Content-Type: multipart/mixed; boundary="-------------0504070211389" This is a multi-part message in MIME format. ---------------0504070211389 Content-Type: text/plain; name="05-125.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-125.keywords" Random Schroedinger operator, Poisson potential ---------------0504070211389 Content-Type: application/x-tex; name="Poisson-paper9.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Poisson-paper9.tex" %#format LaTeX \ifx\documentclass\undefined \documentstyle[12pt]{article} \else \documentclass[12pt]{article} \fi \sloppy %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Adaptation of spaces in eqnarray \makeatletter \renewcommand{\theequation}{\thesection.\arabic{equation}} \@addtoreset{equation}{section} \def\eqnarray{% \stepcounter{equation}% \let\@currentlabel=\theequation \global\@eqnswtrue \global\@eqcnt\z@ \tabskip\@centering \let\\=\@eqncr $$\halign to \displaywidth\bgroup\@eqnsel\hskip\@centering $\displaystyle\tabskip\z@{##}$&\global\@eqcnt\@ne \hfil$\displaystyle{{}##{}}$\hfil &\global\@eqcnt\tw@$\displaystyle\tabskip\z@{##}$\hfil \tabskip\@centering&\llap{##}\tabskip\z@\cr} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Taken From Mathsing \def\bbbr{{\rm I\!R}} %reelle Zahlen \def\bbbn{{\rm I\!N}} %natuerliche Zahlen \def\bbbp{{\rm I\!P}} \def\bbbe{{\rm I\!E}} \def\bbbz{{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}} {\hbox{$\sf\textstyle Z\kern-0.4em Z$}} {\hbox{$\sf\scriptstyle Z\kern-0.3em Z$}} {\hbox{$\sf\scriptscriptstyle Z\kern-0.2em Z$}}}} % \def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}} % \def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle \rm C$}\hbox{\raise 0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm C$}\hbox{\raise 0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\raise 0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\raise 0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}}} % \def\B{\bf B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % eqnum \makeatletter \renewcommand{\theequation}{% \thesection.\arabic{equation}} \@addtoreset{equation}{section} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{remark}{Remark}[section] \newtheorem{definition}{Definition}[section] \newsavebox{\toy} \savebox{\toy}{\framebox[0.65em]{\rule{0cm}{1ex}}} \newcommand{\QED}{\usebox{\toy}} \def\nlni{\par\ifvmode\removelastskip\fi\vskip\baselineskip\noindent} \newenvironment{proof}{\nlni\begingroup\it Proof.\rm}{ \endgroup\vskip\baselineskip} \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 %%%%%%%%%%%%%%%%%%%%% % \small \begin{thebibliography}{99} % \bibitem{Birman-Borzov} M. 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