Content-Type: multipart/mixed; boundary="-------------0708081031802" This is a multi-part message in MIME format. ---------------0708081031802 Content-Type: text/plain; name="07-192.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-192.keywords" Sobolev inequalities; Schroedinger operator; Lieb-Thirring inequalities ---------------0708081031802 Content-Type: application/x-tex; name="dll240707.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="dll240707.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[11pt]{article} \usepackage{amsmath,amsfonts,amsthm,amssymb,setspace} \setlength{\textwidth}{125mm} \setlength{\textheight}{195mm} \newcommand\Tr{{\rm Tr\,}} \newtheorem{thm}{Theorem} \newtheorem{rem}{Remark} \begin{document} \newcommand{\email}[1]{{\sl E-mail:\/} {\texttt{#1}}} \newcommand{\version}{July 24, 2007} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{Lieb-Thirring inequalities with improved constants} \author{Jean Dolbeault, Ari Laptev, and Michael Loss} \date{} \maketitle \thispagestyle{empty} \begin{abstract} Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one-dimension. This allow us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schr\"odinger operators. \end{abstract} \noindent\begin{minipage}{125mm}\linespread{0.9}\selectfont{\small {\sl Key-words:\/} Sobolev inequalities; Schr\"odinger operator; Lieb-Thirring inequalities. {\sl MSC (2000):\/} Primary: 35P15; Secondary: 81Q10}\end{minipage} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} Let $H$ be a Schr\"odinger operator in $L^2(\mathbb{R}^d)$ % \begin{equation}\label{H} H = -\Delta - V \end{equation} % For a real-valued potential $V$ we consider Lieb-Thirring inequalities for the negative eigenvalues $\{\lambda_n\}$ of the operator $H$ % \begin{equation}\label{LTh} \sum |\lambda_n|^\gamma \le L_{d,\gamma} \int_{\mathbb{R}^d} V_+^{d/2+\gamma}(x) \, dx\,, \end{equation} % where $V_+ = (|V| + V)/2$ is the positive part of $V$. Eden and Foias have obtained in \cite{EF} a version of a one-dimensional generalised Sobolev inequality which gives best known estimates for the constants in the inequality \eqref{LTh} for $1\le\gamma<3/2$. The aim of this short article is to extend the method from \cite{EF} to a class of matrix-valued potentials. By using ideas from \cite{LW} this automatically improves on the known estimates of best constants in \eqref{LTh} for multidimensional Schr\"odinger operators. Lieb-Thirring inequalities for matrix-valued potentials for the value $\gamma=3/2$ were obtained in \cite{LW} and also in \cite{BL}. Here we state a result corresponding to $\gamma=1$. %------------------------------------------------------------------------------- \begin{thm}\label{1D-matrix-LTh} Let $V\ge 0$ be a Hermitian $m\times m$ matrix-function defined on~$\mathbb R$ and let $\lambda_n$ be all negative eigenvalues of the operator \eqref{H}. Then % \begin{equation}\label{Th1} \sum |\lambda_n| \le \frac{2}{3\sqrt 3} \int_{\mathbb R}\Tr\left[V^{3/2}(x)\right] \, dx\,. \end{equation} % \end{thm} %------------------------------------------------------------------------------- \begin{rem} The constant $\frac 2{3\sqrt3}$ should be compared with the Lieb-Thirring constant found in \cite{LT} for a class of single eigenvalue potentials and with the constant obtained in \cite{HLW} which is twice as large as the semi-classical one % \begin{equation*} \frac 4{3\sqrt 3\,\pi}<\frac{2}{3\sqrt 3}<2\times \frac{2}{3\pi} = 2\times \frac 1{2\pi} \int_{\mathbb R} (1-\xi^2)_+\,d\xi\,. \end{equation*} % This is about $0,2450\dots<0,3849\dots <0,4244\dots$. \end{rem} \begin{rem} Note that the values of the best constants for the range $1/2<\gamma<3/2$ remain unknown. \end{rem} Let $\mathcal A(x) = (a_1(x),\dots, a_d(x))$ be a magnetic vector potential with real valued entries $a_k\in L^2_{\rm loc}(\mathbb R^d)$ and let % \begin{equation*}\label{H(A)} H(\mathcal A) = (i\,\nabla + \mathcal A)^2 -V\,, \end{equation*} % where $V\ge 0 $ is a real-valued function. Denote the ratio of $2/3\sqrt3$ and the semi-classical constant by % \begin{equation*} R := \frac{2}{3\sqrt 3} \times \left(\frac 2{3\pi}\right)^{-1} = 1.8138 \dots. \end{equation*} By using the Aizenmann-Lieb argument \cite{AL}, a ``lifting" with respect to dimension \cite{LW}, \cite{HLW}, and Theorem \ref{1D-matrix-LTh} we obtain the following result: %------------------------------------------------------------------------------- \begin{thm}\label{d-LTh} The negative eigenvalues of the operator $H(\mathcal A)$ satisfy inequalities % \begin{equation*} \sum |\lambda_n|^\gamma \le L_{d,\gamma} \int_{\mathbb R^d} V^{d/2 + \gamma}(x) \, dx\,, \end{equation*} % where % \begin{equation*} L_{d,\gamma} \le R\times L_{d,\gamma}^{cl} = R\times \frac 1{(2\pi)^d} \int_{\mathbb R^d} (1-|\xi|)_+^\gamma\, d\xi\,. \end{equation*} % \end{thm} %------------------------------------------------------------------------------- \begin{rem} Theorem \ref{d-LTh} allows us to improve on the estimates of best constants in Lieb-Thirring inequalities for Schr\"odinger operators with complex-valued potentials recently obtained in \cite{FLLS}. \end{rem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{One-dimensional generalised Sobolev inequality for matrices} Let $\{\phi_n\}_{n=0}^N$ be an ortho-normal system of vector-functions in $L^2(\mathbb R, \mathbb C^M)$, $M\in\mathbb N$, % \begin{equation*} (\phi_n,\phi_m) = (\phi_n,\phi_m)_{L^2(\mathbb R, \mathbb C^M)} = \sum_{j=1}^M \int_{\mathbb R} \phi_n(x,j)\,\overline{\phi_m(x,j)}\, dx = \delta_{nm}\,, \end{equation*} % where $\delta_{nm}$ is the Kronecker symbol. Let us introduce a $M\times M$ matrix $U$ with entries % \begin{equation*} u_{j,k}(x,y) = \sum_{n=0}^N \phi_n(x,j)\,\overline{\phi_n(y,k)}\,. \end{equation*} % Clearly % \begin{equation}\label{P*=P} U^*(x,y) = U(y,x)\,. \end{equation} % The fact that the functions $\phi_n$ are orthonormal can be written in a compact form % \begin{equation}\label{PP=P} \int_{\mathbb R} U(x,y)\,U(y,z)\, dy = U(x,y)\,. \end{equation} % The latter two properties \eqref{P*=P} and \eqref{PP=P} prove that $U(x,y)$ could be considered as the integral kernel of an orthogonal projection $P$ in $L^2(\mathbb R, \mathbb C^M)$ whose image is the subspace of vector-functions spanned by $\{\phi_n\}_{n=1}^N$. %------------------------------------------------------------------------------- \begin{thm}\label{1D-Sobolev} Let us assume that the vector-function $\phi_n$, $n=1,2,\dots N,$ are from the Sobolev class $H^1(\mathbb R, \mathbb C^M)$. Then % \begin{equation*}\label{1d-sobolev} \int_{\mathbb R} \Tr\Big[U(x,x)^3\Big]\,dx \le \sum_{n=1}^N \sum_{j=1}^M\, \int_{\mathbb R} |\phi_n'(x,j)|^2\,dx\,. \end{equation*} \end{thm} %------------------------------------------------------------------------------- \begin{proof} % \begin{multline}\label{UUU} \frac{d}{dy}\, \Tr\Big[U(x,y)\,U(y,x)\, U(x,x)\Big]\\ = \Tr\! \left[\Big(\frac{d}{dy}\, U(x,y)\Big)\,U(y,x)\, U(x,x)\right] + \Tr\! \left[U(x,y)\,\Big( \frac{d}{dy}\, U(y,x)\Big)\,U(x,x)\right] \end{multline} % By integrating \eqref{UUU} and taking absolute values one obtains % \begin{multline*} \frac 12\,\Tr\Big[ U(x,z)\,U(z,x)\,U(x,x)\Big]\\ \le \frac12\, \int_{-\infty}^z \Big |\,\Tr\Big[ \Big(\frac{d}{dy}\, U(x,y)\Big)\,U(y,x)\, U(x,x)\Big]\\ + \Tr\Big[U(x,y)\,\Big( \frac{d}{dy}\, U(y,x)\Big)\,U(x,x)\Big] \Big |\, dy \end{multline*} % and % \begin{multline*} \frac 12\,\Tr\Big[ U(x,z)\,U(z,x)\,U(x,x)\Big]\\ \le \frac12\,\int_z^\infty \Big|\, \Tr\Big[ \Big(\frac{d}{dy}\, U(x,y)\Big)\,U(y,x)\, U(x,x)\Big]\\ + \Tr\Big[ U(x,y)\,\Big( \frac{d}{dy}\, U(y,x)\Big)\,U(x,x)\Big]\Big|\, dy\,. \end{multline*} % Taking absolute values and adding the two inequalities yields for any $z \in \mathbb{R}$ % \begin{multline} \label{int++UUU} \Big|\,\Tr\Big[ U(x,z)\,U(z,x)\,U(x,x)\Big]\Big|\\ \le \frac12 \, \int_{\mathbb R} \left|\,\Tr\left[\left(\frac{d}{dy}\, U(x,y)\right) U(y,x)\, U(x,x)\right]\right|\, dy\\ + \frac12 \, \int_{\mathbb R} \left|\,\Tr\left[ U(x,y) \left(\frac{d}{dy}\,U(y,x) \right) U(x,x)\right]\right|\, dy\,. \end{multline} % Note that we have reproved Agmon's inequality $$ |f(x)|^2 \le \int_{\mathbb R} |f(y)\,f'(y)|\, dy $$ for traces of matrices. By using properties of traces, the Cauchy-Schwarz inequality for matrix-functions and also properties \eqref{P*=P} and \eqref{PP=P}, we find that for all $z \in \mathbb{R}$ % \begin{multline*} \left(\int_{\mathbb R} \left|\,\Tr\left[ \left(\frac{d}{dy}\, U(x,y)\right)\,U(y,x)\,U(x,x)\right]\right|\, dy\right)^2\\ \le \int_{\mathbb R} \Tr\left[\frac{d}{dy}\, U(x,y)^*\, \frac{d}{dy}\, U(x,y)\right] dy\; \int_{\mathbb R} \Tr \left[U(x,y)^*\,U^2(x,x)\,U(x,y)\right] dy\\ = \int_{\mathbb R} \Tr \left[\frac{d}{dy}\, U(y,x)\, \frac{d}{dy}\, U(x,y)\right] dy\; \int_{\mathbb R} \Tr \left[U^2(x,x)\,U(x,y)\,U(y,x)\right] dy\\ =\int_{\mathbb R} \Tr \left[\frac{d}{dy}\, U(x,y)\,\frac{d}{dy}\, U(y,x)\right] dy\;\;\Tr \left[U(x,x)^3\right] , \end{multline*} % and similarly % \begin{multline*} \left(\int_{\mathbb R} \left|\,\Tr\left[ U(x,y)\,\frac{d}{dy}\,U(y,x)\,U(x,x)\right]\right|\, dy\right)^2\\ \le \int_{\mathbb R} \Tr \left[\frac{d}{dy}\,U(x,y)\, \frac{d}{dy}\,U(y,x)\right] dy\;\;\Tr \left[U(x,x)^3\right]\,. \end{multline*} % Thus, using this, and setting $x=z$ in \eqref{int++UUU}, we arrive at % \begin{equation*} \Big|\,\Tr\Big[ U(x,x)^3\Big] \Big| \le \int_{\mathbb R} \Tr \left[\frac{d}{dy}\,U(x,y)\, \frac{d}{dy}\,U(y,x)\right]\, dy \,. \end{equation*} % Integrating with respect to $x$ we finally obtain % \begin{multline*} \int_{\mathbb R} \Big|\,\Tr\Big[ U(x,x)^3\Big] \Big|\,dx\\ \le \sum_{n,k=1}^N\sum_{i,j=1}^M \int_{\mathbb R} \int_{\mathbb R} \phi_n(x,i)\, \overline{\phi_n'(y,j)}\; \phi_k'(y,j)\, \overline{\phi_k(x,i)} \, dx\,dy\\ = \sum_{n=1}^N\sum_{j=1}^M \int_{\mathbb R} |\phi_n'(x,j)|^2\, dx\,, \end{multline*} % which completes the proof. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Lieb-Thirring inequalities for Schr\"odinger operators with matrix-valued potentials} Let us assume that $V\in C_0^\infty(\mathbb R)$, $V\ge0$, be a $M\times M$ Hermitian matrix-valued potential with entries $\{v_{ij}\}_{i,j=1}^M$. Then the negative spectrum of the Schr\"odinger operator $H = -\frac{d^2}{dx^2} - V$ in $L^2(\mathbb R)$ is finite. Denote by $\{\phi_n\}$ an ortho-normal system eigen-vector functions corresponding to the eigenvalues $\{\lambda_n\}_{n=1}^N$ % \begin{equation*} -\frac{d^2}{dx^2}\,\phi_n - V\phi_n = -\lambda_n\,\phi_n\,. \end{equation*} % Clearly, % \begin{equation*} -\sum_n \lambda_n = \sum_{n,j} \int_{\mathbb R} |\phi_n'(x,j)|^2 \, dx - \Tr\left[ \int_{\mathbb R}V(x)\,U(x,x)\, dx\right] \end{equation*} % and by H\"older's inequality for traces, % \begin{equation*} \int_{\mathbb R} \Tr\left[ V(x)\,U(x,x)\right]\, dx \le \left(\int_{\mathbb R}\Tr\big[ V^{3/2}(x)\big]\, dx \right)^\frac 23 \!\left(\int_{\mathbb R} \Tr\left[ U(x,x)^3\right]\, dx\right)^\frac 13\!, \end{equation*} % so that using Theorem \ref{1D-Sobolev} % \begin{equation*} -\sum_n \lambda_n \ge X - \left(\int_{\mathbb R}\Tr\left[ V^{3/2}(x)\right]\, dx \right)^\frac 23 \,X^\frac 13 \end{equation*} % with $ X := \int_{\mathbb R} \Tr\left[ U(x,x)^3\right] dx$. Minimising the right hand side with respect to $X$ we finally complete the proof of Theorem \ref{1D-matrix-LTh} % \begin{equation*} -\sum_n \lambda_n \ge -\frac{2}{3\sqrt3} \int_{\mathbb R}\Tr\left[V^{3/2}(x)\right]\, dx\,. \end{equation*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent\begin{minipage}{125mm}\linespread{0.9}\selectfont {\it Acknowledgements.} {\small The authors are grateful to the organisers of the meeting \lq\lq Functional Inequalities: Probability and PDE's", Universit\'e Paris-X, June 4-6, 2007, where this paper came to fruition. A.L. thanks the Department of Mathematics of the University Paris Dauphine for its hospitality and also the ESF Programme SPECT. M.~L. would like to acknowledge partial support through NSF grant DMS-0600037.}\end{minipage} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{9} \linespread{0.9} \selectfont\parskip=0pt % \bibitem{AL} Aizenman M. and Lieb E.H.: On semi-classical bounds for eigenvalues of Schr\"odinger operators. Phys. Lett. {\bf 66A}, 427-429 (1978). % \bibitem{BL} Benguria R and Loss M.: {\it A simple proof of a theorem by Laptev and Weidl.} Math. Res. Lett., {\bf 7} (2000), 195--203. % \bibitem{EF} Eden A. and Foias C.: {\it A simple proof of the generalized Lieb-Thirring inequalities in one-space dimension}. J. Funct. Anal. , \textbf{162} (1991), 250-254. % \bibitem{FLLS} Frank R.L., Laptev A. Lieb E.H. and Seiringer R.: {\it Lieb-Thirring inequalities for Schr\"odinger operators with complex-valued potentials}. Lett. Math. Phys. {\bf 77} (2006), 309-316. % \bibitem{HLW}Hundertmark D., Laptev A. and Weidl T.: {\it New bounds on the Lieb-Thirring constants}. Inv. Math. {\bf 140} (2000), 693-704. % \bibitem{LW}Laptev A., Weidl T.: {\it Sharp Lieb-Thirring inequalities in high dimensions.} Acta Math., \textbf{184} (2000), 87-111. % \bibitem{LT}Lieb E.H. and Thirring, W.: {\it Inequalities for the moments of the eigenvalues of the Schr\"{o}dinger Hamiltonian and their relation to Sobolev inequalities.} Studies in Math. Phys., Essays in Honor of Valentine Bargmann., Princeton (1976), 269-303. % \end{thebibliography} \small \noindent {\sc J. Dolbeault:} Ceremade UMR CNRS no. 7534, Universit\'eŽ Paris Dauphine, F-75775 Paris Cedex 16, France. \email{dolbeaul@ceremade.dauphine.fr} \par\smallskip\noindent {\sc A. Laptev:} Department of Mathematics, Imperial College London, London SW7 2AZ, UK, Royal Institute of Technology, 100 44 Stockholm, Sweden. \email{a.laptev@imperial.ac.uk} \par\smallskip\noindent {\sc M. Loss:} School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332-0160, USA. \email{loss@math.gatech.edu} \begin{flushright}{\sl\version}\end{flushright} \end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ---------------0708081031802--