Content-Type: multipart/mixed; boundary="-------------0210022121435" This is a multi-part message in MIME format. ---------------0210022121435 Content-Type: text/plain; name="02-412.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-412.comments" To appear in Proceedings of the UAB 2002 Int'l Conference on Differential Equations and Mathematics Physics ---------------0210022121435 Content-Type: text/plain; name="02-412.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-412.keywords" Scott correction, semi-classical analysis, coherent states ---------------0210022121435 Content-Type: application/x-tex; name="rev2.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="rev2.tex" %----------------------------------------------------------------------- % Beginning of article.tex %----------------------------------------------------------------------- % % AMS-LaTeX 1.2 sample file for book proceedings, based on amsproc.cls. % % Replace amsproc by the documentclass for the target series, e.g. pspum-l. % \documentclass{conm-p-l} \usepackage{amsthm,amsfonts,latexsym,amssymb %showkeys } \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\R{\mathbb R} \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{\Tr}{\mbox{\rm Tr}} \newcommand{\mfr}[2]{{\textstyle\frac{#1}{#2}}} \newcommand{\V}{V^{\rm TF}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} % Absolute value notation \newcommand{\abs}[1]{\lvert#1\rvert} % Blank box placeholder for figures (to avoid requiring any % particular graphics capabilities for printing this document). \newcommand{\blankbox}[2]{% \parbox{\columnwidth}{\centering % Set fboxsep to 0 so that the actual size of the box will match the % given measurements more closely. \setlength{\fboxsep}{0pt}% \fbox{\raisebox{0pt}[#2]{\hspace{#1}}}% }% } \begin{document} \title{New coherent states and a new proof of the Scott correction} % Information for first author \author{Jan Philip Solovej} % Address of record for the research reported here \address{Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark} % Current address %\curraddr{} \email{solovej@math.ku.dk} \thanks{ \copyright 2002 \ by the authors. This article may be reproduced in its entirety for non-commercial purposes.} % Information for second author \author{Wolfgang L Spitzer} \address{Department of Mathematics, University of California, One Shields Avenue, Davis 95616-8366, USA} \email{spitzer@math.ucdavis.edu} %\thanks{Support information for the second author.} % General info \subjclass{81Q20, 35P20} \date{August 31, 2002} %\dedicatory{This paper is dedicated to xxx} \keywords{Scott correction, semi-classical analysis, coherent states} \begin{abstract} We introduce new coherent states and use them to prove semi-classical estimates for Schr\"odinger operators with regular potentials. This can be further applied to the Thomas-Fermi potential yielding a new proof of the Scott correction for molecules. \end{abstract} \maketitle \section{Introduction} In this paper we review a novel proof of the Scott correction for neutral mol\-ecules. So suppose, we have $M$ nuclei of positive charges $Z=(Z_1,\ldots, Z_M)\in\R_+^M$ located at positions $R=(R_1,\ldots,R_M)\in\R^{3N}$. We choose the charge of an electron equal to $-1$, so that neutrality is expressed as $|Z|=\sum_{j=1}^M Z_j = N$, where $N$ is the number of electrons. Further, we use atomic units where $\hbar^2=m$. The interaction of a single electron with all the nuclei is equal to \begin{equation}\label{V} V(Z,R,x) = \sum_{j=1}^M \frac{Z_j}{|x-R_j|} . \end{equation} We now write the molecular non-relativistic Schr\"odinger operator in the form \beax\lefteqn{H(Z,R)=H(Z_1,\ldots,Z_M;R_1,\ldots, R_M) } \\ &=&\sum_{i=1}^N\left(-\mfr{1}{2}\D_i - V(Z,R,x_i)\right) +\sum_{1\le i0$ and $R=|Z|^{-1/3}(r_1,\ldots,r_M)$, with $|r_i - r_j|>r_0>0$, for all $i\ne j$. Then, \begin{equation} E(Z,R) = E^{\mbox\tiny TF} (Z,R) + \mfr{1}{2} \sum_{1\le j\le M} Z_j^2 + {\mathcal O} (|Z|^{2-1/30}), \end{equation} as $|Z|\to\infty$, where the error term ${\mathcal O} (|Z|^{2-1/30})$ besides $|Z|$ depends only on $z_1,\ldots,z_M$, and $r_0$. %The Thomas-Fermi constant, $a^{\mbox\tiny TF}$, equals %$E^{\mbox\tiny TF}(|Z|^{-1}Z, |Z|^{1/3}R)$. \end{theorem} %The proof is established in the last Section. The leading Thomas-Fermi (TF) term, which is of the order $|Z|^{7/3}$ was first rigorously derived in the seminal work by Lieb and Simon~\cite{Lieb-Simon} using the Dirichlet-Neumann bracketing method. The Scott correction, i.e., the term $\mfr{1}{2} \sum_{1\le j\le M} Z_j^2$ was proven by Hughes~\cite{Hughes} (a lower bound), and by Siedentop and Weikard~\cite{Siedentop-Weikard} (both bounds) in the case of atoms. The atomic case is simpler since in TF theory atoms are spherically symmetric. Bach~\cite{Bach} proved the Scott correction for ions. Finally, Ivrii and Sigal~\cite{Ivrii-Sigal} accomplished a proof of the Scott correction for molecules, which was recently extended to matter by Balodis Matesanz~\cite{Matesanz}. Here, we present another proof for molecules. It was later shown by Lieb~\cite{Lieb1} (and independently by Thirring~\cite{Thirring}) how coherent states can be used to give a simple proof of the leading TF term with good upper and lower bounds; see also a recent improvement by Balodis Matesanz and Solovej~\cite{Matesanz-Solovej}. We want to stress that in order to prove an asymptotic expansion for $E(Z,R)$ capturing the Scott term one basically needs to prove a local trace formula for regular potentials (see Theorem~\ref{corollary} with $n=3$) up to the order $h^{-2+\e}$ where $\e$ is any positive number. We accomplish $\e=1/5$. A quick explanation for the $Z^2$-correction goes as follows. Whereas the leading TF term comes from the bulk of electrons, the correction comes {\it only} from electrons close to the nuclei where the Coulomb attraction is unscreened by the presence of the other electrons. From the exact solution of the hydrogen atom one may extract the Scott correction (see~\cite{Lieb1}). Notice, that the Scott correction for molecules is just the sum of the corresponding atomic corrections. This is not the case for the leading term. This review is organized as follows. In Section 2 we recall the main analytic tools and state the main properties of the TF potential. We introduce the new coherent states in Section 3. In Section 4 we sketch the proof of the semi-classical estimates on the sum of negative eigenvalues for regular and the TF potential. In the last Section we present the proof of the main Theorem~\ref{main theorem}. For more details we refer to our paper~\cite{SS}. \section{Preliminaries} \subsection{Some Inequalities} Here we collect the main inequalities which we need in this paper. Various constants are typically denoted by the same letter $C$, and in all cases sharp constants do not play a role. 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 a multiplication operator. If $\psi\in\bigotimes_{i=1}^N L^2(\R^3\times\{-1,1\})$ is an $N$-body wave-function, then 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. %see \cite{Lieb-encyclopediaLT} for a review. \begin{theorem}[Lieb-Thirring inequality] \label{Lieb-Thirring} \begin{description} \item[One-body case] Let $\c$ be a \\ density operator on $L^2(\R^n)$, then we have the Lieb-Thirring (LT) inequality \begin{equation}\label{LTdensity} \Tr\left[-\mfr{1}{2}\Delta \gamma\right]\geq K_n\int\rho_\gamma^{1+2/n}, \end{equation} where $K_n$ is some positive constant. Equivalently, let $V\in L^{1+n/2}(\mathbb R^n)$ and $\c$ a density operator, then \begin{equation}\label{LT} \Tr [(-\mfr{1}{2}\D + V)\c] \ge -L_n \int |V_-|^{1+n/2}, \end{equation} where $x_- := \min\{x,0\}$, and $L_n$ some positive constant. \item[Many-body case] Let $\psi\in \bigwedge_{i=1}^N L^2(\R^{3}\times\{-1,1\})$. 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{description} \end{theorem} The original proofs of these inequalities can be found in \cite{Lieb-Thirring}. From the min-max principle it is clear that the right hand 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 for the Coulomb energy: $$D(f)=D(f,f)=\frac{1}{2}\int\!\!\int \bar{f}(x)|x-y|^{-1} f(y)\, dx dy . $$ It is not difficult to see (by Fourier transformation) that $\|f\| := D(f)^{1/2}$ is a norm. %\end{document} \begin{theorem}[Hardy-Littlewood-Sobolev inequality] There exists a constant $C$ such that \begin{equation} \label{Hardy-Littlewood-Sobolev} D(f)\le C\,\| f \|_{6/5}^2. \end{equation} \end{theorem} The sharp constant $C$ has been found by Lieb \cite{Lieb:sob}, see also \cite{Lieb-Loss}. Finally, we state the two inequalities which we shall 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 the electrostatic inequality providing us with a lower bound. This inequality is due to Lieb \cite{Lieb3}, and was improved in \cite{Lieb-Oxford}. \begin{theorem}[Lieb-Oxford inequality] Let $\psi\in L^2(\R^{3N})$ be normalized, and $\rho_\psi$ its one-electron density. Then, \begin{equation}\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{theorem} 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), \\ \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 $h^{-1}{\mathbf r}$ (and likewise for $h^{3}{\mathbf z}$) we mean that each coordinate is scaled by $h^{-1}$. 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 state 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 $x=r_k$. We have the following estimate for the TF potential. \begin{theorem}[Estimate on TF potential]\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 depends on $z_1,\ldots,z_M$, and $M$. \end{theorem} The relation of TF theory to semi-classical analysis is that the semi-classical density of a gas of non-interacting electrons moving in the TF potential is simply the TF density. More precisely, the semi-classical approximation to the density of the projection onto the eigenspace corresponding to negative eigenvalues of the Hamiltonian $-\frac{1}{2}\Delta-\V$ is $$ 2\int_{\frac{1}{2}p^2-\V({\mathbf z},{\mathbf r},x)\le0}1 \frac{dp}{(2\pi)^3} = 2^{3/2} (3\pi^2)^{-1} (\V)^{3/2}({\mathbf z},{\mathbf r},x)= \rho^{\rm TF}({\mathbf z},{\mathbf r},x). $$ Here, the factor two on the very left is due to the spin degeneracy. Similarly, the semi-classical approximation to the energy of the gas, i.e., to the sum of the negative eigenvalues of $-\frac{1}{2}\Delta-\V$ is \bea\label{eq:sc=tf} 2\int \left(\frac{1}{2}p^2-\V({\mathbf z},{\mathbf r},x)\right)_- \frac{dpdx}{(2\pi)^3}&=& -{\frac {4\sqrt {2}}{15{\pi }^{2}}}\int \,\V({\mathbf z},{\mathbf r},x)^{5/2}dx \\ &=&E^{{\rm TF}}({\mathbf z},{\mathbf r})+D(\rho^{\rm TF}). \nonumber \eea In Section~4 we shall make the semi-classical approximation more precise. \section{New coherent states} Coherent states provide a natural semi-classical description of quantum mechanics. We shall denote these states by $|u,q\rangle$, where $(u,q)\in \R^{2n}$ is a point in phase-space. Their wave-function is given by \begin{equation}\label{old coherent states}\langle x|u,q\rangle = (\pi h)^{-n/4} e^{-(x-u)^2/2h} e^{iqx/h}. \end{equation} Let $\Pi_{u,q} = |u,q\rangle\langle u,q|$ be the projection onto the coherent state $|u,q\rangle$, then they satisfy the completeness condition (in the sense of quadratic forms) \begin{equation}\label{resolution} \int \Pi_{u,q}\, \frac{dudq}{(2\pi h)^n} = {\bf 1}. \end{equation} As functions on phase-space the coherent states are localized on a scale of the order of $h$. We want to broaden this by defining the operator \begin{equation} \label{new coherent states} {\mathcal G}_{u,q}:= \int w(u-u',q-q') \, \Pi_{u',q'}\, du' dq' \end{equation} with $$w(u,q) = \left(\frac{2ha}{1-ha}\right)^n e^{-a/(1-ha)\, (u^2+q^2)}. $$ The new scale is $1/a>h$, which becomes clearer when we look at its kernel, \begin{equation}{\mathcal G}_{u,q}(x,y) = (\pi h)^{-n/2} e^{-a\left(\frac{x+y}{2}-u\right)^2 +iq(x-y)/h -\frac{1}{4h^2a}(x-y)^2} . \label{eq:Gkernel} \end{equation} For simplicity, we have chosen a Gaussian weight, $w$, in the definition of ${\mathcal G}_{u,q}$. We shall use the operators ${\mathcal G}_{u,q}$ as our new coherent states. \footnote{Sometimes (e.g.~see \cite{Lieb1}) it is useful to consider other coherent states where the Gaussian function in (\ref{old coherent states}) is replaced by a general $L^2$ function. Similarly, one could use them to define corresponding generalized coherent states but from a computational point of view the above choice is the simplest. These new coherent states should not be confused with the quantum coherent operators introduced by Lieb and Solovej in \cite{Lieb-Solovej} in order to compare two quantum systems.} Note that if we let $a\to 1/h$ then ${\mathcal G}_{u,q}$ converges to $\Pi_{u,q}$. A straightforward calculation gives the following result. \begin{lemma}[Completeness of new coherent states] These new coherent operators satisfy \begin{equation}\label{eq:resolution} \int {\mathcal G}_{u,q}^2 \,\frac{dudq}{(2\pi h)^n} = {\bf 1} . \end{equation} \end{lemma} This resolution of the identity provides us with a representation of Schr\"odinger operators as phase-space integrals. This will be useful when we prove a lower bound on the sum of the negative eigenvalues of Schr\"odinger operators. \begin{theorem}[Coherent states representation]\label{thm:coherentrepresentation} Consider functions $F$ and $V$ in $C^3(\R^n)$, for which all second and third derivatives are bounded. Let $\s(u,q)=F(q)+V(u)$, then we have for $a<1/h$ and $b=2a/(1+h^2a^2)$ the representation \begin{equation} \label{repr:coherentrepresentation} F(-ih\nabla)+V(\x) = \int \,{\mathcal G}_{u,q}\widehat H_{u,q} {\mathcal G}_{u,q}\frac{du dq}{(2\pi h)^n} + {\mathbf E} \end{equation} as quadratic forms on $C^\infty_0(\R^n)$ with the operator-valued symbol \begin{equation}\label{eq:coherentrepresentation} \widehat H_{u,q}= \s(u,q)+\frac{1}{4b}\D \s(u,q) + \p_u \s(u,q)({\x}-u) + \p_q \s(u,q)(-ih\nabla -q). \end{equation} The error term, ${\mathbf E}$, is a bounded operator with operator norm \begin{equation}\label{error} \|\mathbf E\|\leq Cb^{-3/2}\sum_{|\alpha|=3}\|\partial^\alpha \s\|_\infty+Ch^2b\sum_{|\alpha|=2}\|\partial^\alpha \s\|_\infty . \end{equation} \end{theorem} Starting with the identity (\ref{eq:resolution}), the representation of Schr\"odinger operators as in (\ref{repr:coherentrepresentation}) arises by splitting the product ${\mathcal G}_{u,q}^2$ apart while sandwiching the symbol $\hat{H}_{u,q}$. This operator-valued symbol can be thought of as the first order Taylor expansion of the classical symbol $\s(u,q)$ at $(\hat{x},-ih\nabla)$. Clearly, one could consider higher order expansions but this in not needed here. Also notice that as $a\downarrow 1/h$ the linear term in (\ref{eq:coherentrepresentation}) does not contribute in (\ref{repr:coherentrepresentation}) and one gets the familiar classical approximation $\s(u,q) + \frac{h}{4} \D\s(u,q)$. The representation (\ref{repr:coherentrepresentation}) is symmetric in space and momentum due to the symmetric Gaussian weights in the definition of ${\mathcal G}_{u,q}$. %\end{remark} One major advantage of coherent states is that a positive (upper) symbol leads to a positive operator. This is important when writing down explicit variational states and brings us to consider more generally operators of the form \begin{equation}\label{eq:formf} \int {\mathcal G}_{u,q}\,f({\widehat A}_{u,q})\,{\mathcal G}_{u,q}\,dudq. \end{equation} Here, ${\widehat A}_{u,q}=B_0(u,q)+B_1(u,q)\cdot {\x}-ih B_2(u,q)\cdot\nabla$ is a Hermitian operator which is linear in ${\x}$ and $-ih\nabla$, and $f:\R\to\R$ is any polynomially bounded real function. 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. \section{Proof of semi-classical estimates} \subsection{Regular potentials} The key application of coherent states will be a proof of a semi-classical expansion of the sum of negative eigenvalues of (localized) Schr\"odinger operators. We shall restrict ourselves to localization functions supported in balls. Recall that we use the convention, $x_-=(x)_- = \min\{x,0\}$. %and let us set $\| \psi\|_{C^d}=\sup_{|\alpha|\leq d}\|\partial^\alpha V\|_\infty$. \begin{theorem}[Local 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 the dimension $n$, \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} The constants $C>0$ in the above estimates again depend only on the dimension $n$, the parameters in (\ref{eq:phivdependence}), and the volume of the unit ball in $\R^n$, $\omega_n$. \end{theorem} As mentioned in the Introduction, any power ${\mathcal O}(h^{-n+1+\e})$ with $\e>0$ is sufficient to prove the Scott correction in the main theorem (\ref{main theorem}). The power $\frac{6}{5}$ comes from optimizing the error bound in (\ref{error}) by choosing $b=h^{-\frac{4}{5}}$. \begin{proof}[Sketch of proof] By a simple scaling argument we may restrict ourselves to the unit ball setting $\ell=1$ and the case $f=1$. We start with a sketch of the lower bound. We may assume that $h$ is sufficiently small. Using the representation (\ref{repr:coherentrepresentation}) we have that \begin{eqnarray} {\Tr}[\phi H\phi]_-&\geq& {\Tr}\left[\int \phi \,{\mathcal G}_{u,q}\widehat{H}_{u,q} {\mathcal G}_{u,q}\phi\frac{du dq}{(2\pi h)^n}\right]_-\nonumber \\ &&+ {\Tr}\left[\phi\left(-\epsilon h^2\Delta -C(b^{-3/2}+h^2b)\right) \phi\right]_- \label{eq:LTerror} . \end{eqnarray} Here, $0<\epsilon<1/2$, 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 $\widetilde{\s}(u,q)=(1-\epsilon)q^2+V(u)$. Utilizing the variational principle for the first trace and the LT inequality for the second one we obtain the bound $$ (2\pi h)^n {\Tr}[\phi H\phi]_-\geq \int {\Tr}\left[\phi \,{\mathcal G}_{u,q}\left[\widehat H_{u,q}\right]_- {\mathcal G}_{u,q}\phi\right] {du dq} - C \e^{-n/2}(b^{-3/2}+h^2b)^{1+n/2}. $$ We shall eventually choose $\e=\frac{1}{4}(b^{-3/2}+h^2b)$. Since $\widehat{H}_{u,q}$ is a linear combination of $\x$ and $\nabla$ this operator can be easily rotated into the momentum operator alone (up to some constant term). Then, we conveniently have an expression for the negative part of $\widehat{H}_{u,q}$, and a fortiori, the trace becomes a Gaussian-like integral which we have to estimate. In this integral, the linear function $$ H_{u,q}(v,p)=\widetilde{\s}(u,q)+\frac{1}{4b}\Delta\widetilde{\s}(u,q) +\partial_u\widetilde{\s}(u,q)(v-u) + \partial_q \widetilde{\s}(u,q)(p-q) $$ replaces the operator kernel of $\widehat H_{u,q}$; notice that $(\x,-ih\nabla)$ is simply substituted by $(v,p)$. We can show that if we consider the $u$-integration over $u$ outside the ball $B_2$ of radius 2 then this is bounded below by $-Cb^{-3/2}$. On the other hand, the integration over $B_2$ can be estimated from below by $$\displaystyle\int_{u\in B_2}\phi^2\left(v+h^2ab(u-v)\right) G_b(u-v)G_b(q-p) \left[H_{u,q}(v,p)\right]_- {du dq} dp dv , $$ with $G_b(v)=(b/\pi)^{n/2}\exp(-bv^2)$. Since we are looking for a lower bound we may as well extend the last integral to $\R^n$. Notice that we may now perform the $p$-integration and obtain \begin{eqnarray} (2\pi h)^n {\Tr}[\phi H\phi]_-&\geq&-\frac{2\omega_n}{n+2} (1-\varepsilon)^{-\frac{n}{2}} \int\phi^2\left(v+h^2abu\right) G_b(u)G_b(q)\nonumber \\&&\quad\times \biggl|\left[V(v)+\widetilde{\xi}_v(u,q)- C|u|(b^{-1}+|u|^2)\right]_-\biggr|^{\frac{n}{2}+1} {du dq} dv \nonumber\\&&-C (b^{-3/2}+h^2b),\label{eq:lowerexpansion} \end{eqnarray} where we have introduced the function $$ \widetilde{\xi}_v(u,q) =\frac{1}{4b} \Delta\widetilde{\s}(v,0) -(1-\epsilon)q^2 -\frac{1}{2}\sum_{ij}\partial_i\partial_j V(v)u_iu_j. $$ By expanding we find that \begin{eqnarray*}\lefteqn{ \biggl|\left[V(v)+\widetilde{\xi}_v(u,q)- C|u|(b^{-1}+|u|^2)\right]_-\biggr|^{\frac{n}{2}+1} }\\ &\leq& |V(v)|_-^{\frac{n}{2}+1} +\left(\frac{n}{2}+1\right)|V(v)_-|^{\frac{n}{2}}\widetilde{\xi}_v(u,q)\\ && +C\left(\left|\widetilde{\xi}_v(u,q)\right|+ C|u|(b^{-1}+|u|^2\right)^2\\&&+C|u|(b^{-1}+|u|^2). \end{eqnarray*} We have used that since $n\geq 3$, the function $x\mapsto|x_-|^{\frac{n}{2}+1}$ is $C^2(\R^n)$. Hence, \begin{eqnarray*} (2\pi h)^n {\Tr}[\phi H\phi]_-&\geq&-\frac{2\omega_n}{n+2} (1-\varepsilon)^{-\frac{n}{2}} \int\phi^2\left(v+h^2abu\right) G_b(u)G_b(q)\\&& %\phantom{-\frac{2\omega_n}{n+2}} \times\,\left(|V(v)_-|^{\frac{n}{2}+1} + \left(\frac{n}{2}+1\right)|V(v)_-|^{\frac{n}{2}}\widetilde{\xi}_v(u,q) \right) {du dq} dv \\ &&-C (b^{-3/2}+h^2b). \end{eqnarray*} We now expand $\phi^2$ and use the crucial identities for Gaussian integrals, $$ \int \widetilde{\xi}_v(u,q)G_b(u)G_b(q)dudq=0\quad \hbox{ and } \int u\,G_b(u)du=0 . $$ We arrive at the lower bound, \begin{eqnarray*} (2\pi h)^{n}{\Tr}[\phi H\phi]_-&\geq&-\frac{2\omega_n}{n+2} (1-\varepsilon)^{-\frac{n}{2}} \int\phi(v)^2|V(v)|_-^{\frac{n}{2}+1}dv-C(b^{-3/2}+h^2b)\\ &=&(1-\varepsilon)^{-\frac{n}{2}} \int\phi(v)^2\s(v,p)_-dvdp-C(b^{-3/2}+h^2b). \end{eqnarray*} Finally, we choose $a=h^{-4/5}$ and $\epsilon=b^{-3/2}$. Now we come to the upper bound. We shall show here only the construction of the density matrix $\c$. Let $\chi=\chi_{(-\infty,0]}$ be the characteristic function of $(-\infty,0]$ and \beax\lefteqn{\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. \eeax We then define \begin{equation} \label{trial density} \c = \int {\mathcal G}_{u,q}\,\chi\big[\hat{h}_{u,q} \big]\, {\mathcal G}_{u,q}\,\frac{dudq}{(2\pi h)^n} . \end{equation} Since ${\bf 0}\le\chi\big[\hat{h}_{u,q}\big]\le\bf1$ it is obvious that ${\bf 0} \le\c\le{\bf 1}$. The arguments showing that $${\Tr}[\c\phi H\phi] \le (2\pi h)^{-n}\int \,\phi^2(u) \s(u,q)_- du dq + Ch^{-n+6/5} $$ are then very similar to the above calculations for the lower bound, see \cite{SS}. \end{proof} \subsection{Thomas-Fermi potential} In this Section we shall sketch the proof of the Scott correction for the TF potential. \begin{theorem}[Scott corrected semi-classics]\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 \bea\label{eq:main1} \lefteqn{\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| } \nonumber\\ &&\phantom{\Tr[-h^2\D - V^{\rm TF}]_- - (2\pi h)^{-3} \int (p^2 - V^{\rm TF}(u))_- \sum_{k=1}^M z_k^2} \le C h^{-2+\frac{1}{10}}, \eea 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{theorem} \begin{proof}[Sketch of proof] From Theorem \ref{thm:tfestimate} we know that the TF potential has an inverse fourth power law decay far from the nuclei. Thus, a region outside some ball of radius $R$ (which scales with $h$) should contribute little to the sum of negative energies. For this purpose we introduce a first partition of unity. So let us choose \begin{equation}\label{eq:Rchoice} R=h^{-1/2} , \end{equation} and consider functions $\Phi_\pm\in C^\infty(\R^n)$ such that \begin{enumerate} \item $\Phi_-^2+\Phi_+^2=1$, \item $\Phi_-(x)=1$ if $d(x)2R$. \end{enumerate} Denote ${\mathcal I}=(\nabla\Phi_-)^2+(\nabla\Phi_+)^2$. Then ${\mathcal I}$ is supported on a set whose volume is bounded by $CR^3$ (where as before $C$ depends on $M$) and $\|{\mathcal I}\|_\infty\leq CR^{-2}$. Using the standard IMS localization formula and then the LT inequality we find that \beax\lefteqn{ \Tr[-h^2\Delta-V^{\rm TF}]_-} \\ &\ge&\Tr[\Phi_-(-h^2\Delta-V^{\rm TF}-h^2{\mathcal I})\Phi_-]_- +\Tr[\Phi_+(-h^2\Delta-V^{\rm TF}-h^2{\mathcal I})\Phi_+]_- \\ &\ge&\Tr[\Phi_-(-h^2\Delta-V^{\rm TF}-h^2{\mathcal I})\Phi_-]_- -C (h^2R^{-2} + h^{-3}R^{-7}) . \eeax With the chosen $R$ the last term is of the order $h^{-1/2}$. On the support of $\Phi_-$ we want to use the hydrogenic approximation of the TF potential close (of the order of $h$) to the nuclei, and on the rest the semi-classical estimates from the previous Section. Let us introduce the function \begin{equation}\label{eq:ldefinition} \ell(x)=\mfr{1}{2}\Bigl(1+\sum_{k=1}^M(|x-r_k|^2+ h^2)^{-1/2}\Bigr)^{-1} . \end{equation} Note that $\ell$ is a smooth function with $$ 0<\ell(x)<1,\quad\hbox{and}\quad \|\nabla\ell(x)\|_\infty<1. $$ Now, we fix some localization function $\phi\in C_0^\infty(\R^3)$ with support in the unit ball $\{|x|<1\}$ and such that $\int\phi(x)^2dx=1$. It is not difficult (cf Theorem~22, \cite{SS}) to 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 $\phi$. One can show from (\ref{eq:tfdf}) in Theorem~\ref{thm:tfestimate} that for all $u\in\R^n$ with $d(u)>2h$, \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 $M$. This is the requirement for the semi-classical estimates from Theorem \ref{corollary} to apply with $\ell(u) \to \ell, f(u)\to f$. Another application of the IMS formula shows that \begin{eqnarray} \lefteqn{\hspace*{.7cm}\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^{-1/2} .\nonumber \end{eqnarray} By similar arguments we get corresponding estimates for the hydrogenic operators replacing $V^{\rm TF}$ by $\frac{z_k}{|x-r_k|}-1$ in the above estimates. 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{\hspace*{.4cm}\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 arrive at analoguous upper bounds if we utilize the density matrix $$ \gamma=\int_{d(u)<2R+1}\phi_u\chi\left(\phi_u(-h^2\Delta-V^{\rm TF})\phi_u\right)\phi_u \ell(u)^{-3}du $$ as a trial operator. I.e., \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}\lefteqn{\hspace*{.0cm}\label{eq:hydphiup} \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\nonumber. \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}{|\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}{|\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}), and (\ref{eq:tfphiup}) we have \begin{eqnarray} \lefteqn{\Tr[-h^2\Delta-\V]_- - \sum_{k=1}^M\Tr\Bigl[-h^2\Delta-\frac{z_k}{|\x-r_k|}+1\Bigr]_- } \\ &\geq&\int_{d(u)<2R+1} D_+(u)\ell(u)^{-3}du -C h^{-1/2}\label{eq:Dintegral} \nonumber, \end{eqnarray} and from (\ref{eq:phiuprop}) we get \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} Next, we compute explicitly both the quantum and the semi-classical energies for the Coulomb potential, namely $$ \Tr\Bigl[-h^2\Delta-\frac{z_k}{|\x-r_k|}+1\Bigr]_- =\sum_{1\leq n\leq z_k/(2h)}\left(-\frac{z_k^2}{4h^2}+n^2\right) =-\frac{z_k^3}{12 h^3}+\frac{z_k^2}{8h^2}+{\mathcal O}(h^{-1}) , $$ and $$ (2\pi h)^{-3}\int \Bigl(p^2-\frac{z_k}{|u-r_k|}+1\Bigr)_-dudp =-\frac{z_k^3}{12 h^3}. $$ The first statement of the theorem is thus proven once we establish lower bounds on $D_+(u)-D_{\rm SC}(u)$ and $D_-(u)+D_{\rm SC}(u)$. Here, we have to distinguish between the region $d(u)<2h$ and the semi-classical region, $2h