Content-Type: multipart/mixed; boundary="-------------0111190933330" This is a multi-part message in MIME format. ---------------0111190933330 Content-Type: text/plain; name="01-427.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-427.keywords" discrete spectrum, pseudo-relativistic pair operator, Cwikel-Lieb-Rosenbljum inequality, Lieb-Thirring inequality ---------------0111190933330 Content-Type: application/x-tex; name="relav-new.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="relav-new.tex" %% This LaTeX-file was created by Mon Nov 19 17:13:15 2001 %% LyX 0.12 (C) 1995-1998 by Matthias Ettrich and the LyX Team %% Do not edit this file unless you know what you are doing. \documentclass[12pt]{amsart} \usepackage[T1]{fontenc} \usepackage[latin1]{inputenc} \usepackage{times} \pagestyle{plain} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. \newcommand{\LyX}{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\spacefactor1000} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands. \theoremstyle{plain} \newtheorem{thm}{Theorem} \theoremstyle{plain} \newtheorem{cor}{Corollary} \theoremstyle{plain} \newtheorem{lem}{Lemma} \theoremstyle{plain} \newtheorem{prop}{Proposition} \theoremstyle{remark} \newtheorem{rem}{Remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. \newcounter{const}[section] \newcommand{\co}{c_{\thesection .\refstepcounter{const}\arabic{const}}} \makeatother \begin{document} \title{On the Discrete Spectrum of a Pseudo-Relativistic Two-Body Pair Operator} \author{Semjon Vugalter and Timo Weidl} \address{\textsf{\tiny S. Vugalter: Mathematisches Institut der LMU, Theresienstrasse 39, 80333 Muenchen, Germany. T. Weidl: Universität Stuttgart, Fakultät Mathematik, Pfaffenwaldring 57, D-70569 Stuttgart, Germany.}\tiny } \date{17.11.2001} \subjclass{35P20} \begin{abstract} We prove Cwikel-Lieb-Rosenbljum and Lieb-Thirring type bounds on the discrete spectrum of a two-body pair operator and calculate spectral asymptotics for the eigenvalue moments and the local spectral density in the pseudo-relativistic limit. \end{abstract} \maketitle \section{Introduction} \subsection{Statement of the problem.} In this paper we consider the behaviour of two particles with the masses \( m_{+} \) and \( m_{-} \) in the absence of external fields. The non-relativistic Hamiltonian of such a system is given by \begin{equation} \label{schr} -\frac{1}{2m_{+}}\Delta ^{+}-\frac{1}{2m_{-}}\Delta ^{-}-V(x^{+}-x^{-})\quad \mbox {on}\quad L^{2}(\mathbb R^{2d}), \end{equation} where \( x^{+},x^{-}\in \mathbb R^{d} \) denote the spatial coordinates and \( -V \) stands for the interaction between the particles. Due to translational invariance, this operator is unitary equivalent to the direct integral \( \int ^{\oplus }_{\mathbb R^{d}}h(P)dP \), where \[ h(P)=-\frac{M}{2m_{+}m_{-}}\Delta _{y}-V(y)+\frac{p^{2}}{2M},\quad p=|P|,\] acts on \( L^{2}(\mathbb R^{d}) \). The parameter \( M=m_{+}+m_{-} \) is the total mass of the system and \( P\in \mathbb R^{d} \) is the total momentum. The spectrum of (\ref{schr}) is the union of the spectra of the pair operators \( h(P) \) for all \( P\in \mathbb R^{d} \). Notice that \( h(P) \) depends on \( P \) only by a shift of \( \frac{p^{2}}{2M} \), and the spectra of all \( h(P) \) coincide modulo the respective shift. In other words, the fundamental properties of the pair operator do not depend on the choice of the inertial system of coordinates. On the other hand, if we consider the pseudo-relativistic Hamiltonian \cite{H,LSV} \[ \sqrt{-\Delta ^{+}+m_{+}^{2}}+\sqrt{-\Delta ^{-}+m_{-}^{2}}-V(x^{+}-x^{-}),\] the corresponding decomposition into a direct integral \( \int ^{\oplus }_{\mathbb R^{d}}h_{rel}(P)dP \) gives rise to the pair operators \begin{equation} \label{hrel} h_{rel}(P)=\sqrt{|\mu _{+}P-i\nabla _{y}|^{2}+\mu _{+}^{2}M^{2}}+\sqrt{|\mu _{-}P+i\nabla _{y}|^{2}+\mu _{-}^{2}M^{2}}-V(y), \end{equation} where \( \mu _{\pm }=m_{\pm }M^{-1} \). Obviously these operators show a much more involved dependence on the total momentum \( P\in \mathbb R^{d} \). This implies a non-trivial behaviour of the spectra of \( h_{rel}(P) \) in \( P \). For example, if \( -V \) is a smooth, compactly supported attractive well, the essential spectrum of \( h_{rel}(P) \) coincides with the interval \( [(p^{2}+M^{2})^{1/2},\infty ) \) and the discrete spectrum is finite. However, the distribution of the negative eigenvalues of \begin{equation} \label{qrell} q_{rel}(P)=h_{rel}(P)-\sqrt{p^{2}+M^{2}},\quad p=|P|, \end{equation} depends on \( P \). Even if the attractive force \( -V \) is too weak to induce negative bound states for small \( p \), eigenvalues will appear as \( p \) grows and their total number tends to infinity as \( p\to \infty \). Our paper is devoted to the study of this phenomenom. More precisely, we shall study the following quantities. First, for given \( P \) we chose the system of coordinates such that \( P=(p,0,\dots ,0) \) and we stretch the spatial variables by the factor \( p^{-1} \). Obviously \( p^{-1}q_{rel}(P) \) is unitary equivalent to the operator \begin{equation} \label{opQ} Q(i\nabla ,y)=H_{p}(i\nabla )-V_{p}(y), \end{equation} where \( V_{p}(y)=V(yp^{-1}) \) and \[ H_{p}(\xi )=T_{+}(\xi )+T_{-}(\xi )-\sqrt{1+M^{2}p^{-2}}\] for \[ T_{\pm }(\xi )=\sqrt{|(\eta \mp \mu _{\pm })^{2}+|\zeta |^{2}+\mu ^{2}_{\pm }M^{2}p^{-2},}\] with \( \xi \in {\Bbb R}^{d} \), \( \xi =(\eta ,\zeta ) \) for \( \xi _{1}=\eta \in {\Bbb R} \) and \( (\xi _{2},\dots ,\xi _{d})=\zeta \in {\Bbb R}^{d-1} \), \( \mu _{\pm }>0 \), \( p>0 \). Throughout this paper we focus on the case of higher dimensions \( d\geq 3 \). We will discuss the behaviour of the total number of negative eigenvalues (including multiplicities)\footnote{ By \( \chi _{(0,\infty )} \) we denote the characteristic function of the negative semiaxes. } \[ N_{p}(V)=tr\, \chi _{(-\infty ,0)}(Q_{p}(i\nabla ,y))\] and the sum of the absolute values of the negative eigenvalues\footnote{ For real \( x \) we put \( 2x_{-}=|x|-x \). } \[ S_{p}(V)=tr\, (Q_{p}(i\nabla ,y))_{-}\] of the operator \( Q_{p}(i\nabla ,y) \). In particular, we shall compare these spectral quantities with their classical counterparts \begin{eqnarray} \Xi _{p}=\Xi _{p}(V) & = & (2\pi )^{-d}\int \int _{Q_{p}<0}d\xi dy,\label{phvol0} \\ \Sigma _{p}=\Sigma _{p}(V) & = & (2\pi )^{-d}\int \int (Q_{p}(\xi ,y))_{-}d\xi dy.\label{phsum0} \end{eqnarray} \subsection{The classical picture.} Already the initial analysis of the phase space averages (\ref{phvol0}) and (\ref{phsum0}) shows somewhat unexpected results. Put \( V\geq 0 \). It is not difficult to see, that \( \Xi _{p} \) is finite if and only if \( V\in L^{\frac{d}{2}}(\mathbb R^{d})\cap L^{d}(\mathbb R^{d}) \), while \( \Sigma _{p} \) is finite if and only if \( V\in L^{\frac{d}{2}+1}(\mathbb R^{d})\cap L^{d+1}(\mathbb R^{d}) \). However, within these classes of potentials the quantities \( \Xi _{p} \) and \( \Sigma _{p} \) show various asymptotical orders in \( p \) as \( p\to \infty \). Indeed, we have\footnote{ Below \( \omega _{d} \) is the volume of the \( d \)-dimensional unit ball. } \begin{eqnarray} \Xi _{p}(V)= & \frac{\omega _{d}p^{\frac{d+1}{2}}(1+o(1))}{2^{\frac{3d+1}{2}}\pi ^{d}}\int V^{\frac{d-1}{2}}dy & \quad \mbox {if}\quad V\in L^{\frac{d-1}{2}}\cap L^{d},\label{fo1} \\ \Sigma _{p}(V)= & \frac{\omega _{d}p^{\frac{d-1}{2}}(1+o(1))}{(d+1)2^{\frac{3d-1}{2}}\pi ^{d}}\int _{\mathbb {R}^{d}}V^{\frac{d+1}{2}}dy & \quad \mbox {if}\quad V\in L^{\frac{d+1}{2}}\cap L^{d+1}\label{fo2} \end{eqnarray} as \( p\to \infty \).\footnote{ We point out that the powers of \( V \) in (\ref{fo1}), (\ref{fo2}) are typical for the phase space behaviour of Schrödinger operators in the spatial dimension \( d-1 \). } On the other hand, consider the model potentials \begin{equation} \label{mopo} V_{\theta }(y)=\min \{1,v|y|^{-d/\theta }\}. \end{equation} If \( \frac{d-1}{2}<\theta 0 \). \subsection{Estimates on the counting function. } In section 3 we start the spectral analysis of the operators (\ref{opQ}) and develop Cwikel-Lieb-Rosenbljum type bounds on the counting function \( N_{p}(V) \). The strong inhomogeneity of the symbol prevents us from using ready standard versions of Cwikel inequality \cite{C,BKS}. Instead we apply a modification \cite{W1,W2}, where the estimate follows the phase space distribution as close as possible even for complicated symbols. In particular, we show that for \( p\geq M>0 \) \begin{eqnarray} N_{p}(V) & \leq & c\left( p^{2}\left( 1+\ln pM^{-1}\right) \left\Vert V\right\Vert _{L^{1}}+\left\Vert V\right\Vert _{L^{3}}^{3}\right) ,\quad d=3,\label{esnp1} \\ N_{p}(V) & \leq & c\left( p^{\frac{d+1}{2}}\left\Vert V\right\Vert _{L^{\frac{d-1}{2}}}^{\frac{d-1}{2}}+\left\Vert V\right\Vert _{L^{d}}^{d}\right) ,\quad d\geq 4,\label{esnp2} \\ N_{p}(V) & \leq & c\left( p^{1+\theta }M^{d-1-2\theta }\left\Vert V\right\Vert ^{\theta }_{\theta ,w}+\left\Vert V\right\Vert _{L^{d}}^{d}\right) ,\quad d\geq 3,\label{esnp3} \end{eqnarray} where \( \frac{d-1}{2}<\theta 0 \). Indeed, the massless kinetic energy \( \tilde{H}(\xi ) \) vanishes on the interval between \( e_{+} \) and \( e_{-} \), the first coordinate of the momentum will not contribute in this region and we experience practically a \( d-1 \) dimensional kinetic behaviour. Hence, to establish (\ref{esnp1}) for \( d=3 \) we have to deal with problems resembling spectral estimates for two-dimensional Schrödinger operators. In the massless case virtual bound states will prevent any estimates on \( N_{p}(V) \). The inclusion of a finite mass supresses this effect to some extend, but leads with our method of proof to the additional factor \( (1+\ln pM^{-1}) \) in (\ref{esnp1}) compared to (\ref{fo1}). If the potential \( V \) has a repulsive tail at infinity, the bound (\ref{esnp1}) can be complemented by the estimate \[ N_{p}\leq c(V)p^{2},\quad p\geq M>0,\quad d=3.\] This is carried out in Theorem \ref{tm: A1} in Appendix II. Moreover, combining the techniques of Appendix II and inequality (\ref{esnp1}) it is possible to show that \( N=o(p^{2}\ln pM^{-1}) \) as \( p\to \infty \) for arbitrary \( V\in L^{1}(\mathbb R^{3})\cap L^{3}(\mathbb R^{3}) \). Nevertheless it remains an open problem, up to what extend the logarithmic increase in \( p \) can be removed from (\ref{esnp1}) in general. \subsection{Estimates on the eigenvalue moments.} In section 4 we integrate the estimates (\ref{esnp1})-(\ref{esnp3}) according to the Lieb-Aizenman trick \cite{AL} to obtain Lieb-Thirring type bounds on the sums of the negative eigenvalues and find that for \( p\geq M>0 \) \begin{eqnarray} S_{p}(V) & \leq & c\left( p\left( 1+\ln pM^{-1}\right) \left\Vert V\right\Vert _{L^{2}}+p^{-1}\left\Vert V\right\Vert _{L^{4}}^{4}\right) ,\quad d=3,\label{essp1} \\ S_{p}(V) & \leq & c\left( p^{\frac{d-1}{2}}\left\Vert V\right\Vert _{L^{\frac{d+1}{2}}}^{\frac{d+1}{2}}+p^{-1}\left\Vert V\right\Vert _{L^{d+1}}^{d+1}\right) ,\quad d\geq 4,\label{essp2} \\ S_{p}(V) & \leq & c\left( p^{1-\theta }M^{d+1-2\theta }\left\Vert V\right\Vert ^{\theta }_{\theta ,w}+p^{-1}\left\Vert V\right\Vert _{L^{d+1}}^{d+1}\right) ,\quad d\geq 3,\label{essp3} \end{eqnarray} where \( \frac{d+1}{2}<\theta s\}\right) ,\quad s>0,\label{nuf} \\ f^{*}(t) & = & \inf _{\nu _{f}(s)\leq t}s,\quad t>0.\label{rearr} \end{eqnarray} Note that \( \int |f|^{q}d\nu =\int (f^{*})^{q}dt \) and that \( |f_{1}(x)|\geq |f_{2}(x)| \) for a.e. \( x\in \mathbb {R}^{d} \) implies \( f_{1}^{*}(t)\geq f_{2}^{*}(t) \) for all \( t>0 \). We say that \( f\in L^{q}_{w}(\mathbb {R}^{d}) \) if \[ \left\Vert f\right\Vert _{q,w}=\sup _{t>0}t^{-q^{-1}}f^{*}(t)\] is finite. Beside the quasi-norm \( \left\Vert \cdot \right\Vert _{q,w} \) we shall also use the asymptotical functionals \begin{eqnarray*} \delta _{q}(f) & = & \liminf _{t\to \infty }t^{-q^{-1}}f_{\nu }^{*}(t),\\ \Delta _{q}(f) & = & \limsup _{t\to \infty }t^{-q^{-1}}f_{\nu }^{*}(t), \end{eqnarray*} which are continuous on \( L^{q}_{w}(\mathbb {R}^{d}) \). The function \( \chi _{M} \) will denote the characteristic function of the set \( M \). If \( M=(-\infty ,t)\subset \mathbb {R} \) we write in shorthand \( \chi _{t}=\chi _{(-\infty ,t)} \). Let \( \omega _{d} \) stand for the volume of the unit ball in \( \mathbb R^{d} \). Finally, by \( c \) or \( c_{j.k} \) we denote various constants where we do not keep track of their exact values. In particular, the same notion \( c \) in different equations does not imply that these constants coincide. \section{Uniform Estimates on the Number of Negative Eigenvalues: Cwikel's Inequality Revised} \subsection{Statement of the result.} \label{sec: 3}In this section we discuss a priori bounds on the counting function of the discrete spectrum of the operator \[ Q_{p}(i\nabla ,y)=H_{p}(i\nabla )-V_{p}(y).\] Our goal is to find estimates, which reproduce the behaviour of the phase space \[ \Xi _{p}=\Xi _{p}(V)=(2\pi )^{-d}\int \int _{Q_{p}<0}d\xi dy\] in general, and the asymptotics of \( \Xi _{p} \) for \( p\to \infty \) in particular, as closely as possible. In particular, we shall obtain the following two statements. \begin{thm} \label{sec: tm1}Assume that \( V\geq 0 \), \( V\in L^{\frac{d-1}{2}}(\mathbb {R}^{d})\cap L^{d}(\mathbb {R}^{d}) \) and \( p\geq M>0 \). Then there exists a finite constant \( c=c(d) \), which is independent on \( p \), \( M \) and \( V \), such that \begin{eqnarray} N_{p}(V) & \leq & c\left( p^{2}\left( 1+\ln pM^{-1}\right) \left\Vert V\right\Vert _{L^{1}}+\left\Vert V\right\Vert _{L^{3}}^{3}\right) ,\quad d=3,\label{bd13} \\ N_{p}(V) & \leq & c\left( p^{\frac{d+1}{2}}\left\Vert V\right\Vert _{L^{\frac{d-1}{2}}}^{\frac{d-1}{2}}+\left\Vert V\right\Vert _{L^{d}}^{d}\right) ,\quad d\geq 4.\label{bd13d4} \end{eqnarray} \end{thm} \begin{rem} Note that for \( d=3 \) in contrast to the asymptotical behaviour of the phase space volume \( \Xi _{p}\asymp p^{2}\left\Vert V\right\Vert _{L^{1}} \) as \( p\to \infty \) for \( V\in L^{1}(\mathbb R^{3})\cap L^{3}(\mathbb R^{3}) \), the bound (\ref{bd13}) contains an additional logarithmic factor. This underlines, that formula (\ref{bd13}) has in fact a two-dimensional character, see {[}W2{]}. \end{rem} \vspace{-0mm} \begin{rem} We point out, that in the case \( M=0 \) in the dimension \( d=3 \) one expects infinite many negative eigenvalues for any non-trivial attractive potential \( V\geq 0 \). In contrast to that in higher dimensions the bound (\ref{bd13d4}) holds true in the massless case as well. \end{rem} \begin{thm} \label{sec: tm2}Assume that \( d\geq 3 \), \( V\geq 0 \) and \( V\in L^{\theta }_{w}(\mathbb {R}^{d})\cap L^{d}(\mathbb {R}^{d}) \) for \( \frac{d-1}{2}<\theta <\frac{d}{2} \). Then there exist finite constants \( c_{1}(\theta ) \) and \( c_{2}(\theta ) \) independent on \( p \), \( M \) and \( V \), such that \begin{equation} \label{the11} N_{p}(V)\leq c_{1}(\theta )p^{1+\theta }M^{d-1-2\theta }\left\Vert V\right\Vert ^{\theta }_{\theta ,w}+c_{2}(\theta )\left\Vert V\right\Vert _{L^{d}}^{d} \end{equation} for all \( 00 \). \end{rem} The remaining part of this section is devoted to the proof of Theorem \ref{sec: tm1} and Theorem \ref{sec: tm2}. \subsection{A modification of Cwikel's inequality} Let \( Q_{A,B} \) be an operator of the type \[ Q_{A,B}=B(i\nabla )-A(x)\] on \( L^{2}(\mathbb {R}^{d}) \), where \( A=a^{2} \) and \( B=b^{-2} \) with \( a,b\geq 0 \). Assume that the operator \[ E_{a,b}=a(x)b(i\nabla )\] is compact in \( L^{2}(\mathbb {R}^{d}) \) and let \( \{s_{n}(E_{a,b})\}_{n\geq 1} \) be the non-increasing sequence of the singular values (approximation numbers) of \( E_{a,b} \). According to the Birman-Schwinger principle \cite{B,S} the total multiplicity of the negative spectrum of \( Q_{A,B} \) equals to the number of singular values \( s_{n}(E_{a,b}) \) exceeding one, that is \[ N_{A,B}:=tr\, \chi _{0}(Q_{A,B})=\mbox {card}\, \left\{ n:s_{n}(E_{a,b})>1\right\} .\] Hence, spectral estimates on the operators \( Q_{A,B} \) can be found in terms of estimates on the sequence \( \{s_{n}(E_{a,b})\}_{n\geq 1} \). In particular, if \( a \) and \( b \) satisfy \( a\in L^{r}(\mathbb {R}^{d}) \) and \( b\in L^{r}_{w}(\mathbb {R}^{d}) \) for some \( 21\}.\] For functions \( b(\xi ) \) which are not ``optimal'' members of the weak class \( L^{r}_{w}(\mathbb {R}^{d}) \), the right hand side of (\ref{CLR}) does not capture the respective phase space volumina. We are therefore in need for a suitable generalisation of (\ref{CLR}), which is applicable to a sufficiently wide class of symbols \( b \) and which reflects the phase space character of the estimate even for non-homogeneous symbols. Corresponding results can be found in \cite{W1,W2}. For the problem at hand we shall use the following statement from \cite{W2}. Consider the function \( q(x,\xi )=a(x)b(\xi ) \) on \( \mathbb {R}^{d}\times \mathbb {R}^{d} \) and assume that \( q\in L^{2}(\mathbb {R}^{2d})+L_{0}^{\infty }(\mathbb {R}^{2d}) \). Here \( L_{0}^{\infty }(\mathbb {R}^{2d}) \) stands for the subspace of bounded functions \( q \) satisfying \( q(x,\xi )\to 0 \) as \( |x|+|\xi |\to \infty \). Let \( q^{*} \) be the non-increasing rearrangement of \( q \), see (\ref{rearr}) and put \begin{equation} \label{qqq} \left\langle q\right\rangle (\hat{t})=\left( \hat{t}^{-1}\int _{0}^{\hat{t}}(q^{*}(t))^{2}dt\right) ^{1/2}, \end{equation} which is finite for any \( \hat{t}>0 \). If \( \nu =dxd\xi \) is the Lebesgue measure on \( \mathbb {R}^{2d} \) and the distribution function \( \nu _{q} \) is defined according to (\ref{nuf}), then using integration by parts the quantity (\ref{qqq}) can also be rewritten as follows \begin{equation} \label{qq} \left\langle q\right\rangle (\hat{t})=\left( (q^{*}(\hat{t}))^{2}+\frac{2}{\hat{t}}\int _{q^{*}(\hat{t})}^{\infty }s\nu _{q}(s)ds\right) ^{1/2},\quad \hat{t}>0. \end{equation} The following proposition holds true: \begin{prop} \label{pro1}(\cite{W2}) Assume that \( q(x,\xi )=a(x)b(\xi )\in L^{2}(\mathbb {R}^{2d})+L_{0}^{\infty }(\mathbb {R}^{2d}) \). Then \( E_{a,b}\in S_{\infty }(L^{2}(\mathbb {R}^{d})) \) and the inequality \begin{equation} \label{TWCLR} s_{n}(E_{a,b})\leq 5\left\langle q\right\rangle ((2\pi )^{d}n) \end{equation} holds true for all \( n\in \mathbb {N} \). \end{prop} \begin{rem} In conjunction with the Birman-Schwinger principle the bound (\ref{TWCLR}) implies \begin{equation} \label{BS} \frac{1}{5}\leq \left\langle q\right\rangle \left( (2\pi )^{d}N_{A,B}\right) . \end{equation} \end{rem} \subsection{Cwikel´s inequality for the operator \protect\( H_{p}(\xi )-V_{p}(y)\protect \). Preliminary estimates.} Now we apply Proposition \ref{pro1} to the particular symbol \( q_{p}(x,\xi )=a_{p}(x)b_{p}(\xi ) \) with \( A_{p}(x)=a_{p}^{2}(x)=V_{p}(x)\geq 0 \) and \( B_{p}(\xi )=b_{p}^{-2}(\xi )=H_{p}(\xi ) \). We start with some basic observations. Obviously it holds \[ \nu _{q_{p}}(s)=\nu \left\{ (x,\xi )\in \mathbb {R}^{d}\times \mathbb {R}^{d}|q_{p}(x,\xi )>s\right\} =\Xi _{p}(s^{-2}V),\quad s>0.\] The behaviour of the quantity \( \Xi _{p} \) is analysed in Appendix I. We establish there that according to (\ref{prelest}) and (\ref{Ladef1}) for \( p\geq M \) the two-sided bound \begin{equation} \label{nunu} \nu _{q_{p}}(s)\asymp \nu _{q_{p,1}}(s)+\nu _{q_{p,2}}(s)+\nu _{q_{p,3}}(s) \end{equation} holds true, where \begin{eqnarray} \nu _{q_{p,1}}(s) & = & \frac{p^{\frac{d}{2}+1}}{s^{d}M}\int _{\Omega _{1}(p,s)}V^{\frac{d}{2}}dx\label{nunu1} \\ \nu _{q_{p,2}}(s) & = & \frac{p^{\frac{d+1}{2}}}{s^{d-1}}\int _{\Omega _{2}(p,s)}V^{\frac{d-1}{2}}dx,\label{nunu2} \\ \nu _{q_{p,3}}(s) & = & s^{-2d}\int _{\Omega _{3}(p,s)}V^{d}dx,\label{nunu3} \end{eqnarray} and \begin{eqnarray} \Omega _{1}(p,s) & = & \{x|V(x)\leq s^{2}M^{2}p^{-1}\},\label{O1} \\ \Omega _{2}(p,s) & = & \{x|s^{2}M^{2}p^{-1}s^{2}p\}.\label{O3} \end{eqnarray} Moreover, note that from (\ref{nunu}) and (\ref{O1}), (\ref{O2}) one concludes \[ \nu _{q_{p}}(s)\geq \co \frac{p^{d+1}}{s^{2d}M^{d+1}}\int _{\Omega _{1}(p,s)}V^{d}dx+\co s^{-2d}\int _{\Omega _{2}(p,s)\cup \Omega _{3}(p,s)}V^{d}dx,\quad s>0.\] Since we assume \( p\geq M \), the bound \( \nu _{q_{p}}(s)\geq \co s^{-2d}\left\Vert V\right\Vert ^{d}_{L^{d}} \) holds true. Hence, for the inverse \( q^{*}_{p} \) of \( \nu _{q_{p}} \) we have \begin{equation} \label{O4} q_{p}^{*}(t)\geq \co t^{-\frac{1}{2d}}\left\Vert V\right\Vert _{L^{d}}^{1/2},\quad t>0. \end{equation} \subsection{Potentials \protect\( V\in L^{\frac{d-1}{2}}(\mathbb {R}^{d})\cap L^{d}(\mathbb {R}^{d})\protect \).} For this class of potentials (\ref{nunu}) and (\ref{O1}) imply \[ \nu _{q_{p}}(s)\leq \co \max \left\{ s^{1-d}p^{\frac{d+1}{2}}\left\Vert V\right\Vert ^{\frac{d-1}{2}}_{L^{\frac{d-1}{2}}},s^{-2d}\left\Vert V\right\Vert ^{d}_{L^{d}}\right\} ,\] or \begin{equation} \label{O5} q_{p}^{*}(t)\leq \co \max \left\{ t^{-\frac{1}{d-1}}p^{\frac{d+1}{2(d-1)}}\left\Vert V\right\Vert ^{1/2}_{L^{\frac{d-1}{2}}},t^{-\frac{1}{2d}}\left\Vert V\right\Vert ^{1/2}_{L^{d}}\right\} . \end{equation} Assume now that \( d\geq 4. \) Then (\ref{O5}), (\ref{qqq}) and (\ref{BS}) imply \begin{equation} \label{O6} 1\leq \co (N_{p}(V))^{-\frac{2}{d-1}}p^{\frac{d+1}{d-1}}\left\Vert V\right\Vert _{L^{\frac{d-1}{2}}}+\co (N_{p}(V))^{-\frac{1}{d}}\left\Vert V\right\Vert _{L^{d}}. \end{equation} The analogous bound for the case \( d=3 \) requires some more attention. For this we insert each of the three summand (\ref{nunu1})-(\ref{nunu3}) in (\ref{nunu}) into the integral in (\ref{qq}) and obtain \begin{eqnarray} \frac{1}{\hat{t}}\int ^{\infty }_{q_{p}^{*}(\hat{t})}s\nu _{1,q_{p}}(s)ds & \leq & \co \hat{t}^{-1}M^{-1}p^{\frac{5}{2}}\int _{\mathbb {R}^{3}}dxV^{\frac{3}{2}}(x)\int _{M^{-1}\sqrt{pV(x)}}^{\infty }s^{-2}ds\notag \\ & \leq & \co \hat{t}^{-1}p^{2}\left\Vert V\right\Vert _{L^{1}},\label{p1} \end{eqnarray} \begin{eqnarray} \frac{1}{\hat{t}}\int ^{\infty }_{q_{p}^{*}(\hat{t})}s\nu _{2,q_{p}}(s)ds & \leq & \co \hat{t}^{-1}p^{2}\int _{\mathbb {R}^{3}}dxV(x)\int _{\sqrt{p^{-1}V(x)}}^{M^{-1}\sqrt{pV(x)}}s^{-1}ds\notag \\ & \leq & \co \left\Vert V\right\Vert _{L^{1}}\hat{t}^{-1}p^{2}\ln pM^{-1},\label{p2} \end{eqnarray} as well as \begin{eqnarray*} \frac{1}{\hat{t}}\int ^{\infty }_{q_{p}^{*}(\hat{t})}s\nu _{3,q_{p}}(s)ds & \leq & \co \hat{t}^{-1}\int _{\mathbb {R}^{3}}dxV^{3}(x)\int _{q^{*}_{p}(\hat{t})}^{\infty }s^{-5}ds\\ & \leq & \co \hat{t}^{-1}(q_{p}^{*}(\tau ))^{-4}\left\Vert V\right\Vert _{L^{3}}^{3}. \end{eqnarray*} By (\ref{O4}) the last bound implies \begin{equation} \label{p3} \frac{1}{\hat{t}}\int ^{\infty }_{q_{p}^{*}(\hat{t})}s\nu _{3,q_{p}}(s)ds\leq \co \hat{t}^{-1/3}\left\Vert V\right\Vert _{L^{3}}. \end{equation} If we insert (\ref{O5})-(\ref{p3}) into (\ref{qq}) and (\ref{BS}) we arrive at \begin{equation} \label{bd130} 1\leq \co \max \left\{ (N_{p}(V))^{-1}\left\Vert V\right\Vert _{L^{1}}p^{2}\left( 1+\ln pM^{-1}\right) ,(N_{p}(V))^{-1/3}\left\Vert V\right\Vert _{L^{3}}\right\} . \end{equation} The relations (\ref{O6}) and (\ref{bd130}) imply Theorem \ref{sec: tm1}. \subsection{Potentials \protect\( V\in L^{\theta }_{w}(\mathbb R^{d})\cap L^{d}(\mathbb R^{d})\protect \), \protect\( \frac{d-1}{2}<\theta <\frac{d}{2}\protect \).} First observe, that (\ref{nunu3}) implies \begin{equation} \label{v3co} \nu _{3,q_{p}}(s)\leq \co s^{-2d}\left\Vert V\right\Vert _{L^{d}}^{d},\quad s>0. \end{equation} Furthermore, by (\ref{nunu1}) and (\ref{nunu2}) we have \begin{eqnarray} & & \notag \nu _{1,q_{p}}(s)+\nu _{2,q_{p}}(s)\\ & \leq & \co \int _{\mathbb R^{d}}\min \left\{ \frac{p^{\frac{d}{2}+1}}{s^{d}M}V^{\frac{d}{2}}(x),\frac{p^{\frac{d+1}{2}}}{s^{d-1}}V^{\frac{d-1}{2}}(x)\right\} dx.\label{OO} \end{eqnarray} Assume now \( \left\Vert V\right\Vert _{L_{w}^{\theta }}\leq v \), that is \( V^{*}(t)\leq vt^{-\frac{1}{\theta }} \) for all \( t>0 \). Passing from integration in space to integration of rearrangements (\ref{OO}) turns into \begin{eqnarray*} \nu _{1,q_{p}}(s)+\nu _{2,q_{p}}(s) & \leq & \co \int _{0}^{\infty }\min \left\{ \frac{p^{\frac{d}{2}+1}}{s^{d}M}v^{\frac{d}{2}}t^{-\frac{d}{2\theta }},\frac{p^{\frac{d+1}{2}}}{s^{d-1}}v^{\frac{d-1}{2}}t^{-\frac{d-1}{2\theta }}\right\} dt\\ & \leq & \co (\theta )\frac{p^{\frac{d}{2}+1}}{s^{d}M}v^{\frac{d}{2}}t_{c}^{1-\frac{d}{2\theta }}+\co (\theta )\frac{p^{\frac{d+1}{2}}v^{\frac{d-1}{2}}}{s^{d-1}}t_{c}^{1-\frac{d-1}{2\theta }} \end{eqnarray*} with \( t_{c}=M^{-2\theta }s^{-2\theta }v^{\theta }p^{\theta } \), and \[ \nu _{1,q_{p}}(s)+\nu _{2,q_{p}}(s)\leq \co (\theta )v^{\theta }M^{d-1-2\theta }s^{-2\theta }p^{1+\theta }.\] Together with (\ref{v3co}) this gives \[ \nu _{q_{p}}(s)\leq \co (\theta )\max \left\{ p^{1+\theta }M^{d-1-2\theta }s^{-2\theta }\left\Vert V\right\Vert _{\theta ,w}^{\theta },s^{-2d}\left\Vert V\right\Vert _{L^{d}}^{d}\right\} ,\quad s>0,\] and \[ q^{*}_{p}(t)\leq \co (\theta )\max \left\{ p^{\frac{1}{2}+\frac{1}{2\theta }}M^{\frac{d-1}{2\theta }-1}\left\Vert V\right\Vert _{\theta ,w}^{\frac{1}{2}}t^{-\frac{1}{2\theta }},\left\Vert V\right\Vert _{L^{d}}^{\frac{1}{2}}t^{-\frac{1}{2d}}\right\} ,\quad t>0.\] From (\ref{BS}) we conclude Theorem \ref{sec: tm2}. \section{Uniform estimates on the Eigenvalue Moments: Lieb-Thirring Inequalities Revised.} \subsection{Statement of the results.} \label{sec: 4}Alongside with estimates on the number of negative eigenvalues we shall make use of estimates on the moments of eigenvalues. Given a bound on the counting function \( N_{p}(V) \), estimates on eigenvalue sums can be deduced from the identity \begin{equation} \label{snpnp} S_{p}(V)=\int _{0}^{\infty }N_{p}(V-pu)du. \end{equation} We shall obtain the following estimates. \begin{thm} Assume that \( V\geq 0 \), \( V\in L^{\frac{d+1}{2}}(\mathbb R^{d})\cap L^{d+1}(\mathbb R^{d}) \) and \( 00 \). \end{rem} \subsection{Potentials \protect\( V\in L^{\frac{d+1}{2}}(\mathbb {R}^{d})\cap L^{d+1}(\mathbb {R}^{d})\protect \).} First put \( d=3 \). Standard variational arguments and the Aizenman-Lieb integration \cite{AL} of the bound (\ref{bd13}) give \begin{eqnarray*} S_{p}(V) & \leq & \co p\left( 1+\ln pM^{-1}\right) \int _{0}^{\infty }du\int _{\mathbb R^{d}}(V-pu)_{+}dx\\ & & \qquad +\co \int _{0}^{\infty }du\int _{\mathbb R^{d}}(V-pu)_{+}^{3}dx, \end{eqnarray*} which implies (\ref{bd24}). In higher dimensions a similar integration of (\ref{bd13d4}) implies (\ref{bd245}). \subsection{Potentials \protect\( V_{+}\in L^{\theta }_{w}(\mathbb {R}^{d})\cap L^{d+1}(\mathbb {R}^{d})\protect \) with \protect\( \frac{d+1}{2}<\theta <\frac{d}{2}+1\protect \).} The inequality (\ref{the11}) contains a term with a weak \( L_{w}^{\theta } \)-norm. In contrast to the usual \( L^{p} \)-norms, these weak norms in the bound for the counting function cannot be carried over a respective weak norm in the Lieb-Thirring inequality via the Aizenman-Lieb trick. In fact, for the proof of our results below it shows to be necessary to refine (\ref{the11}) for potentials \( V=(W-pu)_{+} \). Using the same notation as in the previous section in analogy to (\ref{v3co}) we first find that \begin{equation} \label{v3co0} \nu _{q_{p},3}(s)\leq \co s^{-2d}\int _{\mathbb R^{d}}(W(x)-pu)_{+}^{d}dx. \end{equation} On the other hand, in analogy to (\ref{OO}) passing to the integration of rearrangements we find \begin{eqnarray*} \nu _{q_{p},1}+\nu _{q_{p},2} & \leq & \co \int _{\Omega _{1}\cup \Omega _{2}}\min \left\{ \frac{p^{\frac{d}{2}+1}}{Ms^{d}}(W-pu)_{+}^{\frac{d}{2}},\frac{p^{\frac{d+1}{2}}}{s^{d-1}}(W-pu)^{\frac{d-1}{2}}_{+}\right\} dx\\ & \leq & \co \int _{0}^{\infty }\min \left\{ \frac{p^{\frac{d}{2}+1}}{Ms^{d}}(W^{*}-pu)_{+}^{\frac{d}{2}},\frac{p^{\frac{d+1}{2}}}{s^{d-1}}(W^{*}-pu)^{\frac{d-1}{2}}_{+}\right\} dt. \end{eqnarray*} Put \( W\in L_{w}^{\theta } \) and \( \left\Vert W\right\Vert _{w,\theta }\leq v \), that is \( W^{*}(t)\leq vt^{-\frac{1}{\theta }} \) for \( t>0 \). Then we see that \begin{eqnarray*} \nu _{q_{p},1}(s)+\nu _{q_{p},2}(s) & \leq & \co \frac{p^{\frac{d}{2}+1}}{Ms^{d}}\int _{t_{c}}^{\infty }(vt^{-\frac{1}{\theta }}-pu)_{+}^{\frac{d}{2}}dt\\ & & \qquad +\co \frac{p^{\frac{d+1}{2}}}{s^{d-1}}\int _{0}^{t_{c}}(vt^{-\frac{1}{\theta }}-pu)^{\frac{d-1}{2}}_{+}dt, \end{eqnarray*} where \( t_{c}=v^{\theta }(pu+p^{-1}s^{2}M^{2})^{-\theta } \). The later integral transforms into \begin{eqnarray} \nu _{q_{p},1}(s)+\nu _{q_{p},2}(s) & \leq & \notag \co \frac{v^{\theta }p^{\frac{d}{2}+1}}{Ms^{d}}\int ^{p^{-1}s^{2}M^{2}}_{0}(t+pu)^{-\theta -1}t^{\frac{d}{2}}dt\\ & & \qquad +\co \frac{v^{\theta }p^{\frac{d+1}{2}}}{s^{d-1}}\int _{p^{-1}s^{2}M^{2}}^{\infty }(t+pu)^{-\theta -1}t^{\frac{d-1}{2}}dt.\label{lit} \end{eqnarray} Notice that for \( \frac{d+1}{2}<\theta <\frac{d}{2}+1 \) we have \begin{eqnarray} \int _{0}^{a}(t+\tilde{u})^{-\theta -1}t^{\frac{d}{2}}dt & \leq & \co \min \left\{ a^{\frac{d}{2}+1}\tilde{u}^{-\theta -1},\tilde{u}^{\frac{d}{2}-\theta }\right\} ,\label{lit1} \\ \int _{a}^{\infty }(t+\tilde{u})^{-\theta -1}t^{\frac{d-1}{2}}dt & \leq & \co \min \left\{ \tilde{u}^{\frac{d-1}{2}-\theta },a^{\frac{d-1}{2}-\theta }\right\} ,\label{lit12} \end{eqnarray} where the minimum is taken for the first elements of the respective sets if \( 00. \end{eqnarray*} The inverse \( q^{*}_{p} \) of \( \nu _{q_{p}} \) satisfies then the bound \begin{eqnarray*} q^{*}_{p}(t) & \leq & \co t^{-\frac{1}{2d}}\left\Vert (W-pu)_{+}\right\Vert ^{d}_{L^{d}}+\co \min \left\{ t^{\frac{1}{1-d}}v^{\frac{\theta }{d-1}}p^{\frac{d-\theta }{d-1}}u^{\frac{1}{2}-\frac{\theta }{d-1}},\right. \\ & & \qquad \left. t^{-\frac{1}{d}}v^{\frac{\theta }{d}}p^{1+\frac{1-\theta }{d}}M^{-\frac{1}{d}}u^{\frac{1}{2}-\frac{\theta }{d}}+t^{-\frac{1}{2\theta }}v^{\frac{1}{2}}p^{\frac{1+\theta }{2\theta }}M^{\frac{d-1}{2\theta }-1}\right\} \end{eqnarray*} for all \( t>0 \). Hence, if \( d\geq 4 \) we get \begin{eqnarray*} \left\langle q_{p}\right\rangle (t) & \leq & \co t^{-\frac{1}{2d}}\left\Vert (W-pu)_{+}\right\Vert ^{d}_{L^{d}}+\co \min \left\{ \frac{v^{\frac{\theta }{d-1}}p^{\frac{d-\theta }{d-1}}}{t^{\frac{1}{d-1}}u^{\frac{\theta }{d-1}-\frac{1}{2}}},\right. \\ & & \qquad \left. \frac{v^{\frac{\theta }{d}}p^{1+\frac{1-\theta }{d}}u^{\frac{1}{2}-\frac{\theta }{d}}}{t^{\frac{1}{d}}M^{\frac{1}{d}}}+\frac{v^{\frac{1}{2}}p^{\frac{1+\theta }{2\theta }}M^{\frac{d-1}{2\theta }-1}}{t^{\frac{1}{2\theta }}}\right\} \end{eqnarray*} for all \( t>0 \), while for the dimension \( d=3 \) we obtain \begin{eqnarray*} \left\langle q_{p}\right\rangle (t) & \leq & \co t^{-\frac{1}{6}}\left\Vert (W-pu)_{+}\right\Vert ^{\frac{1}{2}}_{L^{3}}+\co \min \left\{ \frac{v^{\frac{1}{2}}p^{\frac{1}{2\theta }+\frac{1}{2}}}{t^{\frac{1}{2\theta }}M^{1-\frac{1}{\theta }}}+\right. \\ & & \left. +\frac{v^{\frac{\theta }{3}}p^{\frac{4-\theta }{3}}u^{\frac{1}{2}-\frac{\theta }{3}}}{t^{\frac{1}{3}}M^{\frac{1}{3}}},\frac{v^{\frac{\theta }{2}}p^{\frac{3-\theta }{2}}u^{\frac{1-\theta }{2}}}{t^{\frac{1}{2}}}\left( 1+\ln _{+}\left( \frac{tu^{\theta }p^{\theta -1}}{v^{\theta }M^{2}}\right) \right) \right\} \end{eqnarray*} as \( t>0 \). In view of (\ref{BS}) we conclude, that it holds either in higher dimesions \begin{eqnarray} N_{p}(W-pu) & \leq & \notag \co \left\Vert (W-pu)_{+}\right\Vert ^{d}_{L^{d}}+\co v^{\theta }\min \left\{ p^{d-\theta }u^{\frac{d-1}{2}-\theta },\right. \\ & & \qquad \left. p^{d+1-\theta }M^{-1}u^{\frac{d}{2}-\theta }+p^{1+\theta }M^{d-1-2\theta }\right\} ,\quad d\geq 4,\label{np0} \end{eqnarray} or \begin{eqnarray} N_{p}(V-pu) & \leq & \co \left\Vert (W-pu)_{+}\right\Vert ^{d}_{L^{d}}+\co v^{\theta }\min \left\{ M^{-1}p^{4-\theta }u^{\frac{3}{2}-\theta }+\right. \notag \\ & & \left. +M^{2-2\theta }p^{1+\theta },u^{-\theta }M^{2}p^{1-\theta }f(C\sqrt{u}pM^{-1})\right\} \quad \mbox {if}\quad d=3,\label{np} \end{eqnarray} where \( C \) is some fixed finite positive constant and \( y=f(x) \) is the inverse function to \( x=\sqrt{y}/(1+\ln _{+}Cy) \) on \( \mathbb {R}_{+} \). Inserting (\ref{np0}) into (\ref{snpnp}) we obtain immediately (\ref{bd24theta}) for \( d\geq 4 \). To settle the case \( d=3 \) we first note that \( f(x)\leq cx^{2}(1+\ln _{+}\sqrt{C}x)^{2} \) if \( c \) is chosen such that \( \sqrt{c}\geq (1+\ln _{+}t)\left( 1+\ln _{+}\frac{t}{1+\ln _{+}t}\right) ^{-1} \) for all \( t>0 \). Hence, the bound (\ref{np}) can be developed as follows \begin{equation} \label{np1} N_{p}(V-pu)\leq \co \left\{ \begin{array}{ccc} v^{\theta }M^{-1}p^{4-\theta }u^{\frac{3}{2}-\theta }+v^{\theta }M^{2-2\theta }p^{1+\theta } & \: \mbox {for}\: & u\leq \frac{M^{2}}{Cp^{2}}\\ v^{\theta }p^{3-\theta }u^{1-\theta }(1+\ln _{+}(\sqrt{Cu}pM^{-1}))^{2} & \: \mbox {for}\: & u>\frac{M^{2}}{Cp^{2}} \end{array}\right. . \end{equation} For \( \theta >2 \), the following identity holds true \begin{eqnarray} & & \int _{a^{-2}}^{\infty }u^{1-\theta }(1+\ln (a\sqrt{u}))^{2}du\notag \\ & = & \left( \frac{1}{\theta -2}+\frac{1}{(\theta -2)^{2}}+\frac{1}{2(\theta -2)^{3}}\right) a^{2\theta -4}\label{np2} \end{eqnarray} for any \( a>0 \). If we integrate (\ref{np1}) in \( u \) for \( 2<\theta <\frac{5}{2} \) and take (\ref{np2}) into account, we arrive at (\ref{bd24theta}). \section{Asymptotics of the Eigenvalue Moments and the Counting Function} \subsection{Statement of the main results.} We turn now to the calculation of the asymptotical behaviour of \( \Sigma _{p}(V) \) and \( N_{p}(V) \) for certain cases. In particular, we shall obtain the following two formulae: \begin{thm} \label{sec: maintm1}Assume that \( V\in L^{\theta }(\mathbb {R}^{3})\cap L^{4}(\mathbb {R}^{3}) \) for some \( \theta <2 \) and that \( V \) has uniformly bounded, continuous second derivatives if \( d=3 \), or that \( V\in L^{\frac{d+1}{2}}(\mathbb {R}^{d})\cap L^{d+1}(\mathbb {R}^{d}) \) if \( d\geq 4 \). Then the asymptotical formula \begin{equation} \label{mainas1} S_{p}(V)=(1+o(1))\Sigma _{p}(V)=\frac{(1+o(1))p^{\frac{d-1}{2}}\omega _{d}}{(d+1)2^{\frac{3d-1}{2}}\pi ^{d}}\int _{\mathbb {R}^{d}}V_{+}^{\frac{d+1}{2}}(y)dy \end{equation} holds true as \( p\to \infty \). \end{thm} \begin{rem} For \( d=3 \) the assumptions on the potential \( V \) in Theorem \ref{sec: maintm1} are more restrictive than the natural one \( V\in L^{2}\cap L^{4} \). The additional logarithmic factor in (\ref{bd24}) prevents one to use this bound to close formula (\ref{mainas1}) to the natural class of potentials. It remains an open problem, whether (\ref{mainas1}) holds actually for all \( V\in L^{2}\cap L^{4} \) if \( d=3 \). \end{rem} \begin{thm} \label{sec: maintm2}Assume that \( U,V\geq 0 \), \( U,V\in L^{\theta }\cap L^{d+1} \) for some \( \theta <\frac{d+1}{2} \) and that \( U \) and \( V \) possess uniformly bounded second derivatives. Put \( U(y;p)=U(p^{-1}y) \). Then \begin{equation} \label{mainas2} \lim _{p\to \infty }p^{-\frac{d+1}{2}}tr\: U(y;p)\chi _{0}(Q_{p}(i\nabla ,y))=\frac{\omega _{d}}{2^{\frac{3d+1}{2}}\pi ^{d}}\int U(x)V^{\frac{d-1}{2}}(x)dx. \end{equation} \end{thm} We mention the following obvious consequence of Theorem \ref{sec: maintm2}: \begin{cor} If \( V\geq 0 \) has uniformly bounded second derivatives and \( V\in L^{\frac{d-1}{2}}\cap L^{d+1} \) then \[ \liminf _{p\to \infty }p^{-\frac{d+1}{2}}N_{p}(V)\geq \frac{\omega _{d}}{2^{\frac{3d+1}{2}}\pi ^{d}}\int V^{\frac{d-1}{2}}dx.\] \end{cor} The remaining part of this paper is devoted to the proof of Theorem \ref{sec: maintm1} and Theorem \ref{sec: maintm2}. Our approach is based on the methods of coherent states. Therefore we first give a short survey of the necessary general material from this subject. \subsection{Coherent States and Berezin-Lieb Inequalities: Preliminaries.} Fix some spherically symmetric, smooth, non-negative function \( f \) with compact support in \( \mathbb {R}^{d} \), such that \( \left\Vert f\right\Vert _{L^{2}(\mathbb {R}^{d})}=1 \). Put \( f_{\epsilon }(x)=\epsilon ^{d/2}f(\epsilon x) \) where \( \epsilon >0 \). For given \( \gamma =\{y,\xi \} \) with \( y,\xi \in \mathbb {R}^{d} \) we define the coherent states \begin{equation} \label{CoHe} \Pi _{\gamma }^{\epsilon }(x)=e^{-i\xi x}f_{\epsilon }(x-y). \end{equation} For any fixed \( \gamma \) and \( \epsilon \) it holds \( \left\Vert \Pi _{\gamma }^{\epsilon }\right\Vert _{L^{2}(\mathbb {R}^{d})}=1 \). Let \( J \) be a non-negative, locally integrable function on \( \mathbb {R}^{d} \) with not more than polynomial growth at infinity. We define the operator \( J(i\nabla )=\Phi ^{*}J\Phi \) in the usual way with \( \Phi \) being the unitary Fourier transformation. Put \( \hat{f}=\Phi f \). In view of our choice of coherent states it is associated with the symbol function \begin{equation} \label{jint} j_{\epsilon }(\gamma )=j_{\epsilon }(\xi )=(J(i\nabla _{x})\Pi _{\gamma }^{\epsilon }(x),\Pi ^{\epsilon }_{\gamma }(x))_{L^{2}(\mathbb R^{d},dx)}=(J\star |\hat{f}_{\epsilon }|^{2})(\xi ). \end{equation} The operator of multiplication by a locally integrable real-valued function \( W \) on \( \mathbb {R}^{3} \) corresponds to the symbol \[ w_{\epsilon }(\gamma )=w_{\epsilon }(y)=(W(x)\Pi _{\gamma }^{\epsilon }(x),\Pi ^{\epsilon }_{\gamma }(x))_{L^{2}(\mathbb R^{d},dx)}=(W\star f_{\epsilon }^{2})(y),\] Here \( (\cdot ,\cdot )_{L^{2}(\mathbb R^{d},dx)} \) is the scalar product in \( L^{2}(\mathbb R^{d}) \) with respect to the variable \( x \) and \( u\star v \) denotes the convolution \[ (u\star v)(x)=\int u(x-x^{\prime })v(x^{\prime })dx^{\prime }.\] If now \( W=W_{1}+W_{2} \), where \( W_{1} \) is uniformly bounded and \( W_{2} \) is form compact with respect to \( J(i\nabla ) \), the operator sum \( J(i\nabla )+W(x) \) can be defined in the form sense. Let \( \psi \) be some non-negative convex function on \( \mathbb {R} \), such that \( \psi (J(i\nabla )+W(x)) \) is trace class. Then the Lieb-Berezin inequality states that (\cite{Be}, see also \cite{LS}) \begin{equation} \label{LB1} \int _{\mathbb {R}^{2d}}\psi (j_{\epsilon }(\xi )+w_{\epsilon }(y))d\gamma \leq tr\, \psi (J(i\nabla )+W(x)). \end{equation} Moreover, if the average of \( \psi (J(\xi )+W(y)) \) in \( \mathbb {R}^{2d} \) with respect to \( d\gamma \) is finite, then \( \psi (j_{\epsilon }(i\nabla )+w_{\epsilon }(x)) \) is trace class and \begin{equation} \label{LB2} tr\, \psi (j_{\epsilon }(i\nabla )+w_{\epsilon }(x))\leq \int _{\mathbb {R}^{2d}}\psi (J(\xi )+W(y))d\gamma \end{equation} Let us finally assume that in addition to this \( J \) or \( W \) are twice continuously differentiable with the following uniform bounds on the matrix norms of the respective Hessians \[ \vartheta (J)=\max _{\xi \in \mathbb {R}^{d}}\left\Vert \left\{ \frac{\partial ^{2}J}{\partial \xi _{l}\partial \xi _{k}}\right\} _{k,l=1}^{d}\right\Vert \quad \mbox {and}\quad \vartheta (W)=\max _{\xi \in \mathbb {R}^{d}}\left\Vert \left\{ \frac{\partial ^{2}W}{\partial \xi _{l}\partial \xi _{k}}\right\} _{k,l=1}^{d}\right\Vert .\] Put \[ \psi (x)=x_{-}=\left\{ \begin{array}{lcl} -x & \quad \mbox {for}\quad & x<0\\ 0 & \quad \mbox {for}\quad & x\geq 0 \end{array}\right. .\] We also recall that \( \chi _{-t} \) is the characteristic function of the interval \( (-\infty ,-t) \). Under the above conditions we have \begin{lem} \label{sec: Lme1}The two-sided bound \begin{eqnarray} \int (J(\xi )+W(y)+\kappa )_{-}d\gamma & \leq & tr\, (J(i\nabla )+W(x))_{-},\label{bdulm1} \\ tr\, (J(i\nabla )+W(x))_{-} & \leq & \int (J(\xi )+W(y))_{-}d\gamma +\Theta _{\kappa }\label{bdolm1} \end{eqnarray} holds true, where \[ \kappa =2\sqrt{\vartheta (J)\vartheta (W)}\left\Vert xf(x)\right\Vert \left\Vert \nabla f\right\Vert \] and \[ \Theta _{\kappa }=\int _{0}^{\kappa }tr\: \chi _{-t}(J(i\nabla )+W(x))dt.\] \end{lem} \begin{proof} Indeed, by Taylors formula we have \[ J(\xi -\xi ^{\prime })=J(\xi )-\xi ^{\prime }\cdot \nabla J(\xi )+\sum _{k,l}\frac{\partial ^{2}J(\tilde{\xi }(\xi ,\xi ^{\prime }))}{\partial \xi _{k}\partial \xi _{l}}\xi ^{\prime }_{k}\xi ^{\prime }_{l},\] where \( \tilde{\xi } \) is some point on the line segment connecting \( \xi \) and \( \xi ^{\prime } \). Inserting this into the integral expression for (\ref{jint}), because of \( \left\Vert \hat{f}_{\epsilon }\right\Vert _{L^{2}(\mathbb {R}^{d})}=1 \) one finds that \[ j_{\epsilon }(\xi )-J(\xi )=-\nabla J(\xi )\cdot \int \xi ^{\prime }|\hat{f}_{\epsilon }(\xi ^{\prime })|^{2}d\xi ^{\prime }+\sum _{k,l}\int \frac{\partial ^{2}J(\tilde{\xi })}{\partial \xi _{k}\partial \xi _{l}}\xi ^{\prime }_{k}\xi ^{\prime }_{l}|\hat{f}_{\epsilon }(\xi ^{\prime })|^{2}d\xi ^{\prime }.\] Since \( \hat{f}_{\epsilon } \) is spherically symmetric, the first integral on the r.h.s. vanishes and \begin{eqnarray} \notag |j_{\epsilon }(\xi )-J(\xi )| & \leq & \vartheta (J)\int |\xi ^{\prime }|^{2}|\hat{f}_{\epsilon }(\xi ^{\prime })|^{2}_{L^{2}(\mathbb {R}^{d})}d\xi ^{\prime }\\ & \leq & \vartheta (J)\epsilon ^{2}\left\Vert \nabla f\right\Vert ^{2}_{L^{2}(\mathbb {R}^{d})}.\label{erj} \end{eqnarray} In a similarly way we get \begin{equation} \label{erw} |w_{\epsilon }(y)-W(y)|\leq \vartheta (W)\epsilon ^{-2}\left\Vert xf(x)\right\Vert ^{2}_{L^{2}(\mathbb {R}^{d})}. \end{equation} Now (\ref{LB1}), (\ref{erj}) and (\ref{erw}) for the optimal choice of \( \epsilon \) give the first inequality of Lemma \ref{sec: Lme1}. On the other hand (\ref{erj}) and (\ref{erw}) imply \[ J(i\nabla )+W(x)+\kappa \geq j_{\epsilon _{0}}(i\nabla )+w_{\epsilon _{0}}(x)\] and \[ tr\, (J(i\nabla )+W(x))_{-}\leq tr\, (j_{\epsilon _{0}}(i\nabla )+w_{\epsilon _{0}}(x))_{-}+tr\: g_{\kappa }(J(i\nabla )+W(x))\] with \( g_{\kappa }(x)=\min \left\{ \kappa ,-x\right\} \) for \( x<0 \) and \( g_{\kappa }(x)=0 \) for \( x\geq 0 \). Since \[ g_{\kappa }(x)=\int _{0}^{\kappa }\chi _{-t}(x)dt,\] the bound (\ref{LB2}) implies the second statement of the Lemma. \end{proof} \section{Moments of negative Eigenvalues. An Estimate from Below.} \subsection{Summary.} We turn here to the study of the asymptotics of eigenvalue moments \[ S(p)=tr\, (Q_{p}(i\nabla ,y))_{-},\quad Q_{p}(i\nabla ,y)=H_{p}(\xi )-V_{p}(y).\] Because of the divergence of the second derivatives of \( H_{p}(\xi ) \) near the points \( e_{\pm }=(\pm \mu _{\pm },0,\dots ,0)\in \mathbb R^{d} \) as \( p\to \infty \), a straightforward application of the bound (\ref{bdulm1}) in Lemma \ref{sec: Lme1} will not lead to the desired results. Therefore we have to implement a suitable smoothing procedure of the symbol first. In this section we consider the bound from below. \subsection{Basic properties of the symbol \protect\( H_{p}(\xi )\protect \)} Consider the functions \[ T_{\pm }(\xi )=\sqrt{(\eta \mp \mu _{\pm })^{2}+|\zeta |^{2}+\mu ^{2}_{\pm }M^{2}p^{-2},}\] with \( \xi \in {\Bbb R}^{d} \), \( \xi =(\eta ,\zeta ) \) for \( \xi _{1}=\eta \in {\Bbb R} \) and \( (\xi _{2},\dots ,\xi _{d})=\zeta \in {\Bbb R}^{d-1} \), \( M=m_{+}+m_{-} \), \( \mu _{\pm }=m_{\pm }M^{-1} \). Here \( m_{\pm } \) and \( p \) are positive parameters. We have \[ H_{p}(\xi )=T_{+}(\xi )+T_{-}(\xi )-\sqrt{1+M^{2}p^{-2}}.\] This is a convex non-negative function, which is rotational symmetric with respect to the \( \eta \)-axes. It achieves a unique, non-degenerate minimum at the point \( \xi =0 \) where \( H_{p}(0)=0 \). The gradient and the Hessian of \( T_{\pm } \) calculate as follows \begin{eqnarray*} \nabla T_{\pm }(\xi ) & = & T_{\pm }^{-1}(\xi )\left( \eta \mp \mu _{\pm },\zeta ^{t}\right) ^{t},\\ (\nabla \nabla ^{t})T_{\pm } & = & T_{\pm }^{-1}\left( {\Bbb I}-(\nabla T_{\pm })(\nabla T_{\pm })^{t}\right) . \end{eqnarray*} Hence, \begin{equation} \label{dh} \left| \frac{\partial H_{p}(\xi )}{\partial \xi _{k}}\right| \leq 2\quad \mbox {and}\quad \left| \frac{\partial ^{2}H_{p}(\xi )}{\partial \xi _{k}\partial \xi _{l}}\right| \leq T_{+}^{-1}(\xi )+T_{-}^{-1}(\xi ) \end{equation} for all \( \xi \in \mathbb {R}^{d} \), \( p,M>0 \) and \( l,k=1,\dots ,d \). \subsection{Smoothing of the symbol.} Let \( g \) be a smooth, spherically symmetric non-negative function on \( {\Bbb R}^{d} \) supported within the unit ball, such that \( \int g(x)dx=1 \). If \( \sigma >0 \) we put \( g_{\sigma }(x)=\sigma ^{-d}g(\sigma ^{-1}x) \), for \( \sigma =0 \) we set \( g_{0}(x)=\delta (\cdot -x) \) and define \begin{eqnarray} H_{p,\sigma }(\xi ) & = & \int _{\mathbb R^{d}}H_{p}(\xi -y)g_{\sigma (\xi )}(y)dy\label{hg} \\ \notag & = & \int _{\mathbb R^{d}}H_{p}(\xi -\sigma (\xi )t)g(t)dt. \end{eqnarray} It holds \begin{lem} \label{sec: pointwise}The functions \( H_{p}(\xi ) \) and \( H_{p,\sigma }(\xi ) \) satisfy the pointwise estimate \begin{equation} \label{hphg} H_{p}(\xi )\leq H_{p,\sigma }(\xi ),\quad \xi \in \mathbb R^{d}. \end{equation} \end{lem} \begin{proof} Note that \( H_{p} \) is convex and the spherically symmetric weight \( g_{\sigma } \) has the total mass \( 1 \). If we represent in (\ref{hg}) the term \( H_{p}(\xi -y) \) in a Taylor series at the point \( \xi \) of order one with a positive quadratic form as remainder term, the inequality (\ref{hphg}) follows immediately. \end{proof} Put \( \tau _{\pm }=\tau _{\pm }(\xi )=|\xi -e_{\pm }| \). Below we chose \begin{equation} \label{sr} \sigma (\xi )=\sigma _{r}(\xi )=\left\{ \begin{array}{lcl} 0 & \mbox {if} & \xi \not \in B_{r}^{+}\cup B_{r}^{-}\\ re^{\varsigma _{-}(\xi ,r)} & \mbox {if} & \xi \in B^{-}_{r}\\ re^{\varsigma _{+}(\xi ,r)} & \mbox {if} & \xi \in B_{r}^{+} \end{array}\right. , \end{equation} where \( 00 \), \( \xi \in \mathbb R^{d} \) and \( k,l=1,\dots \, d \), such that \begin{eqnarray} \left| \frac{\partial H_{p,\sigma }}{\partial \xi _{k}}\right| & \leq & c,\label{dhp1} \\ \left| \frac{\partial ^{2}H_{p,\sigma }}{\partial \xi _{k}\partial \xi _{l}}\right| & \leq & c(1+r^{-1}).\label{dhp2} \end{eqnarray} \end{lem} \begin{proof} Obviously it holds \begin{equation} \label{difH1} \frac{\partial H_{p,\sigma }(\xi )}{\partial \xi _{k}}=\int \sum ^{d}_{j=1}\frac{\partial \nu _{j}}{\partial \xi _{k}}\frac{\partial H_{p}(\nu )}{\partial \nu _{j}}g(t)dt,\quad \nu _{j}=\xi _{j}-\sigma (\xi )t_{j}, \end{equation} and \begin{equation} \label{diff} \frac{\partial ^{2}H_{p,\sigma }(\xi )}{\partial \xi _{k}\partial \xi _{l}}=\int \left\{ \sum _{j=1}^{d}\frac{\partial ^{2}\nu _{j}}{\partial \xi _{k}\partial \xi _{l}}\frac{\partial H_{p}(\nu )}{\partial \nu _{j}}+\sum _{j,i=1}^{d}\frac{\partial \nu _{j}}{\partial \xi _{k}}\frac{\partial \nu _{i}}{\partial \xi _{l}}\frac{\partial ^{2}H_{p}(\nu )}{\partial \nu _{j}\partial \nu _{i}}\right\} g(t)dt. \end{equation} Since \begin{equation} \label{difg} \left| \frac{\partial \sigma _{r}}{\partial \xi _{k}}\right| \leq \co \quad \mbox {and}\quad \left| \frac{\partial ^{2}\sigma _{r}}{\partial \xi _{k}\partial \xi _{l}}\right| \leq \co r^{-1}, \end{equation} from (\ref{difH1}) and the first estimates in (\ref{dh}), (\ref{difg}) we conclude (\ref{dhp1}). To estimate the second derivatives we note , that by (\ref{dh}) and (\ref{difg}) the first part of the integral on the r.h.s. of (\ref{diff}) can be estimated by \( \co (1+r^{-1}) \), while the second term in (\ref{diff}) does not exceed \[ \co \int (T^{-1}_{+}(\nu )+T^{-1}_{-}(\nu ))g(t)dt.\] Note that \( T_{\pm }(\nu )\geq |\tau _{\pm }-\sigma _{r}t| \) and because \( g \) is bounded and of compact support we have \begin{eqnarray*} \int _{\mathbb R^{d}}\frac{g(t)dt}{T_{\pm }(\nu )} & \leq & \co \frac{\tau _{\pm }^{d-1}}{\sigma _{r}^{d}}\int _{\mathbb S^{d-2}}d\phi \int _{0}^{\pi }d\theta \sin \theta \int _{0}^{\frac{\sigma _{r}}{\tau _{\pm }}}\frac{t^{d-1}dt}{\sqrt{1+t^{2}-2t\cos \theta }}\\ & \leq & \co \frac{\tau _{\pm }^{d-1}}{\sigma _{r}^{d}}\int _{0}^{\frac{\sigma _{r}}{\tau _{\pm }}}t^{d-2}(t+1-|t-1|)dt\\ & \leq & \co \min \{\tau _{\pm }^{-1},\sigma _{r}^{-1}\}. \end{eqnarray*} For \( 0\leq \tau _{\pm }\leq r/2 \) the function \( \sigma _{r} \) can be estimated by \( \sigma _{r}\geq e^{-4/3}r \). Hence, \[ \int \frac{g(t)dt}{T_{\pm }(\nu )}\leq c(1+r^{-1})\] and we conclude (\ref{dhp2}). \end{proof} \subsection{The estimate from below.} We are now in the position to obtain the main result of this section. Put \begin{eqnarray*} Q_{p,\sigma _{r}}(\xi ,y) & = & H_{p,\sigma _{r}}(\xi )-V_{p}(y),\\ Q_{p,\sigma _{r}}(i\nabla ,y) & = & H_{p,\sigma _{r}}(i\nabla )-V_{p}(y). \end{eqnarray*} By Lemma \ref{sec: pointwise} we find that \begin{equation} \label{st1} tr\, (Q_{p}(i\nabla ,y))_{-}\geq tr\, (Q_{p,\sigma _{r}}(i\nabla ,y))_{-}. \end{equation} Next we apply the first part of Lemma \ref{sec: Lme1} with \( J=H_{p,\sigma _{r}} \) and \( W=V_{p} \) to this bound. By (\ref{dhp2}) we have \( \vartheta (H_{p,\sigma _{r}})\leq cr^{-1} \) for \( 00 \). \end{lem} \begin{proof} Let \( r\leq \min \{\mu _{-},\mu _{+}\} \). Since \[ G_{p,\delta ,\sigma _{r}}(\xi )=H_{p}((1-\delta )\eta ,\zeta )\quad \mbox {if}\quad \xi \not \in B_{r,\delta }^{+}\cup B_{r,\delta }^{-},\] the bound (\ref{GH}) for that case is an obvious consequence of the local monotonicity of \( H_{p}(\eta ,\zeta ) \) in \( \eta \) for fixed \( p \), \( M \) and \( \zeta \). On the other hand, by (\ref{dh}) it holds \( |\partial H_{p}/\partial \eta |\leq 2 \) and \( |\partial G_{p,\delta }/\partial \eta |\leq 2 \). Hence, if \[ r\leq r(\delta )=(H_{p}(e_{\pm ,\delta })-G_{p}(e_{\pm ,\delta }))/4\] we have \begin{equation} \label{maxest} \min _{\xi ^{\prime }\in B_{r,\delta }^{\pm }}H_{p}(\xi ^{\prime })\geq \max _{\xi ^{\prime \prime }\in B_{r,\delta }^{\pm }}G_{p,\delta }(\xi ^{\prime \prime }). \end{equation} For any \( \xi \in B_{r,\delta }^{\pm } \) and \( t\in \mathbb {R}^{d} \), \( |t|\leq 1 \) it holds \[ |(\xi -t\sigma _{r,\delta }(\xi ))-e_{\pm ,\delta }|\leq |\xi -e_{\pm ,\delta }|+\sigma _{r,\delta }(\xi )\leq r.\] The later inequality follows from the fact that \( x+e^{-(1-x^{2})^{-1}}\leq 1 \) for all \( 0\leq x<1 \). Thus, the argument \( \xi ^{\prime \prime }=\xi -t\sigma _{r,\delta }(\xi ) \) of \( G_{p,\delta } \) in (\ref{gsigma}) satisfies \( \xi ^{\prime \prime }\in B_{r,\delta }^{\pm } \) on the support of \( g \), and we conclude (\ref{GH}) from (\ref{maxest}) and the normalisation of \( g \). It remains to estimate \( r(\delta ) \) from below. Note that \( Mp^{-1}\leq 1 \) and \( 0<\delta <1/2 \). Then \begin{eqnarray*} 4r(\delta ) & = & H_{p}(e_{\pm ,\delta })-H_{p}(e_{\pm })\\ & \geq & \mp \frac{\delta \mu _{\pm }}{1-\delta }\min _{\mu _{\pm }\leq \eta \leq \mu _{\pm }(1-\delta )^{-1}}\frac{\partial }{\partial \eta }H_{p}(\eta ,0,0)\\ & \geq & \frac{\delta }{1-\delta }\frac{1}{\sqrt{\mu _{\mp }^{2}(1+(1-\delta )^{-1})^{2}+1}}\geq C(\mu _{+},\mu _{-})\delta . \end{eqnarray*} This completes the proof. \end{proof} \subsection{The estimate from above.} We put now \begin{eqnarray*} Q_{p,\delta ,\sigma }(\xi ,y) & = & G_{p,\delta ,\sigma }(\xi )-V_{p}(y),\\ Q_{p,\delta ,\sigma }(i\nabla ,y) & = & G_{p,\delta ,\sigma }(i\nabla )-V_{p}(y). \end{eqnarray*} From (\ref{GH}) it follows that for \( \sigma =\sigma _{r,\delta } \) \[ tr(Q_{p}(i\nabla ,y))_{-}\leq tr(Q_{p,\delta ,\sigma _{r,\delta }}(i\nabla ,y))_{-}\] if \( r\leq \min \{\mu _{-},\mu _{+},C(\mu _{+},\mu _{-})\delta \} \). For the eigenvalue sum on the right hand side we can apply (\ref{bdolm1}) in Lemma \ref{sec: Lme1} and we conclude \begin{lem} \label{sec: esabv}Assume that \( 00 \) if \( d=3 \) and \begin{equation} \label{spo2} S_{p}(V)\leq \frac{\Sigma _{p}(V)}{1-\delta }+\co \frac{p^{\frac{d-1}{2}}\vartheta ^{\beta }(V)\left\Vert V_{+}\right\Vert ^{\theta }_{L^{\theta }}}{(1-\delta )p^{3\beta }r^{\beta }}+\co \frac{\left\Vert V_{+}\right\Vert _{L^{d+1}}^{d+1}}{(1-\delta )p} \end{equation} as \( 00 \) if \( d\geq 4 \). Pick now \( \delta (p)=p^{-\epsilon } \) and \( r=r(p)=\min \{\mu _{+},\mu _{-},C(\mu _{+},\mu _{-})\delta \} \) with \( 0<\epsilon <3 \). Since \( \Sigma _{p}(V) \) is of order \( p^{\frac{d-1}{2}} \) for large \( p \), we claim \[ \limsup _{p\to \infty }p^{-1}S_{p}(V)\leq \lim _{p\to \infty }p^{-1}\Sigma _{p}(V).\] \subsection{The estimate from below.} On the other hand, from (\ref{st3}) and from the identity \( H_{p}(\xi )=H_{p,\sigma _{r}}(\xi ) \) for \( \xi \in \mathbb R^{d}\backslash (B_{r}^{+}\cup B_{r}^{-}) \) it follows that \begin{eqnarray*} S_{p}(V) & \geq & \int (Q_{p,\sigma _{r}}+\kappa )_{-}d\gamma \\ & \geq & \Sigma _{p}(V-p\kappa )-\int _{y\in B^{+}_{r}\cup B_{r}^{-}}(Q_{p}+\kappa )_{-}d\gamma . \end{eqnarray*} Next note that at least \( \left[ \frac{\mu _{\pm }-r}{2r}\right] \) disjoint balls of radius \( r \) can be placed into the domains \( [r-\mu _{-},0]\times (-r,r)^{2} \) and \( [0,\mu _{+}-r]\times (-r,r)^{2} \), respectively. Because of \( H_{p}(\eta ,\zeta )\geq H_{p}(\eta ^{\prime },\zeta ) \) for all \( |\eta ^{\prime }|\leq |\eta | \) we can conclude that \begin{eqnarray*} \left[ \frac{\mu _{\pm }-r}{2r}\right] \int _{y\in B_{r}^{\pm }}(Q_{p}+\kappa )_{-}d\gamma & \leq & \int _{\xi \in [r-\mu _{-},1-\mu _{+}]\times (-r,r)^{2}}(Q_{p}+\kappa )_{-}d\gamma \\ & \leq & \Sigma _{p}(V-p\kappa ) \end{eqnarray*} and \begin{equation} \label{spu} S_{p}(V)\geq \left( 1-\frac{1}{\left[ \frac{\mu _{-}-r}{2r}\right] }-\frac{1}{\left[ \frac{\mu _{+}-r}{2r}\right] }\right) \Sigma _{p}(V-p\kappa ). \end{equation} Put now \( r=r(p)=p^{-\alpha } \) with \( 0<\alpha <1 \). Then \( r\to 0 \) and simultaneously \( p\kappa =\co \vartheta ^{1/2}(V)r^{-1/2}p^{-1/2}\to 0 \) as \( p\to \infty \). Thus, it holds \[ \Sigma _{p}(V-p\kappa )\geq \Sigma _{p}(V-\delta )\] for arbitrary \( \delta >0 \) if \( p \) is large enough. Because of the given class of potentials this means \begin{eqnarray*} \liminf _{p\to \infty }S_{p}(V) & \geq & \lim _{p\to \infty }\Sigma _{p}(V-\delta )\\ & \geq & \frac{\omega _{d}}{(d+1)2^{\frac{3d-1}{2}}\pi ^{d}}\int _{\mathbb {R}^{d}}(V(x)-\delta )_{+}^{\frac{d+1}{2}}dx. \end{eqnarray*} Since \( V\in L^{\frac{d+1}{2}} \) we can pass to the limit \( \delta \to 0 \). \subsection{The closure of the asymptotical formula.} If \( d\geq 4 \), we finally apply inequality (\ref{bd245}) in a standard manner to close asymptotics (\ref{mainas1}) to all potentials \( V_{+}\in L^{\frac{d+1}{2}}\cap L^{d+1} \). However, for \( d=3 \) the appropriate Lieb-Thirring inequality (\ref{bd24}) contains the logarithmic factor \( 1+\ln \frac{p}{M} \), which prevents us from carrying out the same procedure in that case. \section{The Proof of Theorem \ref{sec: maintm2}} For the proof of Theorem \ref{sec: maintm2} we follow the main strategy of \cite{ELSS} and apply the bounds (\ref{spo1}), (\ref{spo2})and (\ref{spu}) of the previous section in a more subtle way. For the shortness of notation we shall write \[ Y_{p}=U(y;p)\chi _{0}(Q_{p}(i\nabla ,y))=pU_{p}\chi _{0}(Q_{p}(i\nabla ,y)),\] where in agreement with our previous notation \( U_{p}(y)=p^{-1}U(yp^{-1}) \). \subsection{The estimate from above.} Let \( \{\psi _{p,n}\} \) be an o.n. system of eigenfunctions corresponding to the negative part of \( Q_{p}(i\nabla ,y) \). Then for any \( \epsilon \in (0,1) \) it holds \begin{eqnarray*} tr\: p^{-1}Y_{p} & = & \sum _{n}\int U_{p}(x)|\psi _{p,n}(x)|^{2}dx\\ & \leq & \frac{1}{\epsilon }\left( tr\: (Q_{p}(i\nabla ,y)-\epsilon U_{p})_{-}-tr\: (Q_{p}(i\nabla ,y))_{-}\right) . \end{eqnarray*} Here we make use of the variational property \[ tr\: (Q_{p}(i\nabla ,y)-\epsilon U_{p})_{-}\geq tr\: D(\epsilon U_{p}-Q_{p}(i\nabla ,y))\] for any operator \( 0\leq D\leq 1 \). Put \( V_{\epsilon }=V+\epsilon U \). Then (\ref{spo1}) - (\ref{spu}) imply that \[ tr\: p^{-1}Y_{p}\leq \frac{1}{\epsilon }\left( \Sigma _{p}(V_{\epsilon })-\Sigma _{p}(V-p\kappa )+R(p,\epsilon ,\mu _{\pm },\delta ,V,U)\right) ,\] for all \( 00 \), \( p>0 \). Below we shall study properties of the phase space averages \begin{eqnarray} \Sigma _{p}=\Sigma _{p}(V) & = & (2\pi )^{-d}\int \int (Q_{p}(\xi ,y))_{-}d\xi dy,\label{Sup} \\ \Xi _{p}=\Xi _{p}(V) & = & (2\pi )^{-d}\int \int _{Q_{p}<0}d\xi dy.\label{Nup} \end{eqnarray} Set \begin{equation} \label{Ladef} \Lambda _{p}(y;V)=(2\pi )^{-d}\int _{Q_{p<0}}d\xi . \end{equation} \begin{lem} \label{sec: Lphv}Assume that \( \tau =Mp^{-1}\leq 1 \). Then for any \( y\in \Bbb R^{d} \) it holds \begin{equation} \label{Lambda} \Lambda _{p}(y)=\frac{\omega _{d}W^{\frac{d}{2}}\left( W+\upsilon \right) \left( W+2\upsilon \right) ^{\frac{d}{2}}\left( W^{2}+2W\upsilon +\tau ^{2}(1-4\tilde{\mu }^{2})\right) ^{\frac{d}{2}}}{(4\pi )^{d}\left( W^{2}+2W\upsilon +\tau ^{2}\right) ^{\frac{d+1}{2}}}, \end{equation} where \( W=W(y)=(V_{p}(y))_{+} \) and \( \upsilon =\sqrt{1+\tau ^{2}} \). \end{lem} \begin{proof} Fix some point \( y\in \Bbb R^{d} \). Since \( H_{p}(\xi )\geq 0 \) we have \( Q_{p}(\xi ,y)\geq 0 \) if \( V_{p}(y)\leq 0 \), what settles the statement in that case. Assume now \( V_{p}(y)\geq 0 \). Put \( \tilde{\mu }=(\mu _{-}-\mu _{+})/2 \) and \( \tilde{\eta }=\eta +\tilde{\mu } \). Then \( Q_{p}(\xi ,y)<0 \) is equivalent to \begin{eqnarray} \notag & & 2\sqrt{\left( |\zeta |^{2}+\tilde{\eta }^{2}+\frac{1}{4}+\left( \tilde{\mu }^{2}+\frac{1}{4}\right) \tau ^{2}\right) ^{2}-(\tilde{\eta }-\tilde{\mu }\tau ^{2})^{2}}\\ & < & A^{2}-2\left( |\zeta |^{2}+\tilde{\eta }^{2}+\frac{1}{4}+\left( \tilde{\mu }^{2}+\frac{1}{4}\right) \tau ^{2}\right) ,\label{c} \end{eqnarray} where \( A=V_{p}+\sqrt{1+\tau ^{2}} \). Thus, in particular, the condition \begin{equation} \label{co} |\zeta |^{2}+\tilde{\eta }^{2}0 \). Then \( \tilde{\mu }^{2}<1/4 \) and by (\ref{Lambda}) the quantity \( \Lambda _{p} \) permitts the following two-sided estimate \begin{equation} \label{prelest} \Lambda _{p}(y;V)\asymp \left\{ \begin{array}{lcl} \tau ^{-1}(V_{p}(y))_{+}^{d/2} & \quad \mbox {on}\quad & \Omega _{1}=\{y|V_{p}(y)\leq \tau ^{2}\}\\ (V_{p}(y))^{\frac{d-1}{2}}_{+} & \quad \mbox {on}\quad & \Omega _{2}=\{y|\tau ^{2}\leq V_{p}(y)\leq 1\}\\ (V_{p}(y))_{+}^{d} & \quad \mbox {on}\quad & \Omega _{3}=\{y|V_{p}(y)\geq 1\} \end{array}\right. , \end{equation} or equivalently, \begin{equation} \label{prel1} \Lambda _{p}(y;V)\asymp \min \left\{ \tau ^{-1}(V_{p}(y))_{+}^{\frac{d}{2}},(V_{p}(y))^{\frac{d-1}{2}}_{+}\right\} +(V_{p}(y))_{+}^{d}, \end{equation} which for fixed \( \tilde{\mu } \) is uniform for all \( p \) and \( M \) satisfying \( \tau \leq 1 \). Hence, \( V_{+}\in L^{d/2}(\mathbb {R}^{d})\cap L^{d}(\mathbb {R}^{d}) \) is sufficient and necessary for \begin{equation} \label{Ladef1} \Xi _{p}(V)=\int \Lambda _{p}(y;V)dy=p^{d}\int \Lambda _{p}(px;V)dx \end{equation} to be finite. \subsection{Potentials \protect\( V_{+}\in L^{\frac{d-1}{2}}(\mathbb {R}^{d})\cap L^{d}(\mathbb {R}^{d})\protect \)} For this class of potentials by (\ref{prelest}) the function \( p^{\frac{d-1}{2}}\Lambda _{p}(p\cdot ) \) has an integrable majorant, and by Lebesgues' limit theorem it holds \begin{equation} \label{dp1} \lim _{p\to \infty }p^{-\frac{d+1}{2}}\Xi _{p}=\lim _{p\to \infty }\int p^{\frac{d-1}{2}}\Lambda _{p}(py)dy=\frac{\omega _{d}}{2^{\frac{3d+1}{2}}\pi ^{d}}\int (V_{+}(y))^{\frac{d-1}{2}}dy. \end{equation} \subsection{Potentials \protect\( V_{+}\in L^{\frac{d+1}{2}}(\mathbb {R}^{d})\cap L^{d+1}(\mathbb {R}^{d})\protect \)} We find that the integrand on the r.h.s. of \[ p^{\frac{1-d}{2}}\Sigma _{p}(V)=p^{\frac{1-d}{2}}\int _{0}^{\infty }\Xi _{p}(V-sp)ds=\int _{0}^{\infty }\int _{\mathbb {R}^{d}}p^{\frac{d-1}{2}}\Lambda _{p}(py;V-t)d^{3}ydt\] is for fixed \( \tilde{\mu } \) bounded by a uniform multiple of \[ \max \{(V(y)-t)^{\frac{d-1}{2}}_{+},(V(y)-t)_{+}^{d}\},\] which is integrable on \( [0,\infty )\times \mathbb {R}^{d} \) for \( V_{+}\in L^{\frac{d+1}{2}}(\mathbb {R}^{d})\cap L^{d+1}(\mathbb {R}^{d}) \). Thus, \begin{eqnarray} \lim _{p\to \infty }p^{-\frac{d-1}{2}}\Sigma _{p}(V) & = & \int _{0}^{\infty }dt\int _{\mathbb {R}^{d}}\lim _{p\to \infty }\left( p^{\frac{d-1}{2}}\Lambda _{p}(py;V-t)\right) dy\notag \\ & = & \frac{\omega _{d}}{2^{\frac{1+3d}{2}}\pi ^{d}}\int _{0}^{\infty }dt\int _{\mathbb {R}^{d}}(V(y)-t)^{\frac{d-1}{2}}_{+}dy\notag \\ & = & \frac{\omega _{d}}{(d+1)2^{\frac{3d-1}{2}}\pi ^{d}}\int _{\mathbb {R}^{d}}V_{+}^{\frac{d+1}{2}}(y)dy.\label{sp24} \end{eqnarray} \subsection{Potentials \protect\( V_{+}\in L^{\theta }_{w}(\mathbb {R}^{d})\cap L^{d}(\mathbb {R}^{d})\protect \) with \protect\( \frac{d-1}{2}<\theta <\frac{d}{2}\protect \)} For potentials \( V \) where \( V_{+} \) is ``strictly between'' \( L^{\frac{d-1}{2}}(\mathbb {R}^{d})\cap L^{d}(\mathbb {R}^{d}) \) and \( L^{\frac{d}{2}}(\mathbb {R}^{d})\cap L^{d}(\mathbb {R}^{d}) \) the phase space volume shows a different behaviour in \( p \). Let us study the model potential \begin{equation} \label{vy} V(y)=\min \{1,v|y|^{-d/\theta }\}, \end{equation} where \( \frac{d-1}{2}<\theta <\frac{d}{2} \). Then \( V=V_{+}\in L^{\theta }_{w}(\mathbb {R}^{d}) \) and \( \left\Vert V\right\Vert _{\theta ,w}=c(\theta ,d)v \). The preliminary estimate (\ref{prelest}) shows that \begin{eqnarray*} \Xi _{p} & \asymp & p^{\frac{1-d}{2}}\int _{p^{1-\frac{\theta }{d}}v^{\frac{\theta }{d}}\leq |y|\leq pv^{\frac{\theta }{d}}}dy\\ & & +v^{\frac{d-1}{2}}p^{\frac{(d-1)(d-\theta )}{2\theta }}\int _{pv^{\frac{\theta }{d}}\leq |y|\leq p^{1+\frac{\theta }{d}}v^{\frac{\theta }{d}}M^{-\frac{2\theta }{d}}}|y|^{-\frac{d(d-1)}{2\theta }}dy\\ & & +v^{\frac{d}{2}}p^{1-\frac{d}{2}+\frac{d^{2}}{2\theta }}M^{-1}\int _{|y|\geq p^{1+\frac{\theta }{d}}v^{\frac{\theta }{d}}M^{-\frac{2\theta }{d}}}|y|^{-\frac{d^{2}}{2\theta }}dy\\ & \asymp & p^{\theta +1}v^{\theta }M^{d-1-2\theta }(1+o(1)) \end{eqnarray*} as \( p\to \infty \). After one has established the order of \( \Xi _{p} \) in \( p \), the same estimate now shows that \[ \Xi _{p}=(1+o(1))\int _{|y|>p^{1+c}}\Lambda _{p}(y)dy,\quad 0p^{1+c}}\Lambda _{p}(y)dy\\ & = & (1+o(1))\frac{\omega _{d}}{(4\pi )^{d}}\int _{|y|>p^{1+c}}W^{\frac{d}{2}}\frac{2^{\frac{d}{2}}(2W+\tau ^{2}\hat{\mu })^{\frac{d}{2}}}{(2W+\tau ^{2})^{\frac{d+1}{2}}}dy, \end{eqnarray*} where \( \hat{\mu }=1-4\tilde{\mu }^{2}\in (0,1] \) and \( W(y)=p^{-1}\min \left\{ 1,vp^{\frac{d}{\theta }}|y|^{-\frac{d}{\theta }}\right\} \) as \( p\to \infty \). This implies \[ \Xi _{p}=(1+o(1))\frac{2^{\frac{d}{2}}\omega _{d}p^{d+1}}{(4\pi )^{d}M}\int _{|x|>p^{c}}\left( \frac{v}{p}\right) ^{\frac{d}{2}}|x|^{-\frac{d^{2}}{2\theta }}\frac{\left( 2\frac{vp}{M^{2}}|x|^{-\frac{d}{\theta }}+\hat{\mu }\right) ^{\frac{d}{2}}}{\left( 2\frac{vp}{M^{2}}|x|^{-\frac{d}{\theta }}+1\right) ^{\frac{d+1}{2}}}dx\] or \begin{equation} \label{tpl1} \Xi _{p}=(1+o(1))\frac{2^{\theta }\theta \omega _{d}^{2}}{(4\pi )^{d}}v^{\theta }M^{d-1-2\theta }L(d,\theta ,\hat{\mu })p^{\theta +1}\quad \mbox {as}\quad p\to \infty \end{equation} with \begin{eqnarray*} L(d,\theta ,\hat{\mu }) & = & \int _{0}^{\infty }\frac{(t+\hat{\mu })^{\frac{d}{2}}t^{\frac{d}{2}-\theta -1}dt}{(t+1)^{\frac{d-1}{2}}}\\ & = & \hat{\mu }^{d-\theta }B\left( \frac{d}{2}-\theta ,\theta -\frac{d-1}{2}\right) \! _{2}F_{1}\left( \frac{d}{2}-\theta ,\frac{d+1}{2};\frac{1}{2},1-\hat{\mu }\right) , \end{eqnarray*} where \( _{2}F_{1} \) is Gauss´ hypergeometric function (\cite{PBM} 2.2.6.24 p.303).This result can be generalised to all potentials \( V \) with \( V_{+}\in L^{\theta }_{w}\cap L^{d} \) for \( \frac{d-1}{2}<\theta <\frac{d}{2} \) and \( \Delta _{\theta }(V_{+})=\delta _{\theta }(V_{+})=c(\theta ,d)v \). \subsection{Potentials \protect\( V_{+}\in L^{\theta }_{w}(\mathbb {R}^{d})\cap L^{d}(\mathbb {R}^{d})\protect \) with \protect\( \frac{d+1}{2}<\theta <\frac{d+2}{2}\protect \)} A similar calculation can be carried out for the average \( S_{p}(V) \) if the potential (\ref{vy}) satisfies \( \frac{d+1}{2}<\theta <\frac{d}{2}+1 \) and is therefore ``strictly'' between \( L^{\frac{d+1}{2}}\cap L^{d} \) and \( L^{\frac{d}{2}+1}\cap L^{d} \). First, from (\ref{prelest}) one concludes in general that \begin{eqnarray} \Sigma _{p} & = & \int _{0}^{\infty }\int _{\mathbb {R}^{d}}\Lambda _{p}(y;V-sp)dyds\label{Thl0} \\ & \asymp & \int _{\mathbb {R}^{d}}\Theta _{p}(y;V)dy,\label{Thl} \end{eqnarray} for sufficient large \( p \), where \[ \Theta _{p}(y;V)=(V_{p}(y))_{+}^{d+1}+\tau ^{d+1}\chi _{\Omega _{2}\cup \Omega _{3}}(y)+\tau ^{-1}(V_{p}(y))_{+}^{\frac{d}{2}+1}\chi _{\Omega _{1}}(y)\] with \( \chi \) being the characteristic functions of (unions of) the respective sets \( \Omega _{j} \) defined in (\ref{prelest}). For the potential \( V(y)=\min \{1,v|y|^{-\frac{d}{\theta }}\} \) at hand this gives the preliminary estimate \[ \Sigma _{p}\asymp p^{\theta -1}M^{d+1-2\theta }v^{\theta }\quad \mbox {as}\quad p\to \infty .\] Moreover, the integration in (\ref{Thl}) and therefore in (\ref{Thl0}) can be reduced to \( |y|>p^{1+c} \), \( 0\frac{d+1}{2} \) we finally claim \begin{equation} \label{tmi1} \Sigma _{p}=(1+o(1))\frac{2^{\theta -\frac{3}{2}}\theta \omega ^{2}_{d}}{(4\pi )^{d}}v^{\theta }M^{d+1-2\theta }p^{\theta -1}K(d,\theta ,\hat{\mu }), \end{equation} where \( K(d,\theta ,\hat{\mu }) \) denotes the finite positive constant \[ K(d,\theta ,\hat{\mu })=\int _{0}^{\infty }dww^{-\theta -1}\int _{0}^{w}\frac{u^{\frac{d}{2}}(u+\hat{\mu })^{\frac{d}{2}}}{(u+1)^{\frac{d+1}{2}}}du.\] In fact, this asymptotics holds true for all \( V_{+}\in L^{\theta }_{w}\cap L^{d} \), \( \frac{d+1}{2}<\theta <\frac{d}{2}+1 \), with \( \Delta _{\theta }(V_{+})=\delta _{\theta }(V_{+})=c(\theta ,d)v \). \section{Appendix II: An Estimate \protect\( N_{p}(V)\leq c(V)p^{2}\protect \) in the Dimension \protect\( d=3\protect \).} \subsection{Statement of the result.} In this appendix we show, that for certain short-range potentials with some repulsive tail at infinity the counting function \( N_{p}(V) \) in the dimension \( d=3 \) is bounded by a multiple of \( p^{2} \). This complements the estimate (\ref{esnp1}). As above we concentrate on the case of positive masses \( m_{\pm }>0 \). \begin{thm} \label{tm: A1}Assume that \( d=3 \), \( m_{\pm }>0 \) and that the bounded potential \( V \) satisfies the condition \begin{equation} \label{A.1.1} V(x)\leq -a(1+|x|-b)^{-\gamma },\quad x\in \mathbb R^{3},\: |x|\geq b, \end{equation} for appropriate positive finite constants \( a \), \( b \) and \( \gamma \). Then \begin{equation} \label{A.1.2} N_{p}(V)\leq C(b+1)^{3}p^{2},\qquad p\geq M, \end{equation} where \( C=C(a,\gamma ,\left\Vert V\right\Vert _{L^{\infty }}) \) does not depend on \( p \) and \( b \). \end{thm} \subsection{A localization estimate in spatial coordinates. } Consider the operator \[ T=\sqrt{-\Delta +1}\quad \mbox {on}\quad L^{2}(\mathbb R^{3}).\] Let \( (\cdot ,\cdot ) \) and \( \left\Vert \cdot \right\Vert \) be the scalar product and the norm in \( L^{2}(\mathbb R^{3}) \). For positive \( b \) and \( \gamma \) set \( \varsigma _{\gamma ,b}(x)=(1+|x|-b)^{-\gamma /2} \), \( x\in \mathbb R^{3} \). The proof of Theorem \ref{tm: A1} is based on the following improved localization estimate: \begin{lem} \label{lm A2}For any given positive number \( b \) one can find spherically symmetric functions \( \chi _{1},\chi _{2}\in C^{2}(\mathbb R^{3}) \), which are monotone w.r.t. the radial variable and satisfy \begin{equation} \label{A.2} \chi _{1}(x)=1\quad \mbox {if}\quad |x|\leq b,\quad \chi _{1}(x)=0\quad \mbox {if}\quad |x|\geq b+1,\quad \chi _{1}^{2}+\chi _{2}^{2}=1, \end{equation} such that for any \( \epsilon >0 \) and \( \gamma >0 \) the estimate \begin{equation} \label{A.3} \left| (Tu,u)-\sum _{j=1}^{2}(Tu\chi _{j},u\chi _{j})\right| \leq c\left\Vert u\chi _{1}\right\Vert ^{2}+\epsilon \left\Vert u\chi _{2}\varsigma _{\gamma ,b}\right\Vert ^{2} \end{equation} holds true for all \( u\in C_{0}^{\infty }(\mathbb R^{3}) \) with some appropriate finite constant \( c=c(\gamma ,\epsilon ) \). \end{lem} \begin{proof} For given \( b>0 \) we can obviously chose spherically symmetric cut-off functions \( \chi _{1},\chi _{2}\in C^{2}(\mathbb R^{3}) \), which are monotone in the radial variable and satisfy (\ref{A.2}) as well as \begin{equation} \label{A.4} \chi _{1}(x)\chi _{2}(x)>0\quad \mbox {for}\quad b<|x|0 \). We shall now estimate the quadratic form on the r.h.s. of (\ref{A.5}). Because of symmetry it suffices to estimate the respective integrals over the region \( |x|\leq |y| \) only. Let \( \delta \in (0,1/2) \) be a positive number, which will be specified later. Put \[ b_{\delta }=b+1-\delta \] and define \begin{eqnarray*} O_{1} & = & \{(x,y)|\: |x|\leq |y|\leq b_{\delta }\},\\ O_{2} & = & \{(x,y)|\: |x|\leq b_{2\delta },\: |y|\geq b_{\delta }\},\\ O_{3} & = & \{(x,y)|\: |x|\leq |y|,\: |x|\geq b_{2\delta },\: (x,y)\not \in O_{1}\cup O_{2}\}. \end{eqnarray*} Then \begin{equation} \label{A.9} (Lu,u)_{L^{2}(\mathbb R^{3})}=2\mbox {Re}(I_{1}+I_{2}+I_{3}), \end{equation} where \[ I_{k}=\iint _{O_{k}}L(x,y)u(y)\bar{u}(x)dxdy,\quad k=1,2,3.\] To estimate \( I_{1} \) we notice that \begin{equation} \label{A.10} |\chi _{j}(x)-\chi _{j}(y)|\leq \min \left\{ 1,\co |x-y|\right\} \quad \mbox {for\: all}\quad x,y\in \mathbb R^{3}. \end{equation} From (\ref{A.6}) and (\ref{A.10}) we conclude \[ |L(x,y)|\leq \co |x-y|^{-2}\min \{1,|x-y|^{-2}\}\quad \mbox {for\: all}\quad (x,y)\in O_{1}.\] Hence, it holds \begin{eqnarray*} |I|_{1} & \leq & 2^{-1}\iint _{(x,y)\in O_{1}}(|u(x)|^{2}+|u(y)|^{2})|L(x,y)|dxdy\\ & \leq & \co \int _{|x|\leq b_{\delta }}|u(x)|^{2}dx\int _{\mathbb R^{3}}|x-y|^{-2}\min \{1,|x-y|^{-2}\}dy. \end{eqnarray*} This gives \begin{equation} \label{A.12} |I_{1}|\leq \co \int _{|x|\leq b_{\delta }}|u(x)|^{2}dx\leq \co (\delta )\left\Vert u\chi _{1}\right\Vert ^{2}_{L^{2}(\mathbb R^{3})}. \end{equation} In the last step we used that \( \chi _{1}(x)\geq \co (\delta )>0 \) for all \( |x|\leq b_{\delta } \), what on its turn follows from (\ref{A.4}) and the radial monotonicity of \( \chi _{1} \). We study now the integral \( I_{2} \) and observe that \begin{equation} \label{A.13} |L(x,y)|\leq \co \delta ^{-2}e^{-\kappa |x-y|}\quad \mbox {for}\quad (x,y)\in O_{2}. \end{equation} In view of \[ |u(y)u(x)|\leq 4^{-1}\epsilon _{1}^{-1}|u(x)|^{2}+\epsilon _{1}|u(y)|^{2},\quad \epsilon _{1}>0,\] we find from (\ref{A.13}) that for any given \( \gamma >0 \) and \( \epsilon _{1}>0 \) the bound \begin{eqnarray} |I_{2}| & \leq & \co \delta ^{-2}\epsilon _{1}^{-1}\int _{|x|\leq b_{2\delta }}dx|u(x)|^{2}\int _{|y|\geq b_{\delta }}e^{-\kappa |x-y|}dy\notag \\ & & +\co \frac{\epsilon _{1}}{\delta ^{2}}\int _{|y|\geq b_{\delta }}\frac{dy|u(y)|^{2}}{e^{\frac{\kappa }{2}(|y|-b_{2\delta })}}\int _{|x-y|\geq \delta }e^{-\frac{\kappa }{2}|x-y|}dx\label{A.14.0} \end{eqnarray} holds true. By (\ref{A.4}) and by the radial monotonicity of \( \chi _{1} \) and \( \chi _{2} \) we have \begin{eqnarray*} \chi _{1}(x)\geq \co (\delta )>0 & \quad \mbox {for}\quad & |x|\leq b_{2\delta },\\ \chi _{2}(y)\geq \co (\delta )>0 & \quad \mbox {for}\quad & |y|\geq b_{\delta }>b. \end{eqnarray*} Moreover, it holds \[ e^{-\frac{\kappa }{4}(|y|-b_{2\delta })}\leq \co (\gamma ,\delta )\varsigma _{\gamma ,b}(y),\quad |y|\geq b_{\delta }.\] Hence, the inequality (\ref{A.14.0}) implies \begin{equation} \label{A.14} |I_{2}|\leq \co (\delta ,\epsilon _{1})\left\Vert u\chi _{1}\right\Vert ^{2}_{L^{2}(\mathbb R^{3})}+\co (\gamma ,\delta )\epsilon _{1}\left\Vert u\chi _{2}\varsigma _{\gamma ,b}\right\Vert ^{2}_{L^{2}(\mathbb R^{3})}. \end{equation} Estimating \( I_{3} \) we recall that \begin{equation} \label{A.15.1} \chi _{1}(x)\equiv 0\quad \mbox {and}\quad \chi _{2}(x)\equiv 1\quad \mbox {for\: all}\quad |x|\geq b+1. \end{equation} Since \( \chi _{1},\chi _{2}\in C^{2}(\mathbb R^{2}) \), for any given \( \epsilon _{2}>0 \) we can find an appropriate \( \delta =\delta (\epsilon _{2})\in (0,1/2) \) such that \begin{equation} \label{A.15.0} |\nabla \chi _{1}(x)|^{2}+|\nabla \chi _{2}(x)|^{2}\leq \epsilon _{2},\quad b_{2\delta }\leq |x|\leq b+1. \end{equation} With this value of \( \delta \) the relations (\ref{A.15.1}) and (\ref{A.15.0}) imply \begin{equation} \label{A.15.2} \sum _{j=1}^{2}(\chi _{j}(x)-\chi _{j}(y))^{2}\leq \epsilon _{2}\min \left\{ 4\delta ^{2},|x-y|^{2}\right\} ,\quad b_{2\delta }\leq |x|\leq |y|. \end{equation} Moreover, from (\ref{A.13}), (\ref{A.15.1}) and (\ref{A.15.2}) we conclude that \[ |L(x,y)|\leq \epsilon _{2}\co |x-y|^{-2}\min \left\{ 4\delta ^{2}|x-y|^{-2},1\right\} \min \left\{ e^{-\kappa (|y|-b-1)},1\right\} \] for \( b_{2\delta }\leq |x|\leq |y| \) and \( L(x,y)=0 \) for \( b+1\leq |x|\leq |y| \) . Therefore it holds \begin{eqnarray*} |I_{3}| & \leq & 2^{-1}\iint _{(x,y)\in O_{3}}(|u(x)|^{2}+|u(y)|^{2})|L(x,y)|dxdy\\ & \leq & \epsilon _{2}\co \int _{|x|\leq b+1}|u(x)|^{2}dx\int _{\mathbb R^{3}}|x-y|^{-2}\min \{4\delta ^{2}|x-y|^{-2},1\}dy\\ & & +\epsilon _{2}\co \int _{|y|\geq b_{2\delta }}\frac{|u(y)|^{2}dy}{e^{\kappa (|y|-b-1)}}\int _{\mathbb R^{3}}\frac{\min \{4\delta ^{2}|x-y|^{-2},1\}}{|x-y|^{2}}dx. \end{eqnarray*} Since \( e^{-\frac{\kappa }{2}(|y|-b-1)}\leq \co (\gamma ,\delta )\varsigma _{\gamma ,b}(y) \) for \( |y|\geq b_{2\delta } \) and \( \delta \in (0,1/2) \), we conclude that \begin{equation} \label{A.16} |I_{3}|\leq \epsilon _{2}\co (\gamma ,\delta )(\left\Vert u\chi _{1}\right\Vert _{L^{2}(\mathbb R^{3})}^{2}+\left\Vert u\chi _{2}\varsigma _{\gamma ,b}\right\Vert ^{2}_{L^{2}(\mathbb R^{3})}). \end{equation} We proceed now as follows. For given \( \epsilon >0 \) chose \( \epsilon _{2}>0 \) such that the total constant in front of the bracket in (\ref{A.16}) for given \( b \) and \( \gamma \) does not exceed \( \epsilon /4 \). Fix the corresponding \( \delta (\epsilon _{2})>0 \) for (\ref{A.15.0}) and subsequently (\ref{A.16}) to be satisfied. Finally, fix \( \epsilon _{1}>0 \) such that the total constant in front of the term \( \left\Vert u\chi _{2}\varsigma _{\gamma ,b}\right\Vert _{L^{2}(\mathbb R^{3})}^{2} \) in (\ref{A.14}) for given \( b \), \( \gamma \) and \( \delta (\epsilon _{2}) \) does not exceed \( \epsilon /4 \). Then (\ref{A.5}) together with (\ref{A.10}), as well as (\ref{A.12}), (\ref{A.14}) and (\ref{A.16}) yield (\ref{A.3}). \end{proof} \begin{rem} Let \( t_{rel}=t_{rel}(P) \) be the regularized kinetic part of the operator (\ref{hrel}) on \( L^{2}(\mathbb R^{3}) \), that is \begin{equation} \label{trelop} t_{rel}=\sqrt{|\mu _{+}P-i\nabla |^{2}+m_{+}^{2}}+\sqrt{|\mu _{-}P+i\nabla |^{2}+m_{-}^{2}}-\sqrt{p^{2}+M^{2},} \end{equation} where \( M>0 \), \( \mu _{\pm }=m_{\pm }M^{-1}>0 \), and \( P\in \mathbb R^{3} \), \( p=|P| \). As an immediate consequence of (\ref{A.2}) in Lemma \ref{lm A2} we find that for arbitrary positive \( \epsilon \) and \( \gamma \) it holds \begin{equation} \label{trel} \left| (t_{rel}u,u)-\sum _{j=1}^{2}(t_{rel}u\chi _{j},u\chi _{j})\right| \leq c(\gamma ,\epsilon ,\mu _{j},M)\left\Vert u\chi _{1}\right\Vert ^{2}+\epsilon \left\Vert u\chi _{2}\varsigma _{\gamma ,b}\right\Vert ^{2}. \end{equation} The constant \( c(\gamma ,\epsilon ,\mu _{j},M) \) can be chosen to be independent on \( P \) and \( b \). \end{rem} \subsection{A local estimate in momentum space.} Let \[ t_{rel}(\xi ,P)=\sqrt{|\mu _{+}P-\xi |^{2}+m_{+}^{2}}+\sqrt{|\mu _{-}P+\xi |^{2}+m_{+}^{2}}-\sqrt{p^{2}+M^{2}}\] be the symbol of the operator (\ref{trelop}) where \( M>0 \), \( \mu _{\pm }=m_{\pm }M^{-1}>0 \) and \( P,\xi \in \mathbb R^{3} \). Put \( \xi =(\eta ,\zeta ) \) with \( \eta \in \mathbb R \) and \( \zeta \in \mathbb R^{2} \). We recall that \( P=(p,0,0) \) and \( \mu _{+}+\mu _{-}=1 \). \begin{lem} \label{lm3A}Assume that \( p\geq \nu \geq M \) and that \( \xi =(\eta ,\zeta ) \) satisfies \begin{equation} \label{etazeta} \xi \in W(\nu ,p)=\{\xi |(|\eta |\geq 3p)\}\cup \{\xi |(|\zeta |^{2}\geq \nu p)\}. \end{equation} Then \begin{equation} \label{trellm} t_{rel}(\xi ,P)\geq 2^{-1}3^{-1/2}\nu . \end{equation} \end{lem} \begin{proof} Assume first \( |\eta |\geq 3p\geq 3\nu \geq 3M \). Then \begin{equation} \label{A.18} t_{rel}(\xi ,P)\geq \sqrt{4p^{2}+M^{2}}-\sqrt{p^{2}+M^{2}}\geq 3(\sqrt{5}+\sqrt{2})^{-1}\nu . \end{equation} If instead \( |\zeta |^{2}\geq \nu p \) from \( \nu \geq M \) it follows that \begin{eqnarray*} t_{rel}(\xi ,P) & \geq & \sqrt{p^{2}+|\zeta |^{2}+M^{2}}-\sqrt{p^{2}+M^{2}}\\ & \geq & 2^{-1}|\zeta |^{2}(p^{2}+|\zeta |^{2}+M^{2})^{-1/2}\\ & \geq & 2^{-1}\nu (1+\nu p^{-1}+\nu ^{2}p^{-2})^{-1/2}. \end{eqnarray*} Since \( p\geq \nu \) we conclude \( t_{rel}(\xi ,P)\geq 2^{-1}3^{-1/2}\nu \). Together with (\ref{A.18}) this completes the proof. \end{proof} \subsection{The proof of Theorem \ref{tm: A1}.} Let \[ q_{rel}(P)=t_{rel}(P)-V(y),\quad P=(p,0,0),\] be the operator (\ref{qrell}) for \( d=3 \). Obviously the total multiplicity of the negative eigenvalues of this operator coincides with \( N_{p}(V) \). To verify (\ref{A.1.2}) it suffices to construct a subspace \( G \) in \( L^{2}(\mathbb R^{3}) \) of finite dimension \( \dim G\leq Cb^{3}p^{2} \) such that \begin{equation} \label{qrelpos} (q_{rel}(P)u,u)_{L^{2}(\mathbb R^{3})}\geq 0\quad \mbox {for\: all}\quad u\in G_{0}^{\bot }, \end{equation} where \( G_{0}^{\bot } \) is a \( q_{rel}(P) \)-form dense subset of \( G^{\bot }=L^{2}(\mathbb R^{3})\ominus G \). For given \( b \) construct the cut-off functions \( \chi _{1} \), \( \chi _{2} \) from Lemma \ref{lm A2}. Set \( n=(n_{1},n_{2},n_{3}) \) for \( n_{j}\in \mathbb N_{+} \) and \( x=(x_{1},x_{2},x_{3}) \) with \( x_{j}\in \mathbb R \), \( j=1,2,3 \). Let \( b^{\prime }=b+1 \). We define \[ u_{n}(x)=\left\{ \begin{array}{ccc} b^{\prime -\frac{3}{2}}\prod _{j=1}^{3}\sin \pi n_{j}\left( \frac{1}{2}+\frac{x_{j}}{b^{\prime }}\right) & \: \mbox {for}\: & |x_{j}|\leq b^{\prime },\: j=1,2,3,\\ 0 & & \mbox {otherwise}. \end{array}\right. \] Let \( \tilde{G}=\tilde{G}(\tau _{1},\tau _{\bot }) \) be the linear span of all \( u_{n} \) where \[ n_{1}\leq \tau _{1}b^{\prime }p,\qquad n_{2,3}\leq \tau _{\bot }b^{\prime }p^{1/2},\] and the positive real numbers \( \tau _{1} \), \( \tau _{\bot } \) will be specified below. We put \[ G=G(\tau _{1},\tau _{\bot })=\{u|u=\tilde{u}\chi _{1},\: \tilde{u}\in \tilde{G}\}.\] Obviously we have \[ \dim G=\dim \tilde{G}\leq \tau _{1}\tau ^{2}_{\bot }b^{\prime 3}p^{2}.\] To verify (\ref{qrelpos}) we first notice, that from the boundedness of \( V \), (\ref{A.1.1}) and (\ref{trel}) it follows that \begin{equation} \label{qpos} (q_{rel}(P)u,u)_{L^{2}(\mathbb R^{3})}\geq (t_{rel}(P)u\chi _{1},u\chi _{1})_{L^{2}(\mathbb R^{3})}-\tilde{c}\left\Vert u\chi _{1}\right\Vert _{L^{2}(\mathbb R^{3})}^{2} \end{equation} for all \( u\in C_{0}^{\infty }(\mathbb R^{3}) \). For fixed \( \chi _{1} \) the constant \( \tilde{c}=\tilde{c}(a,\gamma ,\left\Vert V\right\Vert _{L^{\infty }}) \) does not depend on \( p \), \( b^{\prime } \) or \( u \). Let \( W(\nu ,p) \) be the set defined in (\ref{etazeta}) of Lemma \ref{lm3A} for the choice \( \nu =2^{3}3^{1/2}\tilde{c} \). Below we shall show, that for appropriate constants \( \tau _{1}=\tau _{1}(\tilde{c}) \) and \( \tau _{\bot }=\tau _{\bot }(\tilde{c}) \), which do not depend on \( p \), the bound \begin{equation} \label{toprove} \left\Vert \widehat{{u\chi _{1}}}\right\Vert _{W(\nu ,p)}\geq 2^{-1}\left\Vert \widehat{{u\chi _{1}}}\right\Vert _{L^{2}(\mathbb R^{3})},\quad u\bot G(\tau _{1},\tau _{\bot }), \end{equation} holds true. From (\ref{toprove}) and (\ref{trellm}) we conclude that \begin{eqnarray*} (t_{rel}(P)u\chi _{1},u\chi _{1})_{L^{2}(\mathbb R^{3})} & \geq & (t_{rel}(\xi ,P)\widehat{{u\chi _{1}}}(\xi ),\widehat{{u\chi _{1}}}(\xi ))_{L^{2}(W(\nu ,p))}\\ & \geq & 4\tilde{c}\left\Vert \widehat{{u\chi _{1}}}\right\Vert ^{2}_{L^{2}(W(\nu ,p))}\\ & \geq & \tilde{c}\left\Vert u\chi _{1}\right\Vert _{L^{2}(\mathbb R^{3})}^{2}, \end{eqnarray*} where \( u\bot G(\tau _{1},\tau _{\bot }) \) and \( u\in C_{0}^{\infty }(\mathbb R^{3}) \). Together with (\ref{qrelpos}) and (\ref{qpos}) the later bound settles the proof. In the remaining part of this section we establish (\ref{toprove}). Consider some function \( u\bot G \). Then \( u\chi _{1}\bot \tilde{G} \) and consequently \( u\chi _{1}=\sum _{j=1}^{3}\sigma _{j} \), where \( \sigma _{j}=\sum _{n\in \Upsilon _{j}}c_{n}u_{n} \) and \begin{eqnarray*} \Upsilon _{1} & = & \{n|n_{1}\geq \tau _{1}b^{\prime }p\},\\ \Upsilon _{2} & = & \{n|(n_{1}<\tau _{1}b^{\prime }p)\}\cap \{n|(n_{2}\geq \tau _{\bot }b^{\prime }p^{1/2})\},\\ \Upsilon _{3} & = & \{n|(n_{1}<\tau _{1}b^{\prime }p)\}\cap \{n|(n_{2}<\tau _{\bot }b^{\prime }p^{1/2})\}\cap \{n|(n_{3}\geq \tau _{\bot }b^{\prime }p^{1/2})\}. \end{eqnarray*} Put \( \tilde{W}=\mathbb R^{3}\backslash W(\nu ,p) \). Since \[ \left\Vert \widehat{{u\chi _{1}}}\right\Vert _{L^{2}(\tilde{W})}\leq \sum _{j=1}^{3}\left\Vert \hat{\sigma }_{j}\right\Vert _{L^{2}(\tilde{W})},\] for (\ref{toprove}) it suffices to show that \begin{equation} \label{7.4} \left\Vert \hat{\sigma }_{j}\right\Vert _{L^{2}(\tilde{W})}\leq 6^{-1}\left\Vert \widehat{{u\chi _{1}}}\right\Vert _{L^{2}(\mathbb R^{3})},\quad j=1,2,3. \end{equation} We shall verify (\ref{7.4}) for \( j=1 \). The proof for the cases \( j=2,3 \) is similar. Obviously we have \( \hat{\sigma }_{1}=\sum _{n\in \Upsilon _{1}}c_{n}\hat{u}_{n} \) where \[ \hat{u}_{1}(\xi )=b^{\prime -\frac{3}{2}}\prod _{j=1}^{3}e^{i\pi (n_{j}+\frac{1}{2})}\frac{\pi n_{j}}{b^{\prime }}\frac{\sin (\xi _{j}b^{\prime }-\frac{\pi n_{j}}{2})}{\xi _{j}^{2}-\frac{\pi ^{2}n^{2}_{j}}{4b^{\prime 2}}}.\] Since \[ \left\Vert \hat{\sigma }_{j}\right\Vert ^{2}_{L^{2}(\tilde{W})}\leq \int _{|\xi _{1}|<3p}|\hat{\sigma }_{1}(\xi )|^{2}d\xi ,\] after integration in \( \zeta =(\xi _{2},\xi _{3}) \) and using the notation \( \eta =\xi _{1} \), we find that \begin{equation} \label{si1} \left\Vert \hat{\sigma }_{j}\right\Vert ^{2}_{L^{2}(\tilde{W})}\leq \frac{\pi ^{2}}{b^{\prime 3}}\int _{|\eta |<3p}\sum _{\begin{array}{c} n_{2},n_{3}\in \mathbb N_{+}\\ n_{1},n_{1}^{\prime }\geq \tau _{1}b^{\prime }p \end{array}}\frac{|c_{(n_{1},n_{2},n_{3})}c_{(n^{\prime }_{1},n_{2},n_{3})}|n_{1}n^{\prime }_{1}d\eta }{\left| \eta ^{2}-\frac{\pi ^{2}n_{1}^{2}}{4b^{\prime 2}}\right| \left| \eta ^{2}-\frac{\pi ^{2}n_{1}^{\prime 2}}{4b^{\prime 2}}\right| }. \end{equation} Let us assume that \( \tau _{1}^{2}\geq 72\pi ^{-2} \). Then we have \( \frac{\pi ^{2}n_{1}^{2}}{8b^{\prime 2}}\geq 9p^{2}\geq \eta ^{2} \) and \( \frac{\pi ^{2}n_{1}^{\prime 2}}{8b^{\prime 2}}\geq 9p^{2}\geq \eta ^{2} \) in the denominator in the previous sum and thus \begin{equation} \label{si2} \int _{|\eta |\leq 3p}\frac{d\eta }{\left| \eta ^{2}-\frac{\pi ^{2}n_{1}^{2}}{4b^{\prime 2}}\right| \left| \eta ^{2}-\frac{\pi ^{2}n_{1}^{\prime 2}}{4b^{\prime 2}}\right| }\leq \frac{8b^{\prime 2}}{\pi ^{2}n_{1}^{2}}\cdot \frac{8b^{\prime 2}}{\pi ^{2}n^{\prime 2}_{1}}\cdot 6p. \end{equation} Applying Schwarz inequality in the summations over \( n_{1} \) and \( n_{1}^{\prime } \) together with (\ref{si2}) to (\ref{si1}) we obtain \[ \left\Vert \hat{\sigma }_{j}\right\Vert ^{2}_{L^{2}(\tilde{W})}\leq \frac{384b^{\prime }p}{\pi ^{2}}\left( \sum _{n_{1}\geq \tau _{1}b^{\prime }p}n_{1}^{-2}\right) \sum _{n\in \mathbb N_{+}^{3}}|c_{n}|^{2}.\] Since \( \sum _{n_{1}\geq \tau _{1}bp}n_{1}^{-2}\leq 2(\tau _{1}b^{\prime }p)^{-1} \) and \( \sum _{n\in \mathbb N_{+}^{3}}|c_{n}|^{2}=\left\Vert \widehat{{u\chi _{1}}}\right\Vert _{L^{2}(\mathbb R^{3})}^{2} \) a choice of \( \tau _{1}=2\cdot 36\cdot 384\cdot \pi ^{-2}\geq 72\cdot \pi ^{-2} \) will yield (\ref{7.4}) for \( j=1 \). \begin{thebibliography}{ELSS} \bibitem[AL]{AL}M. Aizenman, E. Lieb: ``On semi-classical bounds for eigenvalues of Schrödinger operators.'' \textit{Phys. Lett.} \textbf{66A} (1978) 427-429. \bibitem[Be]{Be}F.A. Berezin: ``Wick and anti-Wick symbols of operators.'' (Russian) \textit{Mat. Sb. (N.S.)} \textbf{86} (128) (1971) 578-610. \bibitem[B]{B}M.S. Birman: ``The spectrum of singular boundary problems.'' (Russian) \textit{Mat. Sb. (N.S.)} \textbf{55} (\textbf{97}) (1961), 125-174. (English) \textit{Amer. Math. Soc. Transl.} \textbf{53} (1966), 23-80. \bibitem[BKS]{BKS}M.Sh. Birman, G.E. Karadzhov, M.Z. Solomyak: ``Boundedness conditions and spectrum estimates for operators \( b(x)a(D) \) and their analogues.'' \textit{Adv. in Sov. Math.} \textbf{7} (1991) 85-106. \bibitem[C]{C}M. Cwikel: ``Weak type estimates for singular values and the number of bound states of Schrödinger operators.'' \textit{Trans. AMS} \textbf{224} (1977) 93-100. \bibitem[ELSS]{ELSS}W.D. Evans, R.T. Lewis, H. Siedentop, J.Ph. Solovej: ``Counting eigenvalues using coherent states with an application to Dirac and Schrödinger operators in the semi-classical limit.'' \textit{Ark. Mat.} \textbf{34} (2) (1996) 265-283. \bibitem[H]{H}I.W. Herbst: ``Spectral theory of the operator \( H=(p^{2}+m^{2})^{1/2}-Ze^{2}/r \).'' \textit{Comm. Math. Phys.} \textbf{53} (3) (1977) 285-294. \bibitem[LSV]{LSV}R.T. Lewis, H. Siedentop, S. Vugalter: ``The essential spectrum of relativistic multi-particle operators.'' \textit{Ann. Inst. H. Poincare Phys. Theor.} \textbf{67} (1) (1997) 1-28. \bibitem[LS]{LS}E. H. Lieb, J.Ph. Solovej: Quantum coherent operators: a generalization of coherent states.'' \textit{Lett. Math. Phys}. \textbf{22} (2) (1991) 145-154. \bibitem[LY]{LY}E. H. Lieb, H.-T. Yau: ``The Stability and Instability of Relativistic Matter.'' \textit{Comm. Math. Phys.} \textbf{118} (1988) 177-213. \bibitem[PBM]{PBM}A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev: ``Integrals and series. Elementary functions.'' (Russian) Nauka, Moscow, 1981 799pp. \bibitem[S]{S}Y.Schwinger:`` On the bound states for a given potential.'' \textit{Proc. Nat. Acad. Sci. USA} \textbf{47} (1961) 122-129. \bibitem[W1]{W1}T. Weidl: ``Cwikel type estimates in nonpower ideals'', \textit{Math Nachrichten} \textbf{176} (1995) 315-334. \bibitem[W2]{W2}T. Weidl: ``Another look at Cwikel's inequality'', in ``Differential Operators and Spectral Theory. M.Sh. Birman's 70th Anniversary Collection'', \textit{AMS Translations Series} 2 \textbf{189} (1999) 247-254. \end{thebibliography} \email{\textsf{\tiny email: wugalter@rz.mathematik.uni-muenchen.de, Timo.Weidl@mathematik.uni-stuttgart.de}\tiny } \end{document} ---------------0111190933330--