.
%
%
In
Section 2,
we prove Theorem 
\ref{admissible potential}.
%
In
Section 3,
we prove Theorems
\ref{location of the spectrum-1}
and
\ref{two dimension-1}
by the strategy outlined above.
%
In
Section 4, we prove Theorem 
\ref{Sigma}.
%
In the Appendix,
we state some basic facts used in this paper.

%%%%%
\section{Admissible potential}
%
In this section,
we introduce the family of admissible potentials, 
in terms of which
we derive a representation of 
$\Sigma$
based on the argument in
\cite{Kir-Mar}.
%
We first define
%
\begin{definition}
%
\label{definition}
%
\begin{eqnarray*}
{\cal A}
&=& \bigcup_{n \ge 1} {\cal A}_n,
\\
%
{\cal A}_n
& = &
\left\{
W(x) = \sum_{j =1}^n c_j f(x- u_j) \; : \;
c_j \in \mbox{supp } {\bf P}_q ,
u_j \in {\bf R}^d , \;
1 \le j \le n
\right\}.
\end{eqnarray*}
%
\end{definition}
%
Usually,
the family of admissible potentials is defined by all the periodic
realizations of  the random potential
$V_{\omega}$, as is done in
\cite{Kir-Mar}.
%
However,
in this paper, we consider superposition of  a finite number of translates 
of
$f$, 
which is more convenient for our purpose.
%
%
\begin{theorem}
{\bf \quad}
%
%
$\Sigma = \overline{
\bigcup_{W \in {\cal A}}
\sigma(- \triangle + W)}$.
%
\label{admissible potential}
\end{theorem}
%
%
\begin{proof}
%
As is noted in the Introduction,
%
\[
\Omega_0 = \{ \omega \in \Omega :
\sigma (H_{\omega}) = \Sigma \}
\]
%
satisfies
${\bf P} (\Omega_0) = 1$. \\
%
(1)
%
Suppose
$\lambda \in \Sigma$.
%
Pick
$\omega \in \Omega_0 \cap \tilde{\Omega}( \ne \emptyset)$
($\tilde{\Omega} \in {\cal F}$, ${\bf P}(\tilde{\Omega})=1$
is the set constructed in Lemma \ref{approximation})
and let
%
\begin{eqnarray*}
V_{N, \omega}(x) =
\sum_{j=1}^{N} q_j  (\omega) f(x - X_j(\omega))
\in {\cal A}_N.
\end{eqnarray*}
%
Then by Lemma
\ref{approximation},
%
$V_{N, \omega} \varphi \stackrel{N \to \infty}{\to}
 V_{\omega} \varphi$
in
$L^2 ({\bf R}^d)$
for
$\varphi \in C_0^{\infty} ({\bf R}^d)$.
 %
Because
$C_0^{\infty} ({\bf R}^d)$
is the common core of
$H_{N, \omega}=-\triangle + V_{N, \omega}$
and
$H_{\omega}$,
$H_{N, \omega} \stackrel{N \to \infty}{\to} H_{\omega}$
in the strong resolvent sense.
%
Therefore for
$\lambda \in \Sigma = \sigma (H_{\omega})$
we can find  sequences
$\{ \lambda_k \}_{k=1}^{\infty}$
and
$\{ N_k \}_{k=1}^{\infty}$
with
$\lambda_k \in \sigma(H_{N_k, \omega})$
s.t.
$\lambda_k \stackrel{k \to \infty}{\to} \lambda$.
%
Hence
$\lambda \in
\overline{
\bigcup_{W \in {\cal A}}
\sigma (-\triangle + W)
}$.\\
%
(2)
Conversely,
pick
$W \in {\cal A}$
and
$\lambda \in \sigma(-\triangle + W)$.
%
Consider the following event\footnote{
%
$|x|_{\infty} = \max_{1 \le j \le d} |x_j|$
for
$x = (x_1, \cdots, x_d) \in {\bf R}^d$
and
$B_N = \{ x \in {\bf R}^d : |x|_{\infty} \le N \}$.
%
}.
%
\begin{eqnarray*}
D_W &=& \Biggl\{
\omega \in \Omega :
\mbox{ for any }
N, k \ge 1,
\mbox{ there exists }
x_0 = x_0(N, k,\omega) \in {\bf R}^d
\\
%
&&\qquad
\; \mbox{s.t.} \;
\| W - V_{\omega} (\cdot + x_0) \|_{L^2(B_N)} < \frac 1k
\Biggr\}.
\end{eqnarray*}
%
%
We show
${\bf P} (D_W) = 1$
in Lemma
\ref{D-Iwatsuka}.
%
Since
$\lambda \in \sigma (-\triangle + W)$,
we can take sequences\footnote{
$\| \cdot \|_p = \| \cdot \|_{L^p({\bf R}^d)}$
($1 \le p \le \infty$)
is the
$L^p ({\bf R}^d)$-norm.
}
%
$\{ \varphi_k \}_{k=1}^{\infty} (\subset  C_0^{\infty}({\bf R}^d))$,
$\| \varphi_k \|_2 = 1$
and
$\{ N_k \}_{k=1}^{\infty} (\subset {\bf N})$
with
$\mbox{supp } \varphi_k \subset B_{N_k}$
s.t.
$\| ( -\triangle + W - \lambda) \varphi_k \|_2 < \frac 1k$.
%
Let
$\omega \in D_W \cap \Omega_0$.
%
Taking
$x_0= x_0(N_k, [\| \varphi_k \|_{\infty} k]  + 1, \omega)$,
we have
%
\begin{eqnarray*}
\| ( -\triangle& + &V_{\omega} - \lambda)
\varphi_k (\cdot- x_0) \|_2
\\
%
& \le &
\| ( -\triangle + W(\cdot- x_0)-\lambda) \varphi_k (\cdot- x_0) \|_2
+
\| \left( V_{\omega} - W(\cdot-x_0)\right) \varphi_k (\cdot-x_0)\|_2
\\
%
& \le &
\frac 1k +
\| \varphi_k \|_{\infty} \| V_{\omega}(\cdot+x_0) - W \|_{L^2(B_{N_k})}
< \frac 2k.
\end{eqnarray*}
%
Therefore
$\lambda \in \sigma (- \triangle + V_{\omega}) = \Sigma$.
%
Since
$\Sigma$
is closed,
$\overline{
\bigcup_{W \in {\cal A}}
\sigma (- \triangle + W)
}
\subset
\Sigma$.
%
\QED
\end{proof}
%
We prove two lemmas mentioned in the proof of
Theorem \ref{admissible potential}.
%
\begin{lemma}
%
There exists
%
$\tilde{\Omega} \in {\cal F}$, ${\bf P}(\tilde{\Omega}) = 1$
s.t. if
$\omega \in \tilde{\Omega}$,
$V_{N, \omega} \varphi \stackrel{N \to \infty}{\to} V_{\omega}\varphi$
in 
$L^2 ({\bf R}^d)$
for any
$\varphi \in C_0^{\infty}({\bf R}^d)$.
%
\label{approximation}
\end{lemma}
%
\begin{proof}
%
This is
a simple application of the argument of the proof of
\cite[Corollary V.3.4]{Ca-La}.
%
Since
$\mbox { supp } \varphi \subset
B_M \subset \bigcup_{ | n |_{\infty} \le M } C_n$
for some
$M \in {\bf N}$
%
($C_n = [-\frac 12, \frac 12)^d + n$, $n \in {\bf Z}^d$), 
%
it suffices 
to show that we can find a set 
$\tilde{\Omega}_M \in {\cal F}$, 
${\bf P}(\tilde{\Omega}_M)=1$
which depends only on
$M$
with
$\| V_{N, \omega} - V_{\omega} \|_{L^2(B_M)}
\stackrel{N \to \infty}{\to} 0$
for
$\omega \in \tilde{\Omega}_M$,
%
by taking
%
$\tilde{\Omega} = \bigcap_{M \ge 1} \tilde{\Omega}_M$
for the proof of Lemma 
\ref{approximation}.
%
Let
%
\[
\tilde{V}_{R, \omega}(x) =
\sum_{ | X_j (\omega) |_{\infty} \ge R}
| q_j (\omega) | | f(x-X_j(\omega)) |.
\]
%
%
Since
$\sharp \{ X_j (\omega) : | X_j (\omega) |_{\infty} < R \} < \infty$
for any
$R$,
it suffices to show
$\| \tilde{V}_{R, \omega} \|_{L^2(B_M)}
\stackrel{R \to \infty}{\to} 0$
whenever
$\omega \in \tilde{\Omega}_M$.
%
By the inequality
%
$
\int_{B_M} | f(x) |^2 dx
\le
\left(
\int_{B_M} | f(x) |^p dx
\right)^{\frac 2p}
\mbox{ vol }(B_M)^{\frac {p-2}{p}}
$
and Chebyshev's inequality, we have
%
%
\begin{eqnarray}
{\bf P}
\Biggl(
&
\int_{B_M}
&
 | \tilde{V}_{R, \omega}(x) |^2 dx
\ge a
\Biggr)
\nonumber
\\
%
& \le &
a^{- \frac r2} \mbox{ vol }(B_M)^{\frac {r(p-2)}{2p}}
{\bf E}
\left[
\left(
\int_{B_M} | \tilde{V}_{R, \omega} (x) |^p dx
\right)^{\frac rp}
\right]
\label{Chebyshev}
\end{eqnarray}
%
for any
$a > 0$.
%
We introduce a measure
%
\[
\mu_{R, \omega} =
\sum_{| X_j (\omega) |_{\infty} \ge R}
| q_j (\omega) | \delta_{X_j (\omega)}
\]
%
so that
%
$\tilde{V}_{R, \omega} (x) =
\int_{{\bf R}^d} | f(x-y) | \mu_{R, \omega} (dy)$.
%
We then have
%
\begin{eqnarray*}
\left(
\int_{B_M} | \tilde{V}_{R, \omega} (x) |^p dx
\right)^{\frac 1p}
%
& \le &
\int_{{\bf R}^d}  \mu_{R, \omega}  (dy)
\left(
\int_{B_M} dx | f(x-y) |^p
\right)^{\frac 1p}
\\
%
& \le &
\sum_{|m|_{\infty} \ge R}
 \mu_{R, \omega}  (C_m)
\| f (\cdot-m) \|_{L^p (B_{M+1})}
\end{eqnarray*}
%
by Minkowski's inequality. 
%
Therefore
%
\begin{eqnarray}
&&
{\bf E}
\left[
\left(
\int_{B_M} dx  | \tilde{V}_{R, \omega} (x) |^p
\right)^{\frac rp}
\right]^{\frac 1r}
\nonumber
\\
%
& \le &
\sum_{|m|_{\infty}\ge R}
{\bf E}
\left[
\Bigl(\mu_{R, \omega} (C_m) \Bigr)^r
\right]^{\frac 1r}
\| f (\cdot-m) \|_{L^p (B_{M+1})}
\nonumber
\\
%
& \le &
\left(
\sup_{R>0}\sup_{m \in {\bf Z}^d}
{\bf E}
\left[
\Bigl( \mu_{R, \omega}  (C_m)\Bigr)^r 
\right]^{\frac 1r}
\right)
S_M
\sum_{|m|_{\infty} \ge R-M-1}
\| f \|_{L^p (C_m)}.
\label{l1Lp}
\end{eqnarray}
%
We set
%
$S_M = \sharp
\{
n \in {\bf Z}^d : |n|_{\infty} \le M+1
\}$
in (\ref{l1Lp}).
%
The fact that 
$\sup_{R>0}\sup_{m \in {\bf Z}^d}
{\bf E}
\left[
\Bigl( \mu_{R, \omega}  (C_m)\Bigr)^r 
\right]< \infty$
is proved in Lemma \ref{H3} at the end of this section.
%
Since
$f \in l^1(L^p({\bf R}^d))$,
the RHS
of (\ref{l1Lp}) tends to
$0$
as
$R \to \infty$.
%
By
(\ref{Chebyshev}), (\ref{l1Lp}),
for each
$k \ge 1$
we can take
$R = R(k, M)$
such that
%
\[
{\bf P}
\left(
B(k,M)
\right)
\le
\frac {1}{k^2}
\quad
\mbox{where}
\quad
B(k,M) =
\left\{
\omega \in \Omega :
\int_{B_M} | \tilde{V}_{R(k, M), \omega} (x) |^2 dx
\ge
\frac 1k
\right\}.
\]
%
Since
$\sum_{k \ge 1} {\bf P}(B(k,M)) < \infty$,
$\tilde{\Omega}_M = \liminf_{k \to \infty} B(k,M)^c$
satisfies 
${\bf P}(\tilde{\Omega}_M) = 1$
and for
$\omega \in \tilde{\Omega}_M$
there exists
$K = K(\omega, M)$
s.t.
$\omega \in B(k,M)^c$
for
$k \ge K(\omega, M)$.
%
Thus
Lemma \ref{approximation}
is proved.
\QED
\end{proof}
%
%%%%%
\begin{lemma}
${\bf P}(D_W) = 1$.
\label{D-Iwatsuka}
\end{lemma}
%
\begin{proof}
%
For
$N, k \ge 1$,
set
%
\begin{eqnarray*}
%
D_W(N, k) &=& \Biggl\{
\omega \in \Omega :
\mbox{there exists }
x_0 = x_0(N, k,\omega) \in {\bf R}^d
\\
%
&&\qquad\qquad
\; \mbox {s.t.} \;
\| W - V_{\omega}(\cdot + x_0) \|_{L^2(B_N)} < \frac 1k
\Biggr\}
\\
%
E_W (N,k)&=& \left\{
\omega \in \Omega :
\| W - V_{\omega} \|_{L^2(B_N)} < \frac 1k
\right\}.
\end{eqnarray*}
%
Since
$D_W = \bigcap_{N, k \ge 1} D_W (N, k)$,
%
and since
$D_W (N,k)$
is
$T$-invariant
and
$E_W (N, k) \subset D_W (N,k)$,
it suffices to show
${\bf P} (E_W(N, k)) > 0$
for any
$N, k \ge 1$.
%
Let
$W(x) = \sum_{j=1}^n c_j f(x-u_j) \in {\cal A}_n$.
%
Without loss of generality, we may suppose
$\{ u_j \}_{j=1}^n \subset B_N$
by taking
$N$
larger if necessary.
%
For
$R, \epsilon > 0$
($R > N$),
we consider the following event.
%
\begin{eqnarray*}
F_W(R,\epsilon)
 =
\{
\omega \in \Omega :
&&
\;
| q_j (\omega)-c_j | < \epsilon, \;
| X_j (\omega) -  u_j | < \epsilon,
j=1, 2, \cdots, n, \\
%
&&
\mbox{there are no other $X_j(\omega)$'s in }
B_{R}
%
\}.
\end{eqnarray*}
%
%
For
$R, \epsilon > 0$,
${\bf P}(F_W(R, \epsilon)) > 0$.
%
It is easy to show
$F_W(R, \epsilon) \cap \tilde{\Omega} \subset E_W (N, k)$
for
$\epsilon > 0$
sufficiently small and
$R > 0$
sufficiently large
($\tilde{\Omega} \in {\cal F}$
is the set constructed in the proof of
Lemma \ref{approximation})
which completes the proof of Lemma \ref{D-Iwatsuka}.
%
\QED
\end{proof}
%
We give the proof of a fact which was used in the proof of 
Lemma \ref{approximation}.
%
\begin{lemma}
%
$\sup_{R>0} \sup_{m \in {\bf Z}^d} 
{\bf E}
\left[
\left(
\mu_{R, \omega}(C_m)
\right)^r
\right]
<\infty$. 
%
\label{H3}
\end{lemma}
%
\begin{proof}
Let 
$\mu_{\omega} = \sum_{j=1}^{\infty}
| q_j (\omega) | \delta_{X_j(\omega)}$.
%
Since
$\mu_{R, \omega}(E) \le \mu_{\omega}(E)$
for any 
$E \in {\cal B}({\bf R})$ 
and since 
${\bf E}[(\mu_{\omega}(C_m))^r]$
is independent of 
$m \in {\bf Z}^d$, 
it suffices to show 
${\bf E}[(\mu_{\omega}(C_0))^r]<\infty$. 
%
Let 
$N_0 (\omega) = 
\sharp \{ j \ge 1 : X_j (\omega)  \in C_0 \}$.
%
We then have
%
\begin{eqnarray*}
{\bf E} [ (\mu_{\omega}(C_0))^r]
&=&
\sum_{k=1}^{\infty}
{\bf E} 
\left[ 
\left(
\sum_{j=1}^k | q_{j}(\omega) |
\right)^r  ; 
N_0 (\omega) = k
\right]
\\
%
& \le &
\sum_{k=1}^{\infty}
{\bf E}
\left[
k^{r-1} \sum_{j=1}^k | q_{j} (\omega) |^r ; 
N_0 (\omega) = k 
\right]
\\
%
&=&
\sum_{k=1}^{\infty}
k^r {\bf E} [ | q_1 (\omega) |^r ]
\frac {\rho^k}{k!} e^{- \rho}
<\infty.
\end{eqnarray*}
%
In the second inequality, we used
$\left(
\sum_{j=1}^k | q_{j}(\omega) | 
\right)^r 
\le 
k^{r-1} \sum_{j=1}^k | q_{j} (\omega) |^r$
by H\"older's inequality.
%
In the last equality, 
we used the assumption (H.3).
%
\QED
\end{proof}
%

%%%%%
\section{Proof of theorems}
%
In this section,
we give proof of Theorems 
\ref{location of the spectrum-1}
and
\ref{two dimension-1}. \\

%
\noindent
{\it Proof of Theorem \ref {location of the spectrum-1} }
%
We first consider the case (1).
%
Since
$W \in {\cal A}$
is relatively compact w.r.t.
$-\triangle$,
$\sigma_{ess} (-\triangle + W) = [0, \infty)$.
%
Thus by Theorem 
\ref{admissible potential},
$[0, \infty) \subset \Sigma$.
%
On the other hand,
if
$f_- = 0$
then
$W \ge 0$
so that
$\Sigma \subset [0, \infty)$.
%
Hence
we have
$\Sigma = [0, \infty)$
if
$f_- = 0$.
%
Next, assume
$f_- \not \ne 0$.
%
We suppose there exists
$b < 0$ with $b \notin \Sigma$, 
and would like to deduce a contradiction.
%
By Theorem 
\ref{admissible potential},
there exists
$\delta > 0$
s.t.
$(b-\delta, b +\delta) \subset \rho (- \triangle + W)$
for any
$W \in {\cal A}$.
%
Take
$c \in \mbox{supp }{\bf P}_q$,
$c > 0$
and set\footnote{
$N_E (H) = \dim \mbox{ Ran }P_E (H)$
and
$P_E (H)$
is the spectral projection onto the Borel set
$E (\subset {\bf R})$
associated to the operator
$H$. }
%
\begin{eqnarray*}
W_{n, {\bf u}} (x)
&=&
\sum_{j=1}^n c f(x - u_j)\in {\cal A}_n,
\quad
%
{\bf u} = (u_1, \cdots, u_n ) \in {\bf R}^{nd}
\\
%
N_b (W_{n, {\bf u}})
& = &
N_{(-\infty, b)}
(-\triangle + W_{n, {\bf u}}).
\end{eqnarray*}
%
For simplicity,
we write
$f$
instead of
$cf$.
%
%
Since
$b$
lies in the common spectral gap of
$-\triangle + W_{n, {\bf u}}$,
Proposition \ref{continuity of eigenvalues}
tells us that
$N_b (W_{n, {\bf u}})$
is continuous w.r.t.
${\bf u}$
and thus
$\gamma (n) = N_b (W_{n, {\bf u}}) \in {\bf N}$
is independent of
${\bf u}$.
%
Here we consider two configurations of
$\{ u_j \}_{j=1}^n$.\\
%

\noindent
(a)
As
$\min_{i \ne j}| u_i - u_j | \to \infty$,
each negative eigenvalue of
$-\triangle + W_{n, {\bf u}}$
approaches to that of
$-\triangle + f$.
%
We thus have, as is confirmed in Proposition \ref{independence}, 
%
\begin{eqnarray}
\gamma (n)
&=&
n N_{(-\infty, b)} (- \triangle + f).
\label{independent}
\end{eqnarray}
%

\noindent
(b)
By taking
${\bf u}={\bf 0}$,
we have\footnote{
$\tau_d$
is the volume of the
$d$-dimensional unit ball. }
%
\begin{eqnarray}
\gamma (n)
&=&
N_{(- \infty, b)} (- \triangle + nf)
\nonumber
\\
%
&=&
\cases{
n^{\frac d2}
\frac {\tau_d}{(2\pi)^d}
\int_{{\bf R}^d} ( f_-(x) )^{\frac d2} dx
(1 + o(1))
&
$(d \ge 3)$\cr
%
o(n)
&
$(d = 1)$ \cr}
\quad
\label{concentrate}
\end{eqnarray}
%
 as 
$n \to \infty$.
%
For the proof of the asymptotics
(\ref{concentrate})
for
$d \ge 3$,
we refer to
\cite[Theorem XIII.80]{Re-Si4}.
%
They prove the asymptotics
(\ref{concentrate})
for
$b = 0$, 
but the proof for
$b< 0$
requires no essential modifications.
%
The asymptotics
(\ref{concentrate})
for
$d=1$
is proved in
Proposition \ref{one dimension}.
%
We note here that, since
$f_- \ne 0$, 
$\gamma(n) = N_{(-\infty, b)}(-\triangle + nf) \ge 1$
for 
$n$
large enough so that 
(\ref{independent}) 
implies 
$N_{(-\infty, b)}(-\triangle +f) \ge 1$. 
%
By
(\ref{independent}), (\ref{concentrate}),
we arrive at the contradiction.

For the case
(2),
we take
$c  \in \mbox{ supp }{\bf P}_q$, $c < 0$.
%
For the case
(3),
%
we take
$c \in \mbox{ supp }{\bf P}_q$
s.t.
$c<0$
if
$f_+ \ne 0$,
and
$c>0$
if
$f_- \ne 0$.
%
Then the rest
of the proof follows similarly as the case (1).
%
\QED\\
%

%%%%%
\noindent
{\it Proof of Theorem \ref{two dimension-1}}
%
$\,$
We first consider the case (1).
%
The proof
for the case of
$f_- = 0$
is the same as that for
$d \ne 2$
and thus we assume
$f_- \ne 0$.
%
%
Suppose that the condition
(b)
holds and take
$c \in \mbox{supp }{\bf P}_q$
s.t.
$c\int_{{\bf R}^2} f_- (x) dx \notin 4 \pi {\bf N}$.
%
We take the admissible potential
$W_{n, {\bf u}} (x) = \sum_{j=1}^n c f(x - u_j)$,
${\bf u}=(u_1, \cdots, u_n) \in {\bf R}^{nd}$.
%
Suppose
there exists
$b <0$
with
$b\notin \Sigma$
and define
$N_b (W_{n, {\bf u}})$, $\gamma(n)$
as in the proof of Theorem 
\ref {location of the spectrum-1}.
%
By the
argument to deduce 
(\ref{independent}), (\ref{concentrate})
in the proof of
Theorem \ref {location of the spectrum-1}, 
we have
%
\begin{eqnarray}
\gamma (n)
&=&
n N_{ (-\infty, b) } (- \triangle + c f), 
\label{concentrate2}
\\
\gamma (n)
&=&
N_{(-\infty, b)} (-\triangle + n c f)
=
n
\left(
\frac {c}{4 \pi}
\int_{{\bf R}^2}   f_-(x)  dx
\right)
(1 + o(1)).
\label{Weyl}
\end{eqnarray}
%
as 
$n \to \infty$. 
%
For the proof of the asymptotics
(\ref{Weyl}), 
we refer to
\cite{Birman-Borzov, Birman-Solomyak}.
%
By
 (\ref{concentrate2}), (\ref{Weyl}), we have
%
\begin{equation}
N_{ (-\infty,b) }(-\triangle + cf) 
=
\frac {c}{4\pi}
\int_{{\bf R}^2} f_- (x) dx.
\label{compare}
\end{equation}
%
Since
%
$\int_{{\bf R}^2}   cf_-(x)  dx \notin 4 \pi{\bf N}$,
%
we have a contradiction.
%

%
We next assume the condition (a) holds and (b) does not : 
$f_+ \ne 0$
and
%
\begin{equation}
c \int_{{\bf R}^2}   f_-(x)  dx
\in 4 \pi{\bf N}
\label{integer}
\end{equation}
%
for any
$c \in \mbox{supp }{\bf P}_q$.
%
Let
%
\[
g (x,v) = c f(x) + c f(x-v),
\quad
v \in {\bf R}^2.
\]
%
Since 
$g_- (x,v) \le c f_- (x) + c f_- (x-v)$
for any 
$x \in {\bf R}^2$, 
and since for some $v$,
$g_- (x,v) < c f_- (x) + c f_- (x-v)$
on a set of positive Lebesgue measure, it is not hard to verify that
%
\begin{equation}
\int_{{\bf R}^2} g_- (x,v) dx
<
2c \int_{{\bf R}^2} f_- (x) dx
\label{difference}
\end{equation}
%
under a suitable choice of
$v = v_0 \in {\bf R}^2$.
%
Moreover 
the difference between both sides can be sufficiently small by continuity 
of LHS of 
(\ref{difference})
in $v$.
%
Hence
%
\begin{equation}
\int_{{\bf R}^2} g_- (x, v_1) dx \notin 4 \pi {\bf N}
\label{non integer}
\end{equation}
%
for some
$v_1 \in {\bf R}^2$.
%
We take the admissible potential in the following form
%
\[
W(x) = \sum_{j=1}^n
g(x - u_j , v_1) \in {\cal A}_{2n},
\]
%
repeat the argument above, and then obtain the contradiction.
%
Therefore,
we proved
Theorem \ref{two dimension-1}
for the case (1).
%
The proof for the case
(2)
is similar.
%
For the case (3),
we consider two possibilities.
%
(i)
If
$f_+, f_- \ne 0$,
and if
$\int_{{\bf R}^2} (cf)_- (x) dx \in 4 \pi {\bf N}$
for any
$c \in  \mbox{supp }{\bf P}_q$,
then we take
$c \in \mbox{ supp }{\bf P}_q$, $c \ne 0$
and let
$g (x,v) =c f(x) + cf(x-v)$.
%
Since
$(c f)_+ \ne 0$,
$\int_{{\bf R}^2} g_- (x,v) dx \notin 4 \pi {\bf N}$
for some
$v \in {\bf R}^2$.
%
(ii)
If
$f \ge 0$
or
$f \le 0$,
without loss of generality, we assume
$f \ge 0$.
%
Take
$c_1, c_2 \in \mbox{supp } {\bf P}_q$
with
$c_1 > 0$, $c_2 < 0$.
%
If
$(-c_2) \int_{{\bf R}^2} f(x) dx \notin 4 \pi {\bf N}$,
the proof has been done.
%
Otherwise, let
$g (x,v) = c_1 f (x) + c_2 f(x-v)$.
%
Then
$\int_{{\bf R}^2} g_- (x,v) dx \notin 4 \pi{\bf N}$
for some
$v \in {\bf R}^2$.
\QED\\
%
%
\begin{remark}
By (\ref{compare}), 
an example of
$f$
with
$\Sigma \ne {\bf R}$
would satisfy the following property
%
\begin{eqnarray*}
N_{ (-\infty, b) } (-\triangle + nf)
&=&
\frac {n}{4 \pi}
\int_{{\bf R}^2}   f_-(x)  dx,
\quad
n \ge 1
\end{eqnarray*}
%
(in the case of 
$\mbox{supp }{\bf P}_q = \{ 1 \}$)
which, we believe, is unlikely to occur.
%
Presumably,
an examination of the second term
(for smooth
$f$,
for instance) of the
Weyl asymptotics
would exclude such possibilities.
%
\label{special property of the counterexample}
\end{remark}
%


%%%%%
\section{A representation of 
$\Sigma$
in two dimensions}
%
In 
$d = 2$, 
if 
$f \le 0$
and if 
$\int_{{\bf R}^2} (cf)_- (x) dx \in 4 \pi {\bf N}$
for any 
$c \in \mbox{supp }{\bf P}_q$, 
we cannot exclude the possibility to have a gap in 
$\Sigma$.
%
In this case, 
we can show that 
$\Sigma$
has at most one gap. 
%
The proof of Theorem \ref {location of the spectrum-1}
implies that we can assume
$\mbox{supp }{\bf P}_q = \{ 1 \}$
without loss of generality.
%
Thus in this section we work under the following conditions. 
%
\begin{eqnarray*}
{\bf (H4)}  
\;
{\it
d=2, 
\,
\mbox{supp }{\bf P}_q = \{ 1 \}, f \le 0}
%
{\it
\mbox{ and }
(4\pi)^{-1}\int_{{\bf R}^2} f_- (x) dx= k \in {\bf N}.
}
\end{eqnarray*}
%
For 
$W \in {\cal A}$, 
let 
%
\[
N(W) = N_{(-\infty, 0)} (-\triangle + W).
\]
%
Let 
$\{ e_j (W) \}_{j=1}^{N(W)}$
be the negative eigenvalues of the operator
$-\triangle + W$
in increasing order counting multiplicity, and let
%
\begin{eqnarray*}
\tilde{e}_{j} (W)
&=& \cases{
e_{j}(W) & $(N(W) \ge j)$ \cr
%
0 & $(N(W) \le j-1)$ \cr}, 
\quad
j \ge 1
\end{eqnarray*}
%
which is equal to the number given by the min-max principle.
\begin{theorem}
%{\bf (A representation of $\Sigma$)}
%
Assume
(H1), (H3) and (H4). 
%
Then 
we have 
\label{Sigma}
%
\[
\Sigma^c
=
\left[
\bigcap_{l \ge 1} \bigcap_{W \in {\cal A}_l}
\left(
\tilde{e}_{lk}(W), \tilde{e}_{lk+1}(W)
\right)
\right]^{int}
\]
%
%
\end{theorem}
%
$[E]^{int}$
is the interior of the set 
$E(\subset {\bf R})$. 
%
If 
$\tilde{e}_{lk}(W) = \tilde{e}_{lk+1}(W)$, 
the set
$\left(
\tilde{e}_{lk}(W), \tilde{e}_{lk+1}(W)
\right)$
is defined to be empty. 
%
Theorem \ref{Sigma} immediately gives the following corollary.
%
%
\begin{corollary}
$\Sigma$
has at most one gap in the negative axis. 
%
\end{corollary}
%
Theorem \ref{Sigma} can be obtained by  
Theorem \ref{admissible potential} and 
Lemma \ref{inclusion} given below, which in turn
is proved by an argument similar to that 
used in the proof of 
Theorem \ref {two dimension-1} 
in view of the assumption (H4)
%
: one prove a equality similar to
(\ref{compare})
where 
$f$
is replaced by
$W \in {\cal A}_l$
and deduce a contradiction.
%
\begin{lemma}
%
Let 
$W \in {\cal A}_l$, $l \ge 1$.
%
Then 
%
$\Sigma^c 
\subset 
( \tilde{e}_{lk} (W), \tilde{e}_{lk+1} (W) )$. 
%
\label{inclusion}
\end{lemma}
%

%%%%% INTRODUCTION %%%%% 
\section{Appendix}
%
In this Appendix, 
we state some basic properties of the negative eigenvalues of the Schr\"
odinger operators, which were used to prove Theorems \ref{location of the 
spectrum-1}, \ref{two dimension-1} and \ref{Sigma}. 
%

Throughout this section, 
we work in general situation and assume that the potential 
$V$
of the Schr\"odinger operator
$-\triangle + V$
satisfies 
%
$V \in L^{r(d)}({\bf R}^d)$
where 
%
\[
r(d) = \cases{
1 & $(d = 1)$ \cr
%
1 + \alpha & $(d=2)$ \cr
%
\frac d2 & $(d \ge 3)$ \cr}
\]
%
$\alpha > 0$
is some positive constant. 
%
Under the assumption 
(H1), 
$f \in L^1({\bf R}^d) \cap L^p({\bf R}^d)$
with 
$p > p(d) \ge r(d)$
%
so that 
${\cal A} \subset L^{r(d)}({\bf R}^d)$.

%
$V$
is relatively form-bounded w.r.t. 
$(-\triangle)$
with bound less than 
$1$. 
%
Hence 
$H = -\triangle + V$
is defined to be the unique self-adjoint operator associated to the 
quadratic form
%
\[
E(\varphi) 
=
\int_{{\bf R}^d} 
\left(
| \nabla \varphi |^2 + V(x) | \varphi (x)|^2
\right) 
dx, \quad
\varphi \in H^1 ({\bf R}^d).
\]
%
If, in addition, 
$V$
is relatively bounded w.r.t. 
$(-\triangle)$
with bound less than $1$
(e.g., 
$V \in L^{r(d)}({\bf R}^d) \cap L^p ({\bf R}^d)$, 
$p =2 (d \le 3)$, $p > 2 (d = 4)$, $p \ge \frac d2 (d \ge 5)$), 
then the domain of 
$H$
is equal to 
$H^2 ({\bf R}^d)$.
%

Moreover, we note 
$\sigma_{ess}(H) = [0, \infty)$
so that the spectrum of 
$H$
in the negative axis consists of eigenvalues of finite multiplicity.
%

%%%
\subsection{Continuity of negative eigenvalues}
%
Let 
$V \in L^{r(d)}({\bf R}^d)$, $\{ V_k \}_{k \ge 1} \subset L^{r(d)} ({\bf R}^
d)$
such that 
$\lim_{k \to \infty} V_k = V$
in 
$L^{r(d)}({\bf R}^d)$
and define
%
\[
H = - \triangle + V, 
\quad
H_k = - \triangle + V_k.
\]
%
%
Let 
$\{ E_n \}_{n=1}^{N(H)}$
(resp. $\{ E^{(k)}_n \}_{n=1}^{N(H_k)}$)
be the negative eigenvalues of 
$H$
(resp. $H_k$)
in increasing order counting multiplicity, 
%
where 
$N(H)=N_{(-\infty, 0)}(H)$
(resp. $N(H_k)=N_{(-\infty, 0)}(H_k)$),  
%
$0 \le N(H) \le \infty$, $0 \le N(H_k) \le \infty$. 
%
\begin{proposition}
{\bf  \mbox{}}\\
%
(1)
Suppose
$M \in {\bf N}$, $M \le N(H)$.
%
Then
$N(H_k) \ge M$
for 
$\| V - V_k \|_{r(d)}$
small enough and 
%
\[
\lim_{k \to \infty} E_j^{(k)} = E_j
\quad
\mbox{for }
\,
1 \le j \le M.
\]
%
(2)
Suppose 
$b < 0$
satisfies 
%
$b \in 
\left[
\rho (H) \cap 
\left(
\bigcap_{k \ge 1} \rho (H_k)
\right)
\right]^{int}$, 
then
%
\label{continuity of eigenvalues}
%
\[
\lim_{k \to \infty} 
N_{(-\infty, b)}(H_k)= N_{(-\infty, b)}(H).
\]
%
\end{proposition}
%
For the proof, we state three lemmas below. 
%
Proposition \ref{continuity of eigenvalues}
follows from 
Lemma \ref{comparison between forms}
with the use of the comparison principle between forms. 
%
Lemma \ref{comparison between forms} 
follows from Lemmas \ref{form bound}, \ref{Sobolev estimate}, which are the immediate consequences of  Sobolev's inequality 
(e.g., \cite{LL}): 
%
\begin{eqnarray*}
&(i) & \; d = 1
\quad
\| f' \|_2^2 + \| f \|^2_2 \ge 2 \| f \|^2_{\infty}
\\
%
& (ii) &\; d = 2 
\quad
\| \nabla f \|^2_2 + \| f \|_2^2 \ge S_{2,q} \| f \|^2_q, 
\quad
2 \le \forall q < \infty
\\
%
& (iii) & \;d \ge 3
\quad
\| \nabla f \|_2^2 \ge S_{d, q} \| f \|_q^2, 
\quad
q = \frac {2d}{d-2}
\end{eqnarray*}
%
where 
$S_{d,q}$
is a positive constant which depends only on 
$d, q$.
%
%
\begin{lemma}
%
If
$V \in L^{r(d)}({\bf R}^d)$, 
then\footnote{
$\| \psi \|_{H^1}
:=
\left(
\| \nabla \psi \|_2^2 + \| \psi \|_2^2
\right)^{\frac 12}$
is the 
$H^1({\bf R}^d)$-norm.}
%
\label{form bound}
%
\begin{eqnarray*}
&&
\left|
\int_{{\bf R}^d} \varphi (x) V(x) \psi(x) dx
\right|
 \le 
C_d \| V \|_{r(d)} \| \varphi \|_{H^1} \| \psi \|_{H^1},
\quad
\varphi, \psi \in H^1({\bf R}^d), 
\\
%
&&
\mbox{where }
C_d = \cases{
\frac 12 & $(d = 1)$ \cr
%
S_{d,q(d)}^{-1} & $(d \ge 2)$ \cr}
\quad
%
q(d) = \cases{
\frac {2(1+\alpha)}{\alpha} & $(d = 2)$ \cr
%
\frac {2d}{d-2} & $(d \ge 3)$ \cr}
\end{eqnarray*}
%
\end{lemma}
%
%
\begin{lemma}
%
For 
$V \in L^{r(d)}({\bf R}^d)$, 
%
let 
%
\[
E(\varphi) = \int_{{\bf R}^d} 
\left(
| \nabla \varphi |^2 
+
 V(x) | \varphi (x) |^2 
 \right)dx,
\quad
\varphi \in H^1({\bf R}^d).
\]
%
If we decompose 
$V = W + U$, 
$W \in L^{\infty}({\bf R}^d)$, 
$U \in L^{r(d)}({\bf R}^d)$, 
we have 
%
\begin{eqnarray*}
& (i) & \; d \le 2 
\quad
\| \nabla \varphi \|_2^2
\le
\frac {1}{1 - C_d \| U \|_{r(d)}}
\left\{
E(\varphi) + 
\left(
\| W \|_{\infty} + C_d \| U \|_{r(d)} 
\right)
\| \varphi \|_2^2
\right\}
\\
%
& (ii) & \; d \ge 3 
\quad
\| \nabla \varphi \|_2^2
\le
\frac {1}{1 - C_d \| U \|_{r(d)}}
\left\{
E(\varphi) + 
\| W \|_{\infty}\| \varphi \|_2^2
\right\}
\end{eqnarray*}
%
%
where 
$\| U \|_{r(d)}$
is taken sufficiently small if necessary so that the denominator in RHS is 
positive : 
$1 - C_d \|U \|_{r(d)} > 0$. 
%
\label{Sobolev estimate}
\end{lemma}
%
\begin{lemma}
%
For 
$V \in L^{r(d)} ({\bf R}^d)$, 
let
%
\begin{eqnarray*}
E(\varphi)
&=&
\int_{{\bf R}^d} 
\left(
| \nabla \varphi (x) |^2 + V(x) | \varphi (x) |^2
\right) dx
\\
%
E_k (\varphi)
&=&
\int_{{\bf R}^d} 
\left(
| \nabla \varphi (x) |^2 + V_k(x) | \varphi (x) |^2
\right) dx
\end{eqnarray*}
%
where 
$\varphi \in H^1 ({\bf R}^d)$, $k \ge 1$. 
%
Then 
for any 
$\epsilon> 0$, 
we can find 
$\delta = \delta(\epsilon) > 0$
s.t. if
$\| V - V_k \|_{r(d)} < \delta$, 
\label{comparison between forms}
%
\[
(1 - \epsilon) E(\varphi)
-
\epsilon \langle \varphi, \varphi \rangle
\le
E_k(\varphi)
%
\le 
(1 + \epsilon) E(\varphi)
+
\epsilon \langle \varphi, \varphi \rangle
\]
%
for 
$\varphi \in H^1({\bf R}^d)$. 
%}
\end{lemma}
%
%
%

%%%%%%%%%%%%%%%
\subsection{Independence of eigenvalues at infinity}
%
Let 
$f \in L^{r(d)} ({\bf R}^d)$
and define
%
\begin{eqnarray*}
H_{\bf u} &=& - \triangle + W_{n, {\bf u}}, 
\quad
%
h = -\triangle + f,
\\
%
W_{n, {\bf u}} (x) 
&=&
\sum_{j=1}^n f(x - u_j), 
\quad
%
{\bf u} = (u_1, u_2, \cdots, u_n) \in {\bf R}^{nd},
\\
%
R({\bf u})
&=&
\min_{i \ne j} | u_i - u_j|. 
\end{eqnarray*}
%
Let 
$\{ e_i \}_{i=1}^{N(h)}$
(resp.
$\{ E_j ({\bf u}) \}_{j=1}^{N(H_{\bf u})}$)
be the negative eigenvalues of 
$h$
(resp. $H_{\bf u}$)
in increasing order counting multiplicity, where
$0 \le N(h) = N_{(-\infty, 0)}(h) \le \infty$, 
$0 \le N(H_{\bf u}) = N_{(-\infty, 0)}(H_{\bf u}) \le \infty$.
%
\begin{proposition}
{\bf  \quad}\\
%
(1)
Suppose 
$M \in {\bf N}$, $M \le N(h)$.
%
Then there is a constant 
$R_0 = R_0 (M) > 0$
such that if 
$R({\bf u}) > R_0$, 
then 
$N(H_{\bf u}) \ge M  n$
and for 
$j = nk + l$ ($0 \le k \le M-1$, $1 \le l \le n$)
we have 
%
\[
\lim_{R({\bf u}) \to \infty}
E_{nk + l} ({\bf u}) = e_{k+1}.
\]
%
(2)
Suppose 
$b \in \rho (h)$, $b < 0$. 
%
Then there is a constant 
$R_1 = R_1(b) > 0$
such that if 
$R({\bf u}) > R_1$, 
 we have 
\label{independence}
%
\[
N_{(-\infty, b)}(H_{{\bf u}})
=
n N_{(-\infty, b)} (h).
\]
%
\end{proposition}
%
We can prove 
Proposition \ref{independence} 
by mimicking the argument in the proof of 
\cite[Theorem 11.1]{CFKS}
with the use of  Lemmas \ref{form bound}, \ref{Estimates of eigenfunctions}(given below) and the so-called IMS localization formula.
%
\begin{lemma}
%
Let  
$\{ \varphi_p \}_{p=1}^{N(h)}$
be the normalized eigenfunctions of 
$h$
corresponding to 
$\{ e_p \}_{p=1}^{N(h)}$, and let 
$\varphi_{p,i} (x) = \varphi_p (x - u_i)$.
%
Then as
$R({\bf u}) \to \infty$, 
we have 
%
\begin{eqnarray*}
&&\langle \varphi_{p, i} , \varphi_{q,j} \rangle
=
\delta_{(p,i) , (q,j)} + o(1)
\\
%
&&\langle \varphi_{p, i} , H_{\bf u} \varphi_{q,j} \rangle
=
e_q \delta_{(p,i) , (q,j)} + o(1)
\end{eqnarray*}
%
where
\label{Estimates of eigenfunctions} 
%
\[
\delta_{(p,i) , (q,j)} = \cases{
1 & $ (p,i) = (q,j) $ \cr
%
0 & otherwise \cr}
\]
%
\end{lemma}
%
Lemma \ref{Estimates of eigenfunctions}
is proved by a standard approximation argument using Lemma \ref{form bound}.
%
%%%%%
\subsection{Number of bound states in one dimension}
%
In this subsection, 
we prove (\ref{concentrate}) for 
$d = 1$.
%
\begin{proposition}
%
Let 
$b < 0$
and 
$f \in L^1({\bf R})$. 
%
Then
\label{one dimension}
%
\[
N_{(-\infty, b]} (-\triangle + nf)
=
o(n)
\mbox{ as } n \to \infty.
\]
%
\end{proposition}
%
\begin{proof}
%%
By the min-max principle,
we may assume
$f \le 0$.
Take
%
$\{ f_k \}_{k=1}^{\infty} \subset C_0({\bf R})$
s.t.
$f_k \to f$
in
$L^1 ({\bf R})$.
%
Take
$\epsilon > 0$
small.
%
We then have
%
\begin{eqnarray*}
&&N_{(-\infty, b]} (-\triangle + n f) \\
& \le &
N_{(-\infty, \frac b2]}  \left(
- (1 - \epsilon)\triangle + n f_k
\right)
+
N_{(-\infty, \frac b2]} \left(
- \epsilon \triangle + n (f -f_k)
\right)
\\
%
%
& \le &
\sqrt{\frac {n}{1 - \epsilon}}
\cdot
\frac {\tau_1}{2 \pi}
\int_{\bf R}
\sqrt{f_k (x)} dx
(1 + o(1))
+
\frac {n}{\epsilon}
\cdot
\frac {1}{2\sqrt{ \frac {|b|}{2 \epsilon} } }
\int_{\bf R} | f(x) - f_k (x) | dx.
\end{eqnarray*}
%
as
$n \to \infty$.
%
In the last inequality,
we used
\cite[Theorem XIII.79]{Re-Si4}
(resp. Lemma \ref{BS-bound-1} given below)
for the first (resp. second) term.
%
Hence
%
%
\begin{eqnarray*}
\limsup_{n \to \infty}
\frac {N_{(-\infty, b]}(-\triangle + n f)}{n}
& \le &
\frac {1}{ \sqrt{2 \epsilon | b |} }
\int_{\bf R} | f (x) - f_k (x) | dx.
\end{eqnarray*}
%
Since
RHS can be made arbitrary small by taking
$k$
sufficiently large, we have
the assertion of the proposition.
\QED
\end{proof}
%
It remains to show the following lemma:

%
\begin{lemma}
%
If
$V \in L^1({\bf R})$,
$E < 0$,
\label{BS-bound-1}
%
\[
N_{(-\infty, E]} (-\triangle + V)
\le
\frac {1}{2 \sqrt{-E}}
\int_{\bf R} | V_- (x) | dx.
\]
%
\end{lemma}
%
\begin{proof}
By the
min-max principle, we may assume
$V \le 0$.
%
Let 
$A = \sqrt{V_-} (-\triangle -E)^{-\frac 12}$, 
$B = (-\triangle -E)^{-\frac 12} \sqrt{V_-}$
and let 
$K_E = \sqrt{V_-} (- \triangle - E)^{-1} \sqrt{V_-}$.
%
Then\footnote{$\| \cdot \|_{HS}$
is the Hilbert-Schmidt norm.} 
%
\[
\mbox{Tr }(K_E)
\le
\| A \|_{HS} \| B \|_{HS}
=
\| B \|_{HS}^2.
\]
%
By direct computation, 
%
\begin{eqnarray*}
\| B \|_{HS}^2
=
\frac {1}{2 \pi} \| g \|_2^2 \| V \|_1
=
\frac {1}{2 \sqrt{-E}} \| V \|_1.
\end{eqnarray*}
%
Therefore, by the Birman-Schwinger principle, 
%
\[
N_{(-\infty, E]}(-\triangle + V)
\le
\mbox{ Tr }(K_E)
\le
\frac {1}{2 \sqrt{-E}}
\int_{\bf R} | V (x) | dx.
\]
%
\QED
\end{proof}
%


\noindent {\bf Acknowledgement }
One of the authors (F.N.) 
would like to thank Professor Hideo Tamura for introducing references 
\cite{Birman-Borzov, Birman-Solomyak} to him. 
%
The work of A.I. is partially supported by JSPS grant 
Kiban C-15540168
and
Kiban C-15540206.
%
The work of F.N. is partially supported by JSPS grant Wakate B-15740049.

%%%%% REFERENCES %%%%%%%%%%%%%%%%%%%%%
%
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%
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English translation in Topics in Math. Phys. {\bf 5}, 1972, 19-30.
%
\bibitem{Birman-Solomyak}
M. Sh. Birman, M. Z. Solomyak, 
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problems (in Russian), Funktsional. Anal. i Prilolzhen. {\bf 4}, 1970, 1-13;
 English translation in Functional Anal. Appl. {\bf 4}, 1970, 1-13. 
%
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%
%
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%
%
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%
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%
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%
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%
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\end{document}

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