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Sobolev inequalities; Schroedinger operator; Lieb-Thirring inequalities
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\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}

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\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*}


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\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}


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\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\'e� 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}

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