Content-Type: multipart/mixed; boundary="-------------0208300813491" This is a multi-part message in MIME format. ---------------0208300813491 Content-Type: text/plain; name="02-357.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-357.keywords" Scott correction, semiclassical analysis, coherent states ---------------0208300813491 Content-Type: application/x-tex; name="aug30-2002.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="aug30-2002.tex" \documentclass[12pt]{article} \usepackage{a4,amsthm,amsfonts,latexsym,amssymb, %showkeys } \font\notefont=cmsl8 %HEADING: \pagestyle{myheadings} \markright{\notefont JPS \& WLS/30-Aug-2002 \hfill} %% \def\a{\alpha} \def\b{\beta} \def\c{\gamma} \def\d{\delta} \def\e{\varepsilon} \def\s{\sigma} \def\w{\omega} \def\D{\Delta} \def\W{\Omega} \def\l{\lambda} \def\p{\partial} \def\x{{\hat{x}}} \def\fv{\frak v} \def\fF{\frak F} \def\r{\rho} \def\cG{{\cal G}} \def\R{\mathbb R} \newtheorem{thm}{Theorem} \newtheorem{lemma}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \newcommand{\bt}{\begin{thm}{\hspace{-.55em}\em{\bf {: }}}} \newcommand{\et}{\end{thm}} \newcommand{\bc}{\begin{cor}{\hspace{-.55em}\em{\bf {: }}}} \newcommand{\ec}{\end{cor}} \newcommand{\bl}{\begin{lemma}{\hspace{-.55em}\em{\bf {: }}}} \newcommand{\el}{\end{lemma}} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\beax}{\begin{eqnarray*}} \newcommand{\eeax}{\end{eqnarray*}} \newcommand{\op}[1]{\mbox{\sf #1}} \newcommand{\komm}[2]{\left[#1,#2\right]} \newcommand{\Tr}{\mbox{\rm Tr}} \newcommand{\mfr}[2]{{\textstyle\frac{#1}{#2}}} \newcommand{\V}{V^{\rm TF}} \textheight=22cm \topmargin=-1cm \textwidth=15.3cm \oddsidemargin -0.1cm \evensidemargin -0.1cm \sloppy \frenchspacing \flushbottom \baselineskip=24pt \begin{document} \title{A NEW COHERENT STATES APPROACH TO SEMICLASSICS WHICH GIVES SCOTT'S CORRECTION \thanks{Work partially supported by an EU TMR grant, by the Danish research foundation center MaPhySto, and by a grant from the Danish research council. \newline \hfill \copyright 2002 \ by the authors. This article may be reproduced in its entirety for non-commercial purposes. } } \author{ \begin{tabular}{ccc} Jan Philip Solovej & &Wolfgang L Spitzer\\ \normalsize Department of Mathematics& &\normalsize Department of Mathematics\\ \normalsize University of Copenhagen& &\normalsize University of California\\ \normalsize Universitetsparken 5& &\normalsize Davis, One Shields Avenue\\ \normalsize DK-2100 Copenhagen, Denmark& &\normalsize CA 95616-8633, USA\\ \normalsize {\it e-mail\/}: solovej@math.ku.dk& &\normalsize {\it e-mail\/}: spitzer@math.ucdavis.edu \end{tabular} } \date{Aug. 30, 2002} \maketitle \tableofcontents \section{Introduction} %\heading{Introduction} There are various highly developed methods for establishing semiclassical approximations. Probably the most refined method is based on pseudo-differential and Fourier integral operator calculi. This extremely technical approach is well suited for getting good or even sharp error estimates. Here, sharp refers to the optimal exponent of the semiclassical parameter in the error term. These sharp estimates however often require strong regularity assumptions on the operators being investigated. A different and very simple method based on coherent states gives the leading order semiclassical asymptotics under optimal regularity assumptions. The method of coherent states was used by Thirring~\cite{Thirring} and Lieb~\cite{Lieb1} to give a very short and simple proof of the Thomas-Fermi energy asymptotics of large atoms and molecules. This asymptotics had been first proved by Lieb and Simon in \cite{Lieb-Simon} using a Dirichlet-Neumann bracketing method. Because of the Coulomb singularity of the atomic potential the pseudo-differential techniques are not immediately applicable to the Thomas-Fermi asymptotics. In fact, although the Coulomb singularity does not affect the leading order Thomas-Fermi asymptotics, in the sense that it is purely semi-classical, it does cause the first correction to be of a non-semiclassical nature. The first correction to the Thomas-Fermi asymptotics was predicted by Scott in \cite{Scott} and was later generalized to molecules and formulated as a clear mathematical conjecture in \cite{Lieb1}. The first mathematical proof of the Scott correction for atoms was given by Hughes~\cite{Hughes} (a lower bound) and by Siedentop and Weikard~\cite{Siedentop-Weikard} (both bounds) by WKB type methods. In \cite{Ivrii-Sigal}, Ivrii and Sigal finally managed to apply Fourier integral operator methods to the atomic problem and proved the Scott correction for molecules. In \cite{Fefferman-Seco}, Fefferman and Seco gave a rigorous derivation of the next correction (after the Scott correction) in the asymptotics of the energy of atoms. This next correction had been predicted by Dirac~\cite{Dirac} and Schwinger~\cite{Schwinger}. As we shall explain below (see Page \pageref{page:classicalcoherent}) one cannot expect to be able to derive the Scott correction using the traditional method of coherent states. In this paper we introduce a new semiclassical approach generalizing the method of coherent states and show that this approach can be used to give a fairly simple derivation of the Scott correction for molecules. The standard coherent states method is based on representing operators on $L^2(\R^n)$ as integrals of the form \begin{equation}\label{eq:classicalcoherentrepresentation} \int_{\R^{2n}} a(u,q)\Pi_{u,q}\frac{dudq}{(2\pi h)^{n}}, \end{equation} where $a(u,q)$ is a function (the symbol of the operator) on the classical phase space $\R^{2n}$ and $\Pi_{u,q}$ is a non-negative operator with the properties $$ \Tr\Pi_{u,q}=1, \quad \int_{\R^{2n}} \Pi_{u,q}\frac{dudq}{(2\pi h)^{n}}={\bf 1}. $$ For the classical coherent states $\Pi_{u,q}$ is the one-dimensional projection $\left|u,q\right\rangle\left\langle u,q\right|$ onto the normalized function \begin{equation}\label{eq:classicalcoherentstates} \langle x|u,q\rangle = (\pi h)^{-n/4} e^{-(x-u)^2/2h} e^{iqx/h}. \end{equation} We generalize this by representing operators in the form \begin{equation}\label{form} \int {\cal G}_{u,q}\,{\widehat A}_{u,q}\,{\cal G}_{u,q}\frac{dudq}{(2\pi h)^{n}}. \end{equation} Here ${\cal G}_{u,q}$ is some self-adjoint operator such that its square plays the role of $\Pi_{u,q}$ and ${\widehat A}_{u,q}=B_0(u,q)+B_1(u,q)\cdot {\x}-ih B_2(u,q)\cdot\nabla$ is a differential operator linear in ${\x}$ and $-ih\nabla$. (We have denoted by $\x$ the position operator.) We shall make an explicit choice of $\cG_{u,q}$ in Sect.~\ref{sec:coherent}. In other words, we allow the symbol in the coherent state operator representation to be not just a real function on phase space but to take values in first order differential operators. If we consider, for example, a Schr\"odinger operator of the form $-h^2\Delta+V(\hat{x})$, where a natural choice of the coherent state symbol would be $a(u,q)=q^2+V(u)$, then the new idea is now to choose the linear approximation $$ {\widehat A}_{u,q}=a(u,q)+\partial_u a(u,q)(\x-u)+\partial_q a(u,q)(-ih\nabla-q). $$ The representation (\ref{form}) will then be a better approximation of the Schr\"odinger operator than (\ref{eq:classicalcoherentrepresentation}) (see Theorem~\ref{thm:coherentrepresentation} for details). In order to explain the Scott correction we consider the non-relativistic Schr\"odinger operator for a neutral molecule \[ H(Z,R)=H(Z_1,\ldots,Z_M;R_1,\ldots, R_M) =\sum_{i=1}^Z\left(-\mfr{1}{2}\D_i - V(Z,R,x_i)\right) +\sum_{i0$ and $R=|Z|^{-1/3}(r_1,\ldots,r_M)$, where $|r_i- r_j|>r_0$ for some $r_0>0$. Then, \begin{equation} E(Z,R) = E^{\rm TF} (Z,R) + \mfr{1}{2} \sum_{1\le j\le M} Z_j^2 + {\cal O} (|Z|^{2-1/30}), \end{equation} as $|Z|\to\infty$, where the error term ${\cal O} (|Z|^{2-1/30})$ besides $|Z|$ depends only on $z_1,\ldots,z_M$, and $r_0$. \end{thm} This is established in lemmas \ref{lower bound} and \ref{upper bound}. In fact, one could improve slightly on the error estimate to the expense of limiting the range of $Z$ and $R$, and vice verse. It turns out that $E^{\rm TF} (Z,R) $ is of order $|Z|^{7/3}$ and the next term $\mfr{1}{2} \sum_{1\le j\le M} Z_j^2$ is the Scott correction. Part of our derivation of Theorem~\ref{main theorem} is similar to the multi-scale analysis in \cite{Ivrii-Sigal} and we adopt their notation. Our semiclassical method, however, is very different. It does not rely on the spectral calculus, but uses only the quadratic form representation of operators. Moreover, we treat the Coulomb singularities completely differently from \cite{Ivrii-Sigal}. In treating the singularities and the region near infinity the Lieb-Thirring inequality plays an essential role. Another virtue of our proof is that it gives an explicit trial state for the energy that is correct to an order including the Scott correction. This is, in fact, how we prove that the Scott correction is correct as an asymptotic upper bound. This paper is organized as follows. In Sect.~\ref{sec:tools} we list for the convenience of the reader the analytic tools that we shall use in a crucial way. In Sect.~\ref{sec:tf} we review Thomas-Fermi theory. In Sect.~\ref{sec:coherent} we introduce the new coherent states. In Sect.~\ref{sec:sc} we apply this new tool to prove the semi-classical expansion of the sum of the negative eigenvalues of a non-singular Schr\"odinger operator localized in some bounded region of space. This is the key application of our new method. The proof for the semi-classical expansion for the Thomas-Fermi potential is presented in Sect.~\ref{sec:sctf}. In Sect.~\ref{sec:scott} we finally prove lower and upper bound for the molecular quantum ground state energy. Some calculations concerning the new coherent states and a theorem on constructing a particular partition of unity are put into the appendices. \section{Preliminaries} \subsection{Analytic tools}\label{sec:tools} In this subsection we collect the main analytic tools which we shall use throughout the paper. We do not prove them here but give the standard references. Various constants are typically denoted by the same letter $C$, although their value might, for instance, change from one to the next line. Let $p\ge1$, then a complex-valued function $f$ (and only those will be considered here) is said to be in $L^p(\R^n)$ if the norm $\|f\|_p := \left(\int |f(x)|^p \,dx\right)^{1/p}$ is finite. For any $1\le p\le t\le q\le\infty$ we have the inclusion $L^p\cap L^q\subset L^t$, since by H\"older's inequality $\| f\|_t \le \|f\|_p^\lambda \|f\|_q^{1-\lambda}$ with $\lambda p^{-1}+(1-\lambda) q^{-1} =t^{-1}$. We call $\c$ a density matrix on $L^2(\R^n)$ if it is a trace class operator on $L^2(\R^n)$ satisfying the operator inequality ${\bf 0}\le\c\le {\bf 1}$. The density of a density matrix $\gamma$ is the $L^1$ function $\rho_\gamma$ such that $\Tr(\gamma\theta)=\int\rho_\gamma(x)\theta(x)dx$ for all $\theta\in C_0^\infty(\R^n)$ considered as multiplication operators. We also need an extension to many-particle states. Let $\psi\in \bigotimes^N L^2(\R^3\times\{-1,1\})$ be an $N$-body wave-function. Its one-particle density $\rho_\psi$ is defined by $$ \rho_\psi(x) = \sum_{i=1}^N \sum_{s_1=\pm1}\cdots\sum_{S_N=\pm1}\int |\psi(x_1,s_1\ldots,x_N,s_N)|^2\,\delta(x_i-x)\,dx_1\cdots x_N. $$ The next inequality we recall is crucial to most of our estimates. \begin{thm}[Lieb-Thirring inequality] \label{Lieb-Thirring} {\bf One-body case:} Let $\c$ be a density operator on $L^2(\R^n)$, then we have the Lieb-Thirring inequality \begin{equation}\label{LTdensity} \Tr\left[-\mfr{1}{2}\Delta \gamma\right]\geq K_n\int\rho_\gamma^{1+2/n} \end{equation} with some positive constant $K_n$. Equivalently, let $V\in L^{1+n/2}(\mathbb R^n)$ and $\c$ a density operator, then \be\label{LT} \Tr [(-\mfr{1}{2}\D + V)\c] \ge -L_n \int |V_-|^{1+n/2}, \ee where $x_- := \min\{x,0\}$, and $L_n$ some positive constant. {\bf Many-body case:} Let $\psi\in \bigwedge^N L^2(\R^{3}\times\{-1,1\})$. Then, \begin{equation}\label{eq:LTmbcase} \left\langle\psi,\sum_{i=1}^N-\mfr{1}{2}\Delta_i\psi\right\rangle\geq 2^{-2/3} K_3\int\rho_\psi^{5/3} . \end{equation} \end{thm} The original proofs of these inequalities can be found in \cite{Lieb-Thirring}. From the min-max principle it is clear that the right side of (\ref{LT}) is in fact a lower bound on the sum of the negative eigenvalues of the operator $-\frac{1}{2}\Delta + V$. We shall use the following standard notation: $$D(f)=D(f,f)=\frac{1}{2}\int\!\!\int \bar{f}(x)|x-y|^{-1} f(y)\, dx dy . $$ It is not difficult to see (by Fourier transformation) that $\|f\| := D(f)^{1/2}$ is a norm. \begin{thm}[Hardy-Littlewood-Sobolev inequality] There exists a constant $C$ such that \be \label{Hardy-Littlewood-Sobolev} D(f)\le C\,\| f \|_{6/5}^2. \ee \end{thm} The sharp constant $C$ has been found by Lieb \cite{Lieb:sob}, see also \cite{Lieb-Loss}. In order to localize into different regions of space we shall use the standard IMS-formula \begin{equation}\label{eq:IMS} -\mfr{1}{2}\theta^2\Delta-\mfr{1}{2}\Delta\theta^2=-\theta\Delta\theta-(\nabla\theta)^2 , \end{equation} which holds, by a straightforward calculation, for all bounded $C^1$-functions $\theta$ (here considered as a multiplication operator). Finally we state the two inequalities which we need to estimate the many-body ground state energy $E(Z,R)$ by an energy of an effective one-particle quantum system. The first one is an electrostatic inequality providing us with a lower bound. This inequality is due to Lieb \cite{Lieb3}, and was improved in \cite{Lieb-Oxford}. \begin{thm}[Lieb-Oxford inequality] Let $\psi\in L^2(\R^{3N})$ be normalized, and $\rho_\psi$ its one-electron density. Then, \be\label{Lieb-Oxford} \left\langle \psi,\sum_{1\le i0$ and $\rho^{{\rm TF}}>0$, and $\rho^{{\rm TF}}$ is the unique solution in $L^{5/3}(\R^3)\cap L^1(\R^3)$ to the TF-equation: \begin{equation}\label{eq:tfeqgeneral} V^{{\rm TF}}({\mathbf z},{\mathbf r},x) = \mfr{1}{2}(3\pi^2)^{2/3} \rho^{{\rm TF}}({\mathbf z},{\mathbf r},x)^{2/3}. \end{equation} \end{thm} %The Thomas-Fermi equation (\ref{eq:tfeqgeneral}) can %be turned into the {\it Thomas-Fermi differential equation} %\begin{equation} \label{eq:tfdiffeqgeneral} % -\mfr{1}{4\pi}\Delta V^{{\rm TF}} ({\mathbf z},{\mathbf r},x)= \sum_{k=1}^M % z_k\d(x-r_k) - % (3\pi^2)^{-1} 2^{3/2} % \left[V^{{\rm TF}}({\mathbf z},{\mathbf r},x)\right]^{3/2}, %\end{equation} %which holds in a distributional sense. Very crucial for a semi-classical approach is the {\it scaling} behavior of the TF-potential. It says that for any positive parameter $h$ \begin{eqnarray}\label{scaling} V^{{\rm TF}}({\mathbf z},{\mathbf r},x) &=& h^{-4} V^{{\rm TF}}(h^{3}{\mathbf z},h^{-1}{\mathbf r},h^{-1}x), \\ \label{scaling:rho} \rho^{{\rm TF}}({\mathbf z},{\mathbf r},x) &=& h^{-6}\rho^{{\rm TF}}(h^{3}{\mathbf z},h^{-1}{\mathbf r},h^{-1}x) \\ E^{\rm TF}({\mathbf z},{\mathbf r})&=& h^{-7}E^{\rm TF}(h^{3}{\mathbf z},h^{-1}{\mathbf r}). \end{eqnarray} By $h^{-1}{\mathbf r}$ we mean that each coordinate is scaled by $h^{-1}$, and likewise for $h^{3}{\mathbf z}$. By the TF-equation (\ref{eq:tfeqgeneral}), the equations (\ref{scaling}) and (\ref{scaling:rho}) are obviously equivalent. Notice that the Coulomb-potential, $V$, has the claimed scaling behavior. The rest follows from the uniqueness of the solution of the TF-energy functional. We shall now establish the crucial estimates that we need about the TF potential. Let \begin{equation}\label{ddefinition} d(x)=\min\{|x-r_k|\ |\ k=1,\ldots,M\} \end{equation} and \begin{equation}\label{fdefinition} f(x)=\min\{d(x)^{-1/2}, d(x)^{-2}\}. \end{equation} For each $k=1,\ldots,M$ we define the function \begin{equation}\label{eq:Wdefinition} W_k({\mathbf z},{\mathbf r},x)=V^{\rm TF}({\mathbf z},{\mathbf r},x) -z_k|x-r_k|^{-1}. \end{equation} The function $W_k$ can be continuously extended to $x=r_k$. The first estimate in the next theorem is very similar to a corresponding estimate in \cite{Ivrii-Sigal}. \begin{thm}[Estimate on $V^{\rm TF}$]\label{thm:tfestimate} Let ${\mathbf z}=(z_1,\ldots,z_M)\in \R_+^M$ and ${\mathbf r}=(r_1,\ldots,r_M)\in \R^{3M}$. For all multi-indices $\alpha$ and all $x$ with $d(x)\ne0$ we have \begin{equation}\label{eq:tfdf} \left|\partial^\alpha_x\V({\mathbf z},{\mathbf r},x)\right|\leq C_\alpha f(x)^2 d(x)^{-|\alpha|}, \end{equation} where $C_\alpha>0$ is a constant which depends on $\alpha$, $z_1,\ldots,z_M$, and $M$. Moreover, for $|x-r_k|0$ here depend on $z_1,\ldots,z_M$, and $M$. \end{thm} \begin{proof} Throughout the proof we shall denote all constants that depend on $\alpha$, $z_1,\ldots,z_M$, $M$ by $C_\alpha$. Constants that depend on $z_1,\ldots,z_M$ we denote by $C$. In this proof we shall omit the dependence on ${\mathbf r}$ and ${\mathbf z}$ and simply write $\V(x)$ and $W_k(x)$. We proceed by induction over $|\alpha|$. If $\alpha=0$ we have the well known bound \cite{Lieb-Simon} that \begin{equation}\label{eq:LSmolecule} 0\leq \max\{\V_{r_k}(x)\ |\ k=1,\ldots,M\}\leq \V(x)\leq \sum_{k=1}^M\V_{r_k}(x), \end{equation} where $\V_{r_k}$ denotes the Thomas-Fermi potential of a neutral atom with a nucleus placed at $r_k\in\R^3$ with nuclear charge $z_k$. This potential satisfies the bounds \cite{Lieb-Simon} \begin{equation}\label{eq:LSatom} C_-\min\{z_k|x-r_k|^{-1}, |x-r_k|^{-4}\} \leq \V_{r_k}\leq C_+\min\{z_k|x-r_k|^{-1}, |x-r_k|^{-4}\}, \end{equation} where $C_\pm>0$ are universal constants (note that by scaling (\ref{scaling}) it is enough to consider the case $z_k=1$). We therefore get that \begin{equation}\label{eq:tfphi} C_-\min\{z_1,\ldots,z_M,1\} f(x)^2\leq \V(x) \leq C_+ M \max\{z_1,\ldots,z_M,1\} f(x)^2. \end{equation} This in particular gives (\ref{eq:tfdf}) for $\alpha=0$. Assume now that (\ref{eq:tfdf}) has been proved for all multi-indices $\alpha$ with $|\alpha|0$. We shall first establish an estimate for the derivatives $\partial^{\alpha}\rho$ of the TF density $\rho$. {F}rom the TF equation we have that $\rho=C(\V)^{3/2}$. Thus $\partial^{\alpha}\rho(x)$ is a sum of terms of the form $$ \V(x)^{3/2-k}\partial^{\beta_1}\V(x)\cdots \partial^{\beta_k}\V(x) $$ where $k=0,\ldots,|\alpha|$ and $|\beta_1|+\ldots+|\beta_k|=|\alpha|$. Thus by the induction hypothesis and (\ref{eq:tfphi}) we have for $|\alpha|< M$ that \begin{equation}\label{eq:tfrho} \left|\partial^{\alpha}\rho(x)\right|\leq C_\alpha f(x)^3d(x)^{-|\alpha|}. \end{equation} We now turn to the potential. Given $\alpha$ with $|\alpha|=M$. Choose some decomposition $\alpha=\beta+\alpha'$, where $|\beta|=1$ and $|\alpha'|=M-1$. For all $y$ such that $|y-x|h$. Note that if we let $a\to 1/h$ then ${\cal G}_{u,q}^2$ converges to the projection $\Pi_{u,q}=|u,q\rangle \langle u,q|$. A straightforward calculation gives the following result. \begin{lemma}[Completeness of new coherent states] These new coherent operators satisfy $$ \int {\cal G}_{u,q}^2 \,\frac{dq}{(2\pi h)^n} =G_b(\x-u),\quad \int {\cal G}_{u,q}^2 \,\frac{du}{(2\pi h)^n} =G_b(-ih\nabla-q), $$ where $\x$ denotes the operator multiplication by the position variable $x$. Here $G_b(v)=(b/\pi)^{n/2}\exp(-bv^2)$ with $b=2a/(1+h^2a^2)$. Note that $G_b$ has integral 1 and hence $$ \int {\cal G}_{u,q}^2 \,\frac{dudq}{(2\pi h)^n} = {\bf 1} . $$ \end{lemma} We shall study operators that can be written in the form (\ref{form}). If ${\widehat A}_{u,q}=B_0(u,q)+B_1(u,q)\cdot {\x}-ih B_2(u,q)\cdot\nabla$ is the operator valued symbol in (\ref{form}) we shall denote by $A_{u,q}$ the linear function $A_{u,q}(v,p)=B_0(u,q)+B_1(u,q)\cdot v+B_2(u,q)\cdot p$. When $A_{u,q}(v,p)$ is independent of $(v,p)$, i.e., if $B_1=B_2=0$ and if $a\to h^{-1}$ we recover the usual coherent states representation of an operator. Thus, on the one hand we do not use as sharp a phase space localization as the one-dimensional coherent state projection since $a<1/h$, but on the other hand, we use a better approximation than if $A_{u,q}$ were just a constant. More generally we shall consider operators of the form \begin{equation}\label{eq:formf} \int {\cal G}_{u,q}\,f({\widehat A}_{u,q})\,{\cal G}_{u,q}dudq, \end{equation} where $f:\R\to\R$ is any polynomially bounded real function. As we shall see in the next theorem the integrand above is a traceclass operator for each $(u,q)$. The integral above is to be understood in the weak sense, i.e., as a quadratic form. We shall consider situations where the integral defines bounded or unbounded operators. \begin{thm}[Trace identity]\label{thm:traceidentity} Let $f:\R\to\R$ and $V:\R^n\to\R$ be polynomially bounded, real measurable functions and $$\hat{A}=B_0 + B_1{\x} -ih B_2\nabla$$ a first order self-adjoint differential operator\footnote{The operator $\hat{A}$ is essentially self-adjoint on Schwartz functions on $\R^n$.} with $B_0\in\R,B_{1,2}\in\R^n$. Then ${\cal G}_{u,q}\,f(\hat{A}) \,{\cal G}_{u,q}\,V({\x})$ is a trace class operator (when extended from $C_0^\infty(\R^n)$) and \begin{eqnarray*} {\Tr}\big[{\cal G}_{u,q}\,f(\hat{A}) \,{\cal G}_{u,q}\,V({\x})\big] &=& \int f(B_0+B_1v+B_2p)\,G_b(v-u)G_b(q-p) \\&&\,\times\,G_{(bh^2)^{-1}}(z)V(v+h^2ab(u-v)+z)dvdpdz. \end{eqnarray*} In particular, $\Tr\Big[{\cal G}_{u,q}^2\Big]=1$. \end{thm} The proof is given in Appendix A. We shall need the following extension of this theorem, where we however only give an estimate on the trace. The proof is again deferred to Appendix A. \begin{thm}[Trace estimates] \label{trace formula} Let $f,\hat{A}$ be as in the previous theorem. Let moreover $\phi\in C^{n+4}(\mathbb R^n)$ be a bounded, real function with all derivatives up to order $n+4$ bounded and $V,F\in C^2(\R^n)$ be real functions with bounded second derivatives. Then, for $h<1$, $10$ and $\s(u,q)=q^2+V(u)$. Then, $$ \left|{\Tr}[\phi H\phi]_-- (2\pi h)^{-n}\int \,\phi^2(u) \s(u,q)_- du dq\right| \leq Ch^{-n+6/5} . $$ The constant $C>0$ here depends only on $n,\|\phi\|_{C^{n+4}}$ and $\|V\|_{C^3}$. [Here $\| V\|_{C^3}= \sup_{|\alpha|\leq3}\|\partial^\alpha V\|_\infty$.] \end{thm} With the classical coherent states the estimate one would normally prove would be that the right side above is $C h^{-n+1}$. We find it instructive to sketch the proof of this here in order to make the comparison with the new method clearer. \label{page:classicalcoherent} We may assume that $V$ is defined on all of $\R^n$ with bounded second and third order derivatives. We shall here assume that $h<1$. {F}rom Theorem~\ref{thm:coherentrepresentation} with $a=1/h$ we have $$H=-h^2\D +V = \int [q^2 +V(u)]\, |u,q\rangle\langle u,q|\, \frac{du dq}{(2\pi h)^n} + {\bf E} , $$ with $\|{\bf E}\| \le C h$. The constant depends on the second and third order derivatives of $V$, which are bounded. We have here used that for $a=b=1/h$ the first order terms in $\x$ and $\nabla$ do not contribute (see (\ref{eq:firstorder}) below). Moreover, the term $\Delta\sigma$ is of order $h$ since $V$ has bounded second order derivatives. The error term ${\bf E}$ can be controlled using the Lieb-Thirring inequality as in (\ref{eq:LTerror}) below. Then from Theorem~\ref{trace formula} we have \begin{eqnarray*} {\Tr}[\phi H\phi]_- &\ge&\int[q^2+V(u)]_-\, \Tr\Big[\phi|u,q\rangle\langle u,q|\phi\Big] \frac{du dq}{(2\pi h)^{n}} - Ch^{-n+1} \\ &=&\int[q^2+V(u)]_- \,\phi^2(u)\frac{du dq}{(2\pi h)^{n}} - Ch^{-n+1} , \end{eqnarray*} where $C$ now also depends on the derivatives of $\phi$. For an upper bound we set $$ \c = \int \chi_{(-\infty,0]}[q^2+V(u)]\,|u,q\rangle\langle u,q|\, \frac{du dq}{(2\pi h)^n}, $$ where $\chi_{(-\infty,0]}$ denotes the characteristic function of the interval $(-\infty,0]$. It is clear that ${\bf 0}\le\c\le{\bf 1}$. When calculating $\Tr[\c \phi H\phi]$ we may again refer to the general theorem \ref{trace formula} with $a=b=1/h$. We obtain \begin{eqnarray*} \Tr[\c \phi H\phi]&=&\int \chi_{(-\infty,0]}[q^2+V(u)] \, \Tr\Big[|u,q\rangle\langle u,q| \phi H\phi\Big]\, \frac{du dq}{(2\pi h)^n} \\ &\leq&\int [q^2+V(u)]_-\,\phi^2(u)\,\frac{du dq}{(2\pi h)^n} + C h^{-n+1}. \end{eqnarray*} As mentioned in the Introduction it is important that we obtain errors bounded by $Ch^{-n+1+\varepsilon}$ for some $\varepsilon>0$ as in Theorem~\ref{local semi-classics}. We shall prove Theorem~\ref{local semi-classics} by again proving upper and lower bounds on ${\Tr}[\phi H\phi]_-$. \begin{lemma}[Lower bound on ${\Tr}(\phi H\phi)_-$] Let $n\geq3$, $\phi\in C_0^{n+4}(\mathbb R^n)$ be supported in a ball $B$ of radius $1$ and assume that $V\in C^3(\overline{B})$. Let $H=-h^2\Delta+V$, $h>0$. Then, $$ {\Tr}[\phi H\phi]_- \ge (2\pi h)^{-n}\int \,\phi^2(u) \s(u,q)_- du dq -Ch^{-n+6/5} . $$ The constant $C>0$ here depends only on $n,\|\phi\|_{C^{n+4}}$ and $\|V\|_{C^3}$. \end{lemma} \begin{proof} Since $\phi$ has support in the ball $B$ we may without loss of generality assume that $V\in C^3_0(\R^3)$ with the support in a ball $B_2$ of radius $2$ and that the norm $\| V\|_{C^3}$ refers to the supremum over all of $\R^n$. We shall not explicitly follow how the error terms depend on $\|\phi\|_{C^3}$ and $\|V\|_{C^3}$. All constants denoted by $C$ depend on $n,\|\phi\|_{C^3}$, $\|V\|_{C^3}$. First note that by the Lieb-Thirring inequality we have that $$ {\Tr}[\phi H\phi]_- \ge C\|\phi\|_\infty^2\int_{u\in B} \, \s(u,q)_-\frac{du dq}{(2\pi h)^n} \geq-C h^{-n}. $$ Consider some fixed $0<\tau<1$ (independent of $h$). If $h\geq \tau$ then $$ {\Tr}[\phi H\phi]_- \ge \int \, \phi^2(u) \s(u,q)_- \frac{du dq}{(2\pi h)^n} -C\tau^{-6/5}h^{-n+6/5}. $$ We are therefore left with considering $h<\tau$. Of course one should really try to find the optimal value of $\tau$ (depending on $\phi$, and $V$) we shall however not do that. In studying the case $h<\tau$ it will be necessary to assume that the choice of $\tau$ is small enough. We therefore now assume that $h<\tau$ and that $\tau$ is small. {F}rom Theorem~\ref{thm:coherentrepresentation} we have that \begin{eqnarray} {\Tr}[\phi H\phi]_-&\geq& {\Tr}\left[\int \phi \,{\cal G}_{u,q}\widehat{H}_{u,q} {\cal G}_{u,q}\phi\frac{du dq}{(2\pi h)^n}\right]_-\nonumber \\ &&+ {\Tr}\left[\phi\left(-\varepsilon h^2\Delta -C(b^{-3/2}+h^2b)\right) \phi\right]_- \label{eq:LTerror} \end{eqnarray} where $0<\varepsilon<1/2$ and $$ \widehat H_{u,q}=\widetilde{\s}(u,q)+\frac{1}{4b}\D \widetilde{\s}(u,q) + \p_u\widetilde{\s}(u,q)({\x}-u) + \p_q \widetilde{\s}(u,q)(-ih\nabla -q) $$ with $\widetilde{\s}(u,q)=(1-\varepsilon)q^2+V(u)$. We shall choose $a$ depending on $h$ satisfying $\tau^{-1}\leq a0$ and $\s(u,q)=q^2+V(u)$. Then there exists a density matrix $\gamma$ on $L^2(\R^n)$ such that \begin{equation} {\Tr}[\phi H\phi\gamma]\le \int \, \phi^2(u) \s(u,q)_-\frac{du dq}{(2\pi h)^n} + Ch^{-n+6/5} \label{eq:lemmaupper}. \end{equation} Moreover, the density of $\gamma$ satisfies \begin{equation}\label{eq:rhogammaprop1} \left|\rho_\gamma(x)-(2\pi h)^{-n}\omega_n \left|V(x)_-\right|^{n/2}\right|\leq Ch^{-n+9/10}, \end{equation} for (almost) all $x\in B$ and \begin{equation}\label{eq:rhogammaprop2} \left|\int\phi(x)^2\rho_\gamma(x)dx-(2\pi h)^{-n}\omega_n\int\phi(x)^2 \left|V(x)_-\right|^{n/2}dx\right|\leq Ch^{-n+6/5}, \end{equation} where $\omega_n$ is the volume of the unit ball in $\R^n$. The constants $C>0$ in the above estimates depend only on $n, \|\phi\|_{C^{n+4}}$, and $\|V\|_{C^3}$. \end{lemma} \begin{proof} As in the lower bound we choose some fixed $0<\tau<1$. We have for $h\ge \tau$ that for some $C>0$ $$\int \phi^2(u) \s(u,q)_-\,\frac{du dq}{(2\pi h)^n} + C\tau^{-6/5} h^{-n+6/5}\geq 0$${and}$$ \left|(2\pi h)^{-n}\omega_n V(x)_-\right|^{n/2}\leq C\tau^{-6/5}h^{-n+6/5}, $$ If $h\geq\tau$ we may therefore choose $\gamma=0$. We may therefore now assume that $h<\tau$ and if necessary that $\tau$ is small enough depending only on $\phi$, and $V$. Also as in the lower bound we may assume that $V\in C_0^3(\R^n)$ with support in the ball $B_{3/2}$ concentric with $B$ and of radius $3/2$. In analogy to the previous proof for the lower bound we now for each $(u,q)$ define an operator $\hat{h}_{u,q}$ by $$ \hat{h}_{u,q}=\left\{\begin{array}{cl} \s(u,q)+ \frac{1}{4b}\Delta\s(u,q) + \p_u\s(u,q)({\x}-u) + \p_q {\s}(u,q)(-ih\nabla -q)&,\hbox{if } u\in B_2\\ 0&,\hbox{if } u\not\in B_2 \end{array}\right.. $$ The corresponding function is $$ {h}_{u,q}(v,p)=\left\{\begin{array}{cl} \s(u,q) + \frac{1}{4b}\Delta\s(u,q) + \p_u\s(u,q)(v-u) + \p_q {\s}(u,q)(p -q)&,\hbox{if } u\in B_2\\ 0&,\hbox{if } u\not\in B_2 \end{array}\right. . $$ Recall that $b=2a/(1+h^2a^2)$ (i.e., in particular $a\leq b\leq 2a$) and as in the lower bound we shall choose $a=\max\{h^{-4/5},\tau^{-1}\}$ Similar to (\ref{eq:atildeapp}) we have for $u\in B_2$ that \begin{eqnarray} \left|h_{u,q}(v,p)-\s(v,p)-\xi_{v}(u-v,q-p)\right| \leq C|u-v|(b^{-1}+|u-v|^2),\label{eq:happ} \end{eqnarray} where $$ \xi_{v}(u,q)=\frac{1}{4b}\Delta\s(v,0)-q^2 -\mfr{1}{2}\sum_{i,j}\p_i\p_j V(v)u_iu_j. $$ We have here used that $\Delta\s(v,p)$ is independent of $p$. If we let $\chi=\chi_{(-\infty,0]}$ be the characteristic function of $(-\infty,0]$ we now define \be \label{trial density} \c = \int {\cal G}_{u,q}\,\chi\big[\hat{h}_{u,q} \big]\, {\cal G}_{u,q}\,\frac{dudq}{(2\pi h)^n} . \ee Since ${\bf 0}\le\chi\big[\hat{h}_{u,q}\big]\le\bf1$ it is obvious that ${\bf 0} \le\c\le{\bf 1}$. Moreover, by Theorem~\ref{thm:traceidentity} and (\ref{eq:happ}), $\gamma$ is easily seen to be a traceclass operator with density $$ \rho_\gamma(x)= \int\chi\left(h_{u,q}(v,p)\right)G_b(u-v)G_b(p-q)G_{(bh^2)^{-1}}(x-v-h^2ab(u-v))dvdp \frac{dudq}{(2\pi h)^n}. $$ If we change variables $u\to u+v$, $q\to q+p$ and perform the $p$-integration we find that \begin{eqnarray}\label{eq:rhogamma} \rho_\gamma(x)&=& \omega_n\int_{u\in B_2-v} \Xi(v,u,q)G_b(u)G_b(q) G_{(bh^2)^{-1}}(x-v-h^2abu)dv \frac{dudq}{(2\pi h)^n}\nonumber\\ &=&\omega_n\hspace{-20pt}\int\limits_{(1-h^2ab)u\in B_2-v}\hspace{-20pt} \Xi(v-h^2abu,u,q)G_b(u) G_b(q)G_{(bh^2)^{-1}}(x-v)dv\frac{dudq}{(2\pi h)^n} \end{eqnarray} where $\Xi(v,u,q)=\omega_n^{-1}\int\chi(h_{(u+v,q+p)}(v,p))\,dp\geq0$. From equation (\ref{eq:happ}) we have \begin{eqnarray} \biggl|\Xi(v,u,q)^{2/n}-\biggl|\biggl(V(v)+\xi_v(u,q) \biggr)_-\biggr|\biggr|\leq C|u|(b^{-1}+|u|^2), \label{eq:Westimate} \end{eqnarray} for all $v,q\in\R^n$ and $u\in B_2-v$. Since $$ |\xi_v(u,q)-\xi_{v-h^2abu}(u,q)|\leq C h^2ab|u|(b^{-1}+|u|^2) $$ we therefore also have \begin{eqnarray*} \Biggl|\Xi(v-h^2abu,u,q)^{2/n}-\biggl|\biggl(V(v)+\eta_v(u,q) \biggr)_-\biggr|\Biggr|&\leq& Ch^4a^2b^2|u|^2\\ &&{}+ C(1+h^2ab)|u|(b^{-1}+|u|^2), \end{eqnarray*} where $$ \eta_v(u,q)=\xi_v(u,q)-h^2ab\nabla V(v)\cdot u . $$ Hence from (\ref{eq:rhogamma}) \begin{eqnarray} \Biggl|\rho_\gamma(x)^{\frac{2}{n}} -\biggl(\omega_n\hspace{-20pt}\int\limits_{(1-h^2ab)u\in B_2-v}\hspace{-20pt} \biggl|\biggl(V(v)+\eta_v(u,q) \biggr)_-\biggr|^{\frac{n}{2}}G_b(u) G_b(q)G_{(bh^2)^{-1}}(x-v)dv\frac{dudq}{(2\pi h)^n}\biggr)^{\frac{2}{n}} \Biggr|\nonumber\\ \leq Ch^{-2}(h^4a^2b+b^{-3/2})\leq C h^{-2+6/5}\label{eq:rhogamma2}, \end{eqnarray} where $C$ may depend on $\tau$. We now use that for all $x,y\in\R$ and all $n\geq3$ we have \begin{equation} \left||x_-|^{\frac{n}{2}}-|y_-|^{\frac{n}{2}} + \mfr{n}{2}|y_-|^{\frac{n}{2}-1}(x-y)\right|\leq \left\{\begin{array}{cl}C|x-y|^{\frac{3}{2}},&n=3\\ C(|x|^{\frac{n}{2}-2}+|y|^{\frac{n}{2}-2})|x-y|^2,&n\geq4 \end{array}\right.\label{eq:3/2holder} \end{equation} where $C$ depends on $n$. This gives for $n=3$ (it is left to the reader to write down the estimates for $n\geq 4$) \begin{equation}\label{eq:3/2holder2} \Biggl|\biggl|\biggl(V(v)+\eta_v(u,q)\biggr)_-\biggr|^{\frac{3}{2}}- |V(v)_-|^{\frac{3}{2}} + \mfr{3}{2}|V(v)_-|^{\frac{1}{2}}\eta_v(u,q)\Biggr| \leq C|\eta_v(u,q)|^{\frac{3}{2}}. \end{equation} It is now again crucial that $\int\eta_v(u,q)G_b(u)G_b(q)dudq=0$ and hence for $v\in\hbox{supp}(V)\subseteq B_{3/2}$ \begin{eqnarray} \biggl|\int\limits_{(1-h^2ab)u\in B_2-v} \eta_v(u,q)G_b(u)G_b(q)dudq\biggr| \leq Ce^{-b/5}\leq Ch^{6/5} . \label{eq:2ndorderexact} \end{eqnarray} Combining (\ref{eq:rhogamma2}), (\ref{eq:3/2holder2}), (\ref{eq:2ndorderexact}), and $|\eta_v(u,q)|\leq C(b^{-1}+|u|^2+|q|^2+h^2ab|u|)$ we obtain \begin{eqnarray} \lefteqn{\biggl|\rho_\gamma(x)-(2\pi h)^{-3}\omega_3\int|V(v)_-|^{3/2} G_{(bh^2)^{-1}}(x-v)dv \biggr|}\nonumber\\ &\leq& Ch^{-3}(e^{-b/5}+h^3a^{3/2}b^{3/4}+b^{-3/2}+h^{6/5})\leq Ch^{-3+6/5},\label{eq:rhogamma*} \end{eqnarray} where we have again removed the condition $(1-h^2ab)u\in B_2-v$ paying a price of $Ch^{-3}e^{-b/5}$. A simple Taylor expansion of $\phi^2$ gives $$ \left|\phi(x)^2-\int\phi(v)^2G_{(bh^2)^{-1}}(x-v)dv\right|\leq C bh^2\leq Ch^{6/5}, $$ where we have again used that $\int vG_{(bh^2)^{-1}}(v)dv=0$. This immediately gives (\ref{eq:rhogammaprop2}). Finally, using again (\ref{eq:3/2holder}) we get $$ \left||V(x+v)_-|^{\frac{3}{2}}-|V(x)_-|^{\frac{3}{2}} + \mfr{3}{2}|V(x)_-|^{\frac{1}{2}}\nabla V(x)\cdot v\right|\leq C (|v|^{\frac{3}{2}}+|v|^2), $$ and hence from (\ref{eq:rhogamma*}) \begin{eqnarray*} \biggl|\rho_\gamma(x)-(2\pi h)^{-3}\omega_n|V(x)_-|^{3/2} \biggr|\leq Ch^{-3}(h^{6/5}+(bh^2)^{3/4}) \leq Ch^{-3+9/10} . \end{eqnarray*} We must now calculate ${\Tr}(\c\phi H\phi)={\Tr}(\c\phi (-h^2\Delta)\phi)+{\Tr}(\c\phi V\phi)$ for $n\ge3$. {F}rom the argument leading to (\ref{eq:rhogammaprop2}) we have \begin{equation}\label{eq:laplace0} (2\pi h)^n{\Tr}(\c\phi V\phi)\leq -\omega_n\int\phi(x)^2|V(x)_-|^{\frac{n}{2}+1}dx +Ch^{-n+6/5}. \end{equation} {F}rom Theorem \ref{trace formula} we have \begin{eqnarray*} (2\pi h)^n{\Tr}(\c\phi(-h^2\Delta) \phi) &=& \int \,\chi({h}_{u,q}(v,p))\,G_b(u-v)G_b(q-p) \Big[E_2+\\&&(\phi(v+h^2ab(u-v))^2+E_1) (p+h^2ab(q-p))^2 \Big]dudq dv dp, \end{eqnarray*} where $E_1, E_2$ are functions such that $\|E_1\|_\infty,\|E_2\|_\infty\leq Ch^2b$. Since $$ \int \,\chi({h}_{u,q}(v,p))\,G_b(u-v)G_b(q-p) (1+p^2)dudq dv dp\leq C, $$ (note that it is important here that ${h}_{u,q}(v,p)=0$ unless $u\in B_2$) we get \begin{eqnarray*} \lefteqn{(2\pi h)^n{\Tr}(\c\phi (-h^2\Delta)\phi)}&&\\ &\le& \int \,\chi(h_{u,q}(v,p))\,G_b(u-v)G_b(q-p)\phi(v+h^2ab(u-v))^2\\&&\times\, (p+h^2ab(q-p))^2 dudq dv dp +Cbh^2. \end{eqnarray*} {F}rom (\ref{eq:happ}) we may now conclude that \begin{eqnarray} (2\pi h)^n{\Tr}(\c\phi ( -h^2\Delta)\phi) &\leq&\int \,\chi(\s(v,p)+\xi_v(u,q)-C|u|(b^{-1}+|u|^2))\,G_b(u)G_b(q) \nonumber\\ &&\times\, \phi(v+h^2abu)^2 (p+h^2abq)^2dudq dv dp+Cbh^2\label{eq:laplace1}. \end{eqnarray} We now perform the $p$-integration in (\ref{eq:laplace1}) and arrive at \begin{eqnarray} (2\pi h)^n{\Tr}(\c\phi ( -h^2\Delta)\phi) &\leq&\frac{n}{n+2}\omega_n\int \, \left|\left(V(v)+\xi_v(u,q) -C|u|(b^{-1}+|u|^2)\right)_-\right|^{\frac{n}{2}+1} \nonumber\\&&\times\,G_b(u)G_b(q) \phi(v+h^2abu)^2 dudq dv+Cbh^2,\label{eq:laplace2} \end{eqnarray} where we have used that the integral over the term containing $q\cdot p$ vanishes and the integral over the term containing $(h^2abq)^2$ is bounded by $h^4a^2b\leq h^2b$. We now expand the integrand in (\ref{eq:laplace2}) in the same way as we did the integrand in (\ref{eq:lowerexpansion}). We finally obtain $$ (2\pi h)^n{\Tr}(\c\phi ( -h^2\Delta)\phi) \leq\frac{n}{n+2}\omega_n\int \, \left|V(v)_-\right|^{\frac{n}{2}+1} \phi(v)^2dv+Ch^{6/5}, $$ which together with (\ref{eq:laplace0}) gives (\ref{eq:lemmaupper}). \end{proof} We shall need the generalization of Theorem~\ref{local semi-classics} and Lemma~\ref{lm:upperbound} to a ball of arbitrary radius. We also require to know how the error term depends more explicitly on the potential. \begin{cor}[Rescaled semi-classics] \label{corollary} Let $n\geq 3$, $\phi\in C^{n+4}_0(\R^n)$ be supported in a ball $B_\ell$ of radius $\ell>0$. Let $V\in C^3(\bar{B}_\ell)$ be a real potential. Let $H=-h^2\D +V$, $h>0$ and $\sigma(u,q) = q^2 + V(u)$. Then for all $h>0$ and $f>0$ we have \begin{equation} \label{eq:phiHphilf} \left|\Tr[\phi H\phi]_- - (2\pi h)^{-n}\int \phi(u)^2 \sigma(u,q)_-\, du dq \right| \le C h^{-n+6/5} f^{n+4/5}\ell^{n-6/5}, \end{equation} where the constant $C$ depends only on \begin{equation}\label{eq:phivdependence} \sup_{|\a|\le n+4}\|\ell^{|\a|}\p^\a\phi\|_{\infty},\quad\hbox{ and }\quad \sup_{|\a|\le3}\|f^{-2}\ell^{|\a|}\p^\a V\|_{\infty}. \end{equation} Moreover, there exists a density matrix $\gamma$ such that \begin{equation} \Tr[\phi H\phi\gamma]\leq (2\pi h)^{-n}\int \phi(u)^2 \sigma(u,q)_-\, du dq +C h^{-n+6/5} f^{n+4/5}\ell^{n-6/5}\label{eq:gammaproplf} \end{equation} and such that its density $\rho_\gamma(x)$ satisfies \begin{equation}\label{eq:rhogammaproplf1} \left|\rho_\gamma(x)-(2\pi h)^{-n}\omega_n \left|V(x)_-\right|^{n/2}\right|\leq Ch^{-n+9/10}f^{n-9/10}\ell^{-9/10}, \end{equation} for (almost) all $x\in B_\ell$ and \begin{equation}\label{eq:rhogammaproplf2} \left|\int\phi(x)^2\rho_\gamma(x)dx-(2\pi h)^{-n}\omega_n\int\phi(x)^2 \left|V(x)_-\right|^{n/2}dx\right|\leq Ch^{-n+6/5}f^{n-6/5}\ell^{n-6/5}, \end{equation} where the constants $C>0$ in the above estimates again depend only on the parameters in (\ref{eq:phivdependence}). \end{cor} \begin{proof} This is a simple rescaling argument. Introducing the unitary operator $(U\psi)(x)=\ell^{-n/2}\psi(\ell^{-1} x)$ we see that $\phi H\phi$ is unitarily equivalent to the operator $$ U^*\phi H\phi U=f^2\phi_\ell(-h^2f^{-2}\ell^{-2}\Delta + V_{f,\ell})\phi_\ell, $$ where $\phi_\ell(x)=\phi(\ell x)$, and $V_{f,\ell}(x)=f^{-2}V(\ell x)$. Thus $$\Tr[\phi H\phi]_-=f^{2} \Tr[\phi_\ell(-h^2f^{-2}\ell^{-2}\Delta + V_{f,\ell})\phi_\ell]_-. $$ Note that $\phi_\ell$ and $V_{f,\ell}$ are defined in a ball of radius $1$ and that for all $\a$ $$ \|\p^\a\phi_\ell\|_\infty =\|\ell^{|\a|}\p^\a\phi\|_\infty,\quad \hbox{ and }\quad \|\p^\a V_{f,\ell}\|_{\infty}= \|f^{-2}\ell^{|\a|}\p^\a V\|_{\infty}. $$ It follows from Theorem~\ref{local semi-classics} that \begin{equation} \left|\Tr[\phi H\phi]_- - (2\pi hf^{-1}\ell^{-1})^{-n}\int \phi_\ell(u)^2 f^2\sigma_{f,\ell}(u,q)_-\, du dq \right| \le C f^2 (hf^{-1}\ell^{-1})^{-n+6/5}, \end{equation} where $\sigma_{f,\ell}(u,q)= q^2-V_{f,\ell}(u)$ and where the constant $C$ only depends on the parameters in (\ref{eq:phivdependence}). A simple change of variables gives $$ (2\pi hf^{-1}\ell^{-1})^{-n}\int \phi_\ell(u)^2 f^2\sigma_{f,\ell}(u,q)_-\, du dq= (2\pi h)^{-n}\int \phi(u)^2 \sigma(u,q)_-\, du dq. $$ Thus (\ref{eq:phiHphilf}) follows. To find the appropriate density matrix $\gamma$. We begin with the corresponding density matrix $\gamma_{f,\ell}$ for $\phi_\ell(-h^2f^{-2}\ell^{-2}\Delta + V_{f,\ell})\phi_\ell$, i.e. the density matrix, which according to Lemma~\ref{lm:upperbound} satisfies the three estimates \beax \Tr\left[\phi_\ell(-h^2f^{-2}\ell^{-2}\Delta + V_{f,\ell}) \phi_\ell\gamma_{f,\ell}\right]&\leq& (2\pi hf^{-1}\ell^{-1})^{-n} \int\phi^2_\ell(u) \sigma_{f,\ell}(u,q)_- \, du dq \\&&+C(hf^{-1}\ell^{-1})^{-n+6/5}, \eeax $$ \left|\rho_{\gamma_{f,\ell}}(x)-(2\pi hf^{-1}\ell^{-1})^{-n} \omega_n|V_{f,\ell}(x)_-|^{n/2}\right| \leq C(hf^{-1}\ell^{-1})^{-n+9/10} , $$$$ \left|\int\phi_\ell^2\rho_{\gamma_{f,\ell}} -(2\pi hf^{-1}\ell^{-1})^{-n}\omega_n\int\phi_\ell(x)^2 |V_{f,\ell}(x)_-|^{n/2}dx\right|\leq C(hf^{-1}\ell^{-1})^{-n+6/5}. $$ The density matrix $\gamma=U\gamma_{f,\ell}U^*$ whose density is $\rho_\gamma(x)=\ell^{-n}\rho_{\gamma_{f,\ell}}(x/\ell)$ then satisfies the properties (\ref{eq:gammaproplf}--\ref{eq:rhogammaproplf2}). \end{proof} \section{Semiclassics for the Thomas-Fermi potential}\label{sec:sctf} We shall consider the semiclassical approximation for a Schr\"odinger operator with the Thomas-Fermi potential $\V({\mathbf z},{\mathbf r},x)$, i.e., $-h^2\Delta-\V$. We shall throughout this section simply write $\V(x)$ instead of $\V({\mathbf z},{\mathbf r},x)$. Recall that $\V(x)>0$. The main result we shall prove here is the Scott correction to the semiclassical expansion for this potential. \begin{thm}[Scott corrected semiclassics]\label{TF} For all $h>0$ and all $r_1,\ldots,r_M\in\R^3$ with $\min_{k\ne m}|r_m-r_k|>r_0>0$ we have \begin{equation}\label{eq:main1} \left|\Tr[-h^2\D - V^{\rm TF}]_- - (2\pi h)^{-3} \int (p^2 - V^{\rm TF}(u))_- \,du dp- \frac{1}{8h^2} \sum_{k=1}^M z_k^2\right| \le C h^{-2+\frac{1}{10}}, \end{equation} where $C>0$ depends only on $z_1,\ldots,z_M$, $M$, and $r_0$. Moreover, we can find a density matrix $\gamma$ such that \begin{equation}\label{eq:maingamma1} \Tr \left[(-h^2\Delta - V^{\rm TF})\gamma\right] \leq \Tr \left[-h^2\Delta - V^{\rm TF}\right]_-+C h^{-2+1/10}, \end{equation} and such that \begin{equation}\label{eq:maingamma2} D\left(\rho_\gamma-\frac{1}{6\pi^2h^3}(V^{\rm TF})^{3/2}\right)\leq Ch^{-5+4/5} \end{equation} and \begin{equation}\label{eq:maingamma3} \int \rho_\gamma\leq \frac{1}{6\pi^2h^3}\int V^{\rm TF}(x)^{3/2}dx+C h^{-2+1/5}, \end{equation} with $C$ depending on the same parameters as before. \end{thm} Note that if we choose $h=2^{-1/2}$ we have from (\ref{eq:tfeqgeneral}) that $(6\pi^2h^3)^{-1}(\V)^{3/2}=\rho^{\rm TF}/2$. The factor $1/2$ on the right is due to the fact that we have not included spin degeneracy in Theorem~\ref{TF}. In order to prove this theorem we shall compare with semiclassics for hydrogen like atoms. \begin{lemma}[Hydrogen comparison]\label{lm:hydrogencom} For all $h>0$ and all $r_1,\ldots,r_M\in\R^3$ with $\min_{k\ne m}|r_m-r_k|>r_0>0$ we have \begin{eqnarray} \lefteqn{\Biggl|\Tr \left[-h^2\Delta-V^{\rm TF}(\x)\right]_- -(2\pi h)^{-3}\int \left(p^2-V^{\rm TF}(u)\right)_-dudp}&&\nonumber\\ &&-\sum_{k=1}^M\left(\Tr \Bigl[-h^2\Delta-\frac{z_k}{|\x-r_k|}+1\Bigr]_- -(2\pi h)^{-3}\int \Bigl(p^2-\frac{z_k}{|u-r_k|}+1\Bigr)_-dudp\right) \Biggr| \nonumber\\ &\leq& C h^{-2+1/10},\label{eq:hydcom} \end{eqnarray} where $C>0$ depends only on $z_1,\ldots,z_M$, $M$ and $r_0$. \end{lemma} The first estimate in Theorem~\ref{TF} follows from Lemma~\ref{lm:hydrogencom} combined with the exact calculations for hydrogen $$ \Tr\Bigl[-h^2\Delta-\frac{z_k}{|\x-r_k|}+1\Bigr]_- =\sum_{1\leq n\leq z_k/(2h)}(-\frac{z_k^2}{4h^2}+n^2) =-\frac{z_k^3}{12 h^3}+\frac{z_k^2}{8h^2}+{\cal O}(h^{-1}) $$ and $$ (2\pi h)^{-3}\int \Bigl(p^2-\frac{z_k}{|u-r_k|}+1\Bigr)_-dudp =-\frac{32\pi^2 z_k^3}{15(2\pi h)^{3}}\frac{\Gamma(7/2)\Gamma(1/2)}{\Gamma(4)} =-\frac{z_k^3}{12 h^3}. $$ Before giving the proof of Lemma~\ref{lm:hydrogencom} we introduce the function \begin{equation}\label{eq:ldefinition} \ell(x)=\mfr{1}{2}\Bigl(1+\sum_{k=1}^M(|x-r_k|^2+\ell_0^2)^{-1/2}\Bigr)^{-1} \end{equation} where $0<\ell_0<1$ is a parameter that we shall choose explicitly in (\ref{eq:ell0choice}) below. Note that $\ell$ is a smooth function with $$ 0<\ell(x)<1,\quad\hbox{and}\quad \|\nabla\ell(x)\|_\infty<1. $$ Note also that in terms of the function $d(x)$ from (\ref{ddefinition}) we have \begin{equation}\label{eq:elldcom} \mfr{1}{2}(1+M)^{-1}\ell_0\leq \mfr{1}{2}(1+M(d(x)^2+\ell_0^2)^{-1/2})^{-1} \leq\ell(x)\leq\mfr{1}{2}(d(x)^2+\ell_0^2)^{1/2}. \end{equation} Note in particular that we have \begin{equation}\label{eq:elldcom2} \ell(x)\geq\mfr{1}{2}(1+M)^{-1}\min\{d(x),1\}. \end{equation} We fix a localization function $\phi\in C_0^\infty(\R^3)$ with support in $\{|x|<1\}$ and such that $\int\phi(x)^2dx=1$. According to Theorem~\ref{partition} we can find a corresponding family of functions $\phi_u\in C_0^\infty(\R^3)$, $u\in\R^3$, where $\phi_u$ is supported in the ball $\{|x-u|<\ell(u)\}$ with the properties that \begin{equation}\label{eq:phiuprop} \int\phi_u(x)^2\ell(u)^{-3}du=1\quad\hbox{and}\quad \|\partial^\alpha\phi_u\|_\infty\leq C\ell(u)^{-|\alpha|}, \end{equation} for all multi-indices $\alpha$, where $C>0$ depends only on $\alpha$ and $\phi$. Moreover, from (\ref{eq:tfdf}) in Theorem~\ref{thm:tfestimate} we know that for all $u\in \R^n$ with $d(u)>2\ell_0$ the TF-potential $V^{\rm TF}$ satisfies \begin{equation}\label{eq:tflf} \sup_{|x-u|<\ell(u)}|\p^\alpha V^{\rm TF}(x)|\leq Cf(u)^2\ell(u)^{-|\alpha|}, \end{equation} where $C>0$ depends only on $\alpha$, $z_1,\ldots,z_M$, and $M$. We have here used the fact that if $d(u)>2\ell_0$ then $\ell(u)\leq \sqrt{5}d(u)/4$ and hence for all $x$ with $|x-u|<\ell(u)$ we have (note that $d(u)\le d(x)+|x-u|$ and $\sqrt{5}/4<1$) $$ \ell(u)2$. \end{enumerate} Let \begin{equation}\label{eq:Rchoice} R=h^{-1/2} \end{equation} and define $\Phi_\pm(x)=\theta_\pm(d(x)/R)$. Then $\Phi_-^2+\Phi_+^2=1$. Denote ${\cal I}=(\nabla\Phi_-)^2+(\nabla\Phi_+)^2$. Then ${\cal I}$ is supported on a set whose volume is bounded by $CR^3$ (where as before $C$ depends on $M$) and $$ \|{\cal I}\|_\infty\leq CR^{-2}. $$ Using the IMS-formula (\ref{eq:IMS}) we find that $$ -h^2\Delta-V^{\rm TF}=\Phi_-(-h^2\Delta-V^{\rm TF}-h^2{\cal I})\Phi_- +\Phi_+(-h^2\Delta-V^{\rm TF}-h^2{\cal I})\Phi_+ $$ {F}rom the Lieb-Thirring inequality the estimates on ${\cal I}$ and the bound $V^{\rm TF}(x)\leq Cd(x)^{-4}$ (see (\ref{eq:tfdf}) with $\alpha=0$) we find $$ \Tr[-h^2\Delta-V^{\rm TF}]_-\geq \Tr[\Phi_-(-h^2\Delta-V^{\rm TF}-h^2{\cal I})\Phi_-]_- -C(h^{-3}R^{-7} + h^{2} R^{-2}). $$ On the support of $\Phi_-$ we now use the localization functions $\phi_u$. Again using the IMS formula (\ref{eq:IMS}) we obtain from (\ref{eq:phiuprop}) that \begin{eqnarray*} \lefteqn{\Phi_-\left(-h^2\Delta-V^{\rm TF}-h^2{\cal I}\right)\Phi_-}&&\\ &\geq& \int \Phi_-\phi_u\left(-h^2\Delta-V^{\rm TF}-Ch^2(\ell(u)^{-2} +R^{-2})\right)\phi_u\Phi_-\ell(u)^{-3}du. \end{eqnarray*} We have here used that if the supports of $\phi_u$ and $\phi_{u'}$ overlap then $|u-u'|\leq\ell(u)+\ell(u')$ and thus $$ \ell(u)\leq\ell(u')+\|\nabla\ell\|_\infty(\ell(u)+\ell(u')). $$ Therefore, since $\|\nabla\ell\|_\infty<1$, we have that $\ell(u)\leq C\ell(u')$ and thus $\ell(u')^{-2}\leq C\ell(u)^{-2}$. {F}rom the variational principle we now get \begin{eqnarray} \lefteqn{\Tr[-h^2\Delta-V^{\rm TF}]_-}&&\label{eq:tfphilow}\\\nonumber &\geq&\int_{d(u)<2R+1}\Tr[\phi_u\left(-h^2\Delta-V^{\rm TF}-Ch^2\ell(u)^{-2}\right)\phi_u]_- \ell(u)^{-3}du\\&&-C(h^{-3}R^{-7} + h^2R^{-2}),\nonumber \end{eqnarray} where we have restricted the integral according to the support of $\Phi_-$ and $\phi_u $ and used that, since we may assume that $h$ is so small that $R>C$, then $\ell(u)^{-2}\geq CR^{-2}$. Note that there is no need to write $\Phi_-$ on the right, since in general $\Tr(\Phi_-A\Phi_-)_-\geq \Tr A_-$ for any selfadjoint operator $A$. In a very similar manner we get corresponding estimates for the hydrogenic operators. In particular, if we choose $h$ so small that $R>\max_k\{z_k\}$ then on the support of $\Phi_+$ we have $-z_k|x-r_k|^{-1}+1\geq0$. Thus we have \begin{eqnarray} \lefteqn{\Tr\Bigl[-h^2\Delta-\frac{z_k}{|\hat{x}-r_k|}+1\Bigr]_-}&& \label{eq:hydphilow}\\ &\geq&\int_{d(u)<2R+1}\Tr\Bigl[\phi_u\Bigl(-h^2\Delta-\frac{z_k}{|\hat{x}-r_k|}+1 -Ch^2\ell(u)^{-2}\Bigr)\phi_u\Bigr]_- \ell(u)^{-3}du\nonumber\\&&-Ch^{2} R^{-2}.\nonumber \end{eqnarray} We shall now get upper bounds similar to (\ref{eq:tfphilow}) and (\ref{eq:hydphilow}). If we again denote by $\chi$ the characteristic function of the interval $(-\infty,0]$ we see from (\ref{eq:phiuprop}) that $$ \gamma=\int_{d(u)<2R+1}\phi_u\chi\left(\phi_u(-h^2\Delta-V^{\rm TF})\phi_u\right)\phi_u\ell(u)^{-3}du $$ defines a density matrix. If we use it as a trial density matrix to get an upper bound we obtain \begin{eqnarray} \Tr[-h^2\Delta-V^{\rm TF}]_-&\leq& \Tr[(-h^2\Delta-V^{\rm TF})\gamma]\nonumber\\ &=&\int\limits_{d(u)<2R+1}\Tr[\phi_u\left(-h^2\Delta-V^{\rm TF}\right)\phi_u]_- \ell(u)^{-3}du.\label{eq:tfphiup} \end{eqnarray} Similarly, \begin{eqnarray} \Tr\Bigl[-h^2\Delta-\frac{z_k}{|\hat{x}-r_k|}+1\Bigr]_-\leq \int\limits_{d(u)<2R+1}\!\!\!\!\Tr\Bigl[\phi_u\Bigl(-h^2\Delta-\frac{z_k}{|\hat{x}-r_k|}+1\Bigr)\phi_u\Bigr]_- \ell(u)^{-3}du.\label{eq:hydphiup} \end{eqnarray} We now introduce the quantities \begin{eqnarray*} D_+(u)&:=&\Tr[\phi_u\left(-h^2\Delta-V^{\rm TF}-Ch^2\ell(u)^{-2}\right)\phi_u]_-\nonumber\\&&- \sum_{k=1}^M\Tr\Bigl[\phi_u\Bigl(-h^2\Delta-\frac{z_k}{|\hat{x}-r_k|}+1\Bigr)\phi_u\Bigr]_- , \\ D_-(u)&:=&\sum_{k=1}^M\Tr\Bigl[\phi_u\Bigl(-h^2\Delta -\frac{z_k}{|\hat{x}-r_k|}+1-Ch^2\ell(u)^{-2}\Bigr)\phi_u\Bigr]_- \nonumber\\&&- \Tr[\phi_u(-h^2\Delta-V^{\rm TF})\phi_u]_- ,\\ \noalign{and} D_{\rm SC}(u)&:=&(2\pi h)^{-3}\int\phi_u(x)^2(p^2-\V(x))_-dpdx\nonumber\\&&- (2\pi h)^{-3}\sum_{k=1}^M\int\phi_u(x)^2\Bigl(p^2-\frac{z_k}{|x-r_k|}+1\Bigr)_-dpdx . \end{eqnarray*} Then from (\ref{eq:tfphilow}), (\ref{eq:hydphilow}),(\ref{eq:tfphiup}), and (\ref{eq:hydphiup}) we have \begin{eqnarray} \Tr[-h^2\Delta-\V]_- - \sum_{k=1}^M\Tr\Bigl[-h^2\Delta-\frac{z_k}{|\hat{x}-r_k|}+1\Bigr]_- &\geq&\int_{d(u)<2R+1} D_+(u)\ell(u)^{-3}du\nonumber\\&& -C(h^2R^{-2}+h^{-3}R^{-7})\label{eq:Dintegral} \end{eqnarray} and \begin{eqnarray} \sum_{k=1}^M\Tr\Bigl[-h^2\Delta-\frac{z_k}{|\hat{x}-r_k|}+1\Bigr]_--\Tr[-h^2\Delta-\V]_- &\geq&\int_{d(u)<2R+1} D_-(u)\ell(u)^{-3}du\nonumber\\ &&-Ch^2R^{-2},\label{eq:D-integral} \end{eqnarray} and from (\ref{eq:phiuprop}) \begin{eqnarray} (2\pi h)^{-3}\int(p^2-\V(x))_-dpdx- (2\pi h)^{-3}\sum_{k=1}^M\int\Bigl(p^2-\frac{z_k}{|x-r_k|}+1\Bigr)_-dpdx \nonumber\\= \int D_{\rm SC}(u)\ell(u)^{-3}du.\label{eq:Dscintegral} \end{eqnarray} The same estimates which led to (\ref{eq:tfphilow}) and (\ref{eq:hydphilow}) give \begin{eqnarray}\label{eq:DscR} \left|\int D_{\rm SC}(u)\ell(u)^{-3}du-\int_{d(u)<2R+1} D_{\rm SC}(u)\ell(u)^{-3}du\right| \leq Ch^{-3}R^{-7}. \end{eqnarray} We shall prove the lemma by establishing lower bounds on $D_+(u)-D_{\rm SC}(u)$ and $D_-(u)+D_{\rm SC}(u)$. We consider first $u$ with $d(u)\leq 2\ell_0$, where $\ell_0$ is the parameter that occurs in the definition (\ref{eq:ldefinition}) of $\ell$. We choose \begin{equation}\label{eq:ell0choice} \ell_0=h, \end{equation} where we assume that $h$ is small enough to ensure that $\ell_0<1$. In fact, we may also assume that $\ell_0\max_k\{z_k\}+1$ then we have that for all $k=1,\ldots,M$ that $$ \Tr\Bigl[\phi_u\Bigl(-h^2\Delta-\frac{z_k}{|\hat{x}-r_k|}+1\Bigr)\phi_u\Bigr]_-=0 \quad\hbox{and}\quad \int\phi_u(x)^2\Bigl(p^2-\frac{z_k}{|x-r_k|}+1\Bigr)_-dpdx=0. $$ On the other hand, if $2\ell_02\ell_0$ we have \begin{eqnarray} \Bigl|\Tr\Bigl[\phi_u\Bigl(-h^2\Delta -\frac{z_k}{|\hat{x}-r_k|}+1\Bigr)\phi_u\Bigr]_-- (2\pi h)^{-3}\int\phi_u^2(x) \Bigl(p^2-\frac{z_k}{|x-r_k|}+1\Bigr)_-dxdp\Bigr|\nonumber\\ \leq C h^{-3}h^{6/5}f(u)^{19/5}\ell(u)^{9/5}.\label{eq:dulargesc2} \end{eqnarray} Hence from (\ref{eq:dulargelower}), (\ref{eq:dulargesc1}), and (\ref{eq:dulargesc2}) we have for all $u$ with $2\ell_02\ell_0$ we have $$ \int\phi_u(x)^2\rho_{\gamma_u}(x)dx\leq \frac{1}{6\pi^2h^3}\int \phi_u(x)^2V^{\rm TF}(x)^{3/2}dx+C h^{-2+1/5}f(u)^{9/5}\ell(u)^{9/5} $$ and from (\ref{eq:hydL1L5/3}) and (\ref{eq:TFL1L5/3}) we get for $d(u)\leq 2\ell_0$ that $$ \int \phi_u(x)^2\rho_{\gamma_u}(x)dx\leq\frac{1}{6\pi^2h^3}\int \phi_u(x)^2V^{\rm TF}(x)^{3/2}dx+Ch^{-3}\ell_0^{3/2}. $$ Hence using the first property of $\phi_u$ in (\ref{eq:phiuprop}) we obtain \begin{eqnarray*} \int\rho_{\gamma}(x)dx&\leq& \frac{1}{6\pi^2h^3}\int V^{\rm TF}(x)^{3/2}dx+ Ch^{-3}\int_{d(u)\leq2\ell_0}\ell_0^{3/2}\ell(u)^{-3}du\\&& +C\int_{2\ell_0R}\left\|\frac{1}{6\pi^2h^3}(V^{\rm TF})^{3/2}\phi_u^2\right\|_{6/5}\ell(u)^{-3}du. \end{eqnarray*} If we use (\ref{eq:hydL1L5/3}) and (\ref{eq:TFL1L5/3}) when $u$ with $d(u)\leq 2\ell_0$, (\ref{eq:rhogammaproplf1}) when $2\ell_0R$ we obtain \begin{eqnarray*} \left\|\rho_\gamma-\frac{1}{6\pi^2h^3}(V^{\rm TF})^{3/2}\right\|_{6/5}&\leq& Ch^{-3}\ell_0 +Ch^{-21/10}\!\!\!\!\!\!\int\limits_{2\ell_0R}f(u)^3\ell(u)^{-1/2}du. \end{eqnarray*} Using as before the properties (\ref{fdefinition}) and (\ref{eq:elldcom2}) we see that the first integral above is bounded by a constant and the second integral (since we may assume that $R>1$) is bounded by $R^{-3}$. Thus using the Hardy-Littlewood-Sobolev inequality (\ref{Hardy-Littlewood-Sobolev}) we obtain $$ D\left(\rho_\gamma-\frac{1}{6\pi^2h^3}(V^{\rm TF})^{3/2}\right)\leq C\left\|\rho_\gamma-\frac{1}{6\pi^2h^3}(V^{\rm TF})^{3/2}\right\|_{6/5}^2\leq C(h^{-5+4/5}+h^{-6}R^{-6}). $$ Finally we turn to proving (\ref{eq:maingamma1}). {F}rom the definition of $\gamma$, (\ref{eq:phiHphilf}) and (\ref{eq:gammaproplf}) we obtain \begin{eqnarray} \Tr \left[(-h^2\Delta - V^{\rm TF})\gamma\right] &=&\int_{d(u) 2\ell_0 \end{array} \right.. $$ If $d(u)\leq2\ell_0$ we choose $\e= c\ell_0^{-1} h^2$ (as we did just before (\ref{eq:D-Dusmall})) and we get \begin{eqnarray} \Tr[\phi_u\left(-h^2\Delta-V^{\rm TF}-Ch^2\ell(u)^{-2}\right)\phi_u]_- &\geq& \Tr[\phi_u\left(-h^2\Delta-V^{\rm TF}\right)\phi_u]_-\nonumber \\&&{} -Ch^{-1}\ell_0^{-1/2}\label{eq:Hgammausmall} \end{eqnarray} If $d(u)>2\ell_0$ we choose $\e= h^2\ell(u)^{-2}f(u)^{-2}$ (as we did just after (\ref{eq:D+lowerfinal})) and we get \begin{eqnarray} \Tr[\phi_u\left(-h^2\Delta-V^{\rm TF}-Ch^2\ell(u)^{-2}\right)\phi_u]_- &\geq& \Tr[\phi_u\left(-h^2\Delta-V^{\rm TF}\right)\phi_u]_-\nonumber \\&&{} -Ch^{-1}f(u)^3\ell(u)\label{eq:Hgammaularge} \end{eqnarray} Thus from (\ref{eq:Hgamma1}), (\ref{eq:Hgamma2}), (\ref{eq:Hgammausmall}), and (\ref{eq:Hgammaularge}) we get \begin{eqnarray*} \Tr \left[(-h^2\Delta - V^{\rm TF})\gamma\right]&\leq& \Tr[-h^2\Delta-V^{\rm TF}]_- +C(h^{-1}\ell_0^{-1/2}+h^2R^{-2}+h^{-3}R^{-7})\\&& +C\int\limits_{d(u)1$. Here the constant depends on the bound on $\phi$, and its first $n+4$ derivatives. Thus the relevant contribution to the integral (\ref{eq:5fourier}) coming from the error term $R_F$ can be estimated by \begin{eqnarray*} &&\Biggl|\int R_F(p+h^2ab(q-p),\eta) \phi(z+w(v,u)) \phi(z'+w(v,u)) e^{i\eta(z-z')/h}\\&&\times\, e^{-b\left(\frac{z-z'}{2}\right)^2 -\frac{1}{4h^2b} (z+z')^2}\, dz dz'd\eta\Biggr|\\ &&\leq Cb^{-n/2} \int(1+\eta^2/(bh^2))^{-k}\eta^2e^{-\frac{1}{h^2b} \tilde{z}^2}\, d\tilde{z}d\eta \leq Cb^{n/2}h^{2n} bh^2, \end{eqnarray*} where we have chosen $k$ so as to make the integral finite. We can always do this without violating $1\leq k\leq (n/2)+2$. Thus after taking the limit $\delta\to0$ we obtain the statement of the theorem. The case when $F=0$ and $V\ne0$ is similar but much simpler since we may start with Theorem~\ref{thm:traceidentity} and expand $V$. \end{proof} \begin{proof}[Proof of theorem \ref{thm:coherentrepresentation}] Since $\s$ is a sum of a function of $q$ and a function of $u$ it is enough to consider only one of the terms, say, $V$. Let as before $G_b(x)=(b/\pi)^{n/2}e^{-b x^2}$. It follows immediately from (\ref{eq:Gkernel}) that \begin{eqnarray} \int {\cal G}_{u,q}^2\frac{dq}{(2\pi h)^n}&=&G_b({\x}-u),\nonumber\\ \int {\cal G}_{u,q}({\x}-u){\cal G}_{u,q}\frac{dq}{(2\pi h)^n} &=&(1-h^2ab)({\x}-u)G_b({\x}-u).\label{eq:firstorder} \end{eqnarray} As a consequence we have \begin{eqnarray*} \lefteqn{V({\x}) -\int {\cal G}_{u,q} \Bigl(V(u)+\mfr{1}{4b}\Delta V(u)+\nabla V(u) \cdot ({\x}-u) \Bigr){\cal G}_{u,q}\frac{du dq}{(2\pi h)^n}}\hspace{4truecm}&& \\&=& \int G_b({\x}-u)\Bigl(V({\x})-\Bigl(V(u)+\mfr{1}{4b}\Delta V(u) \\&&{}+(1-h^2ab)\nabla V(u) \cdot ({\x}-u)\Bigr) \Bigr)du. \end{eqnarray*} Using Taylors' formula we have $$ V({x})=V(u)+\nabla V(u)\cdot ({x}-u) +{\textstyle\frac{1}{2}}\sum_{ij}\partial_i\partial_j V({x})({x}_i-u_i)({x}_j-u_j)-{\cal R}_1({x},u) $$ where $$ {\cal R}_1({x},u)={\textstyle\frac{1}{2}}\int_0^1\sum_{i,j,k} \partial_i\partial_j\partial_kV(u+t(x-u)) ({x}_i-u_i)({x}_j-u_j)({x}_k-u_k)(1-(1-t)^2)dt. $$ Since $\int x_ix_jG_b(x)dx=\frac{1}{2b}\delta_{ij}$ we have \begin{eqnarray*} \lefteqn{V({\x}) -\int {\cal G}_{u,q} \left(V(u)+\mfr{1}{4b}\Delta V(u)+\nabla V(u) \cdot ({\x}-u) \right){\cal G}_{u,q}\frac{du dq}{(2\pi h)^n}}&& \\&=&\int G_b({\x}-u) \Bigl(\mfr{1}{4b}\left(\Delta V({\x})-\Delta V(u) \right) +h^2ab\nabla V(u)\cdot ({\x}-u)-{\cal R}_1({\x},u) \Bigr)du\\ &=&\int G_b({\x}-u) \left(\mfr{1}{4b}\left(\Delta V({\x})-\Delta V(u) \right) -\mfr{1}{2}h^2a\Delta V(u)-{\cal R}_1({\x},u) \right)du \end{eqnarray*} where the last identity follows by integration by parts. The theorem now follows easily since $a\leq b$ and $$ \Delta V({x})-\Delta V(u)=\int_0^1\sum_i\partial_i\Delta V(u+t({x}-u))({x}_i-u_i)dt. $$ \end{proof} \section{Appendix: A localization theorem} \begin{thm} \label{partition} Consider $\phi\in C^\infty_0(\mathbb R^n)$ with support in the ball $\{|x|\leq1\}$ and satisfying $\int\phi^2(x)\,dx=1$. Assume that $\ell:\R^n\to \R$ is a $C^1$ map satisfying $0<\ell(u)\leq1$ and $\|\nabla\ell\|_\infty<1$. Let $J(x,u)$ be the Jacobian of the map $u\mapsto \frac{x-u}{\ell(u)}$, i.e. $$ J(x,u)= \ell(u)^{-n}\left|\det\left[\frac{(x_i-u_i)\partial_j\ell(u)}{\ell(u)} + \delta_{ij}\right]_{ij}\right|. $$ We set $\phi_{u}(x):=\phi\Big(\frac{x-u}{\ell(u)}\Big)\sqrt{J(x,u)} \ell(u)^{n/2}$. Then, for all $x\in \R^n$ \begin{equation}\label{eq:philoc} \int_{\R^n} \phi_{u}^2(x)\ell(u)^{-n}\, du = 1 \end{equation} and for all multi-indices $\alpha$ we have \begin{equation}\label{eq:phiuestimate} \|\p^\alpha\phi_{u}\|_\infty \leq \ell(u)^{-|\alpha|} C_\alpha\max_{|\beta|\leq|\alpha|}\|\p^\beta\phi\|_\infty, \end{equation} where $C_\alpha$ depends only on $\alpha$. \end{thm} \begin{proof} In order to prove (\ref{eq:philoc}) it is of course enough to consider the case $x=0$. The identity follows from the change of variables formula if we can show that the map $F:\R^n\to\R^n$ given by $F(u)=-u/\ell(u)$ is a bijection of $F^{-1}\left(\{|x|\leq1\}\right)$ onto $\{|x|\leq1\}$. The map is, in fact, onto $\R^n$ since $F(0)=0$ and $|F(u)|\geq |u|$. Hence for all $u\in \R^n$ there exists $t\in\R$ with $-1\leq t\leq 0$ such that $F(tu)=u$. That the map is also injective on $F^{-1}\left(\{|x|\leq1\}\right)$ follows since for $u\ne0$ we may write $F(tu)=-g(t)u$ and the map $g:\mathbb R\to\mathbb R$ is monotone increasing for all $t$ for which $|g(t)||u|\leq1$. In fact, $g(t) = t/\ell(tu)$ and thus \begin{eqnarray*} g'(t)&=&\ell(tu)^{-1}-t\ell(tu)^{-2}\nabla\ell(tu)\cdot u =\ell(tu)^{-1}[1-t\ell(tu)^{-1}\nabla\ell(tu)\cdot u]\\ &\geq& \ell(tu)^{-1}[ 1-\|\nabla\ell\|_\infty |g(t)||u|]>0. \end{eqnarray*} Note that $\phi_u(x)=\widetilde{\phi}_u\left(\frac{x-u}{\ell(u)}\right)$, where $$ \widetilde{\phi}_u(x)=\phi(x)\left|\det\left[x_i\p_j\ell(u)+\delta_{ij} \right]_{ij}\right|. $$ The estimates (\ref{eq:phiuestimate}) follow since $ \|\partial^\alpha\widetilde{\phi}_u\|_\infty\leq C\max_{|\beta|\leq|\alpha|}\|\p^\beta\phi\|_\infty. $ \end{proof} \begin{thebibliography}{30} \bibitem{Dirac} P.A.M.~Dirac: {\em Note on exchange phenomena in the Thomas-Fermi atom}, Proc.~Cambridge Phil.~Soc., {\bf 26}, 376--385 (1930) \bibitem{Fefferman-Seco} C.~Fefferman and L.A.~Seco: {\em On the energy of a large atom}, Bull.~AMS, {\bf 23}, 2, 525--530 (1990) \bibitem{Hughes} W.~Hughes: {\em An atomic energy bound that gives Scott's correction}, Adv.~Math., {\bf 79}, 213--270 (1990) \bibitem{Ivrii-Sigal} V.I.~Ivrii and I.M.~Sigal: {\em Asymptotics of the ground state energies of large Coulomb systems}, Ann.~of Math., (2) {\bf 138}, 243--335 (1993) \bibitem{Lieb-Thirring} E.H.~Lieb and W.E.~Thirring: {\em Inequalities for the moments of the eigenvalues of the Schr\"odinger Hamiltonian and their relation to Sobolev inequalities,} in {\em Studies in mathematical physics}, (E.~Lieb, B.~Simon, and A.~S.~Wightman, eds.), Princeton Univ. 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