Content-Type: multipart/mixed; boundary="-------------0003281509240" This is a multi-part message in MIME format. ---------------0003281509240 Content-Type: text/plain; name="00-132.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-132.comments" This is an article in the Kluwer Encyclopedia of Mathematics, Supplement vol.II, p. 311-313 (2000) ---------------0003281509240 Content-Type: text/plain; name="00-132.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-132.keywords" Lieb-Thirring inequalities ---------------0003281509240 Content-Type: application/x-tex; name="lieb-thirring8-encyclopedia.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lieb-thirring8-encyclopedia.tex" %Lieb-Thirring Inequalities \documentclass[twocolumn,notitlepage,]{article} \baselineskip=4.5ex %\pagestyle{empty} %\usepackage{rotating} %\setlength{\topmargin}{-1truein} %\setlength{\textwidth}{7truein} %\setlength{\textheight}{9.5truein} %\setlength{\parindent}{8mm} \frenchspacing %\setlength{\oddsidemargin}{-.25truein} %\setlength{\evensidemargin}{-.25truein} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\sr{{\cal R}} \def\1{{\bf 1}} %\def\d{\rm d} \def\x{{\bf x}} \def\A{{\bf A}} \def\Tr{{\rm Tr}} \def\C{{\bf C}} \def\R{{\bf R}} \def\E{{\cal E}} \def\TF{{\rm TF}} \def\MTF{{\rm MTF}} \def\mfr#1/#2{\hbox{${{#1} \over {#2}}$}} \def\const.{{\rm const.}} \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\beqa{\begin{eqnarray}} \def\eeqa{\end{eqnarray}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \setlength{\unitlength}{1.0cm} %\title{\bf \\ %A rigorous derivation %\footnotetext{\copyright 1998 by Elliott H. Lieb} \def\boxit#1{\thinspace\hbox{\vrule\vtop{\vbox{\hrule\kern1pt \hbox{\vphantom{\tt/}\thinspace{\tt#1}\thinspace}}\kern1pt\hrule}\vrule} \thinspace} \font\eightit=cmti8 \def\ve{{\varepsilon}} \def\C{{\bf C}} \def\D{{\rm D}} \def\E{{\rm E}} \def\R{{\bf R}} \def\Z{{\bf Z}} \def\L{{\Lambda}} \def\l{{\lambda}} \def\d{{\rm d}} \def\Q{{{\rm C}_{\Lambda}}} \def\X{{\underline{X}}} \def\Tr{{\rm Tr}} %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% {\bf LIEB-THIRRING INEQUALITIES} -- Inequalities concerning the negative eigenvalues of the {\bf Schr\"odinger operator} $$ H = - \Delta + V(x) $$ on $L^2(\R^n), n \geq 1$. With $e_1 \leq e_2 \leq \cdots < 0$ denoting the negative eigenvalues of $H$ (if any), the Lieb-Thirring inequalities state that for suitable $\gamma \geq 0$ and constants $L_{\gamma,n}$ \begin{equation} \sum_{j \geq 1} |e_j|^{\gamma} \leq L_{\gamma,n} \int_{\R^n} V\_ (x)^{\gamma + n/2} \ \d x \end{equation} with $V\_ (x) := {\rm{max}} \{ - V (x),0 \}$. When $\gamma = 0$ the left side is just the number of negative eigenvalues. Such an inequality (1) can hold if and only if \begin{eqnarray} \gamma \geq \frac{1}{2} & {\rm{for}} & {\rm{n}} = 1 \nonumber \\ \gamma > 0 & {\rm{for}} & {\rm{n}} = 2\\ \gamma \geq 0 & {\rm{for}} & {\rm{n}} \geq 3 \ . \nonumber \end{eqnarray} The cases $\gamma > \frac{1}{2}, n = 1, \gamma > 0, n \geq 2$, were established by E.H. Lieb and W.E. Thirring~\cite{LT} in connection with their proof of {\bf stability of matter}. The case $\gamma = \frac{1}{2}, n = 1$ was established by T. Weidl~\cite{TW}. The case $\gamma = 0$, $n \geq 3$ was established independently by M. Cwikel~\cite{MC}, Lieb~\cite{EL} and G.V. Rosenbljum~\cite{GR} by different methods and is known as the CLR bound; the smallest known value for $L_{0,n}$ is in~\cite{EL},~\cite{EL2}. Closely associated with the inequality (1) is the {\it semi-classical approximation} for $\sum|e|^{\gamma}$, which serves as a heuristic motivation for (1). It is ({\it cf.}~\cite{LT}). \begin{eqnarray*} \sum_{j\geq1} |e|^{\gamma} &\approx& (2 \pi)^{-n} \int_{\R^n \times \R^n} \left[ p^2 + V(x) \right]^{\gamma}_{\_} \d p \d x\\ & =& L_{\gamma,n}^{c} \int_{\R^n} V \_ (x)^{\gamma + n/2} \ \d x \end{eqnarray*} with $$ L_{\gamma,n}^{c} = 2^{-n} \pi^{-n/2} \Gamma (\gamma + 1)/\Gamma (\gamma + 1 + n/2) \ . $$ Indeed, $L_{\gamma, n}^{c} < \infty$ for all $\gamma \geq 0$ whereas (1) holds only for the range given in (2). It is easy to prove (by considering $V(x) = \lambda W (x)$ with $W$ smooth and $\lambda \rightarrow \infty$) that $$ L_{\gamma, n} \geq L_{\gamma, n}^{c} $$ An interesting, and mostly open problem is to determine the sharp value of the constant $L_{\gamma, n}$, especially to find those cases in which $L_{\gamma,n} = L_{\gamma, n}^{c}$. M. Aizenman and Lieb~\cite{AL} proved that the ratio $R_{\gamma,n} = L_{\gamma,n} / L_{\gamma, n}^{c}$ is a monotonically non-increasing function of $\gamma$. Thus, if $R_{\Gamma,n} =1$ for some $\Gamma$ then $L_{\gamma,n} = L_{\gamma,n}^{c}$ for all $\gamma \geq \Gamma$. The equality $L_{\frac{3}{2},n} = L_{\frac{3}{2},n}^c$ was proved for $n=1$ in ~\cite{LT} and for $n>1$ in \cite{LW} by A. Laptev and T. Weidl. (See also \cite{BL}.) The following sharp constants are known: \begin{eqnarray*} L_{\gamma,n} &=& L_{\gamma,n}^{c} \qquad {\rm{all}}~ \gamma \geq 3/2,~\cite{LT},~\cite{AL},~\cite{LW} \\ L_{1/2, 1} &=& 1/2 \qquad \qquad \qquad \cite{HLT}\\ \end{eqnarray*} There is strong support for the conjecture~\cite{LT} that \begin{equation} L_{\gamma,1} = \frac{1}{\sqrt{\pi} (\gamma - \frac{1}{2})} \frac{\Gamma ( \gamma + 1)}{\Gamma (\gamma + 1/2)} \left( \frac{\gamma - \frac{1}{2}}{\gamma + \frac{1}{2}} \right)^{\gamma + 1/2} \end{equation} for $\frac{1}{2} < \gamma < \frac{3}{2}$. Instead of considering all the negative eigenvalues as in (1), one can consider just $e_1$. Then for $\gamma$ as in (2) $$ |e_1|^{\gamma} \leq L_{\gamma, n}^1 ~ \int_{\R^n} V\_(x)^{\gamma + n/2} \d x \ . $$ Clearly, $L_{\gamma, n}^1 \leq L_{\gamma, n}$, but equality can hold, as in the cases $\gamma = 1/2$ and $ 3/2$ for $ n = 1$. Indeed, the conjecture in (3) amounts to $L_{\gamma,1}^1 = L_{\gamma, 1}$ for $1/2 < \gamma < 3/2$. The sharp value (3) of $L_{\gamma, n}^1$ is obtained by solving a differential equation~\cite{LT}. It has been conjectured that for $n \geq 3, L_{0, n} = L_{0, n}^1$. In any case, B. Helffer and D. Robert showed that for all $n$ and all $\gamma < 1$, $L_{\gamma, n} > L_{\gamma, n}^c$. The sharp constant $L_{0, n}^1, n \geq 3$ is related to the sharp constant $S_n$ in the {\bf Sobolev inequality} \begin{equation} \parallel \nabla f \parallel_{L^2(\R^n)} \geq S_n \parallel ~ f ~ \parallel_{L^{2n/(n-2)}(\R^n)} \end{equation} by $L_{0, n}^1 = (S_n)^{-n}$. By a `duality argument'~\cite{LT} the case $\gamma =1$ in (1) can be converted into the following bound for the Laplacian, $\Delta$. This bound is referred to as a Lieb-Thirring kinetic energy inequality and its most important application is to the {\bf stability of matter}~\cite{EL3}, \cite{LT}. Let $f_1, f_2, \ldots$ be {\it any} orthonormal sequence (finite or infinite) in $L^2 (\R^n)$ such that $\nabla f_j \in L^2 (\R^n)$ for all $j \geq 1$. Associated with this sequence is a `density' \begin{equation} \rho(x) = \sum_{j \geq 1} | f_j (x) |^2 \ . \end{equation} Then, with $K_n := n(2/L_{1, n})^{2/n} (n+2)^{-1-2/n} \ ,$ \begin{equation} \sum_{j \geq 1} \int_{\R^n} | \nabla f_j (x)|^2 \d x \geq K_n \int_{\R^n} \rho(x)^{1 + 2/n} \d x \ . \end{equation} This can be extended to {\it antisymmetric} functions in $L^2 (\R^{nN})$. If $\Phi = \Phi (x_1, \ldots , x_N)$ is such a function we define, for $x \in \R^n$, $$ \rho(x) = N \int_{\R^{n(N-1)}} | \Phi (x, x_2, \ldots , x_N)|^2 \d x_2 \ldots \d x_N \ . $$ Then, if $\int_{\R^{nN}} | \Phi |^2 = 1$, \begin{equation} \int_{R^{nN}} | \nabla \Phi|^2 \geq K_n \int_{\R^n} \rho (x)^{1 + 2/n} \d x \ . \end{equation} Note that the choice $\Phi = (N!)^{-1/2} \det f_j(x_k)|^{N}_{j, k=1}$ with $f_j$ orthonormal reduces the general case (7) to (6). If the conjecture $L_{1, 3} = L_{1, 3}^c$ is correct then the bound in (7) equals the {\bf Thomas-Fermi} kinetic energy {\it ansatz}, and hence it is a challenge to prove this conjecture. In the meantime, see ~\cite{EL2}, ~\cite{BS} for the best available constants to date (1998). Of course, $\int (\nabla f)^2 = \int f (-\Delta f)$. Inequalities of the type (7) can be found for other powers of $-\Delta$ than the first power. The first example of this kind, due to I. Daubechies~\cite{ID}, and one of the most important physically, is to replace $-\Delta$ by $\sqrt{-\Delta}$ in $H$. Then an inequality similar to (1) holds with $\gamma + n/2$ replaced by $\gamma + n$ (and with a different $L_{\gamma, n_{1}}$, of course). Likewise there is an analogue of (7) with $ 1 + 2/n$ replaced by $1 + 1/n$. All proofs of (1) (except \cite{HLT} and \cite{TW})actually proceed by finding an upper bound to $N_E (V)$, the number of eigenvalues of $H = -\Delta + V(x)$ that are below $-E$. Then, for $\gamma > 0$, $$ \sum |e|^{\gamma} = \gamma \int_{0}^{\infty} N_E (V) E^{\gamma - 1} \d E $$ Assuming $V = -V\_$ (since $V_+$ only raises the eigenvalues), $N_E(V)$ is most accessible via the positive semidefiniate {\it Birman-Schwinger kernel} ({\it cf.} \cite{BSi}) $$ K_E (V) = \sqrt{V\_}~ ( -\Delta + E)^{-1} \sqrt{V\_} \ . $$ $e < 0 $ is an eigenvalue of $H$ if and only if 1 is an eigenvalue of $K_{|e|}(V)$. Furthermore, $K_E(V)$ is {\it operator} monotone decreasing in $E$, and hence $N_E(V)$ equals the number of eigenvalue of $K_E(V)$ that are greater than 1. An important generalization of (1) is to replace $- \Delta$ in $H$ by $|i \nabla + A (x)|^2$, where $A(x)$ is some arbitrary vector field in $\R^n$ (called a magnetic vector potential). Then (1) still holds but it is not known if the sharp value of $L_{\gamma, n}$ changes. What is known is that all {\it presently} known values of $L_{\gamma, n}$ are unchanged. It is also known that $(- \Delta + E)^{-1}$, as a kernel in $\R^n \times \R^n$, is pointwise greater than the absolute value of the kernel $(| i \nabla + A |^2 + E)^{-1}$. There is another family of inequalities for orthonormal functions, which is closely related to (1) and to the CLR bound~\cite{EL4}. As before, let $f_1, f_2, \ldots , f_N$ be $N$ orthonormal functions in $L^2(\R^n)$ and set \begin{eqnarray*} u_j &=& ( - \Delta + m^2)^{-1/2} f_j\\ \rho (x) &=& \sum_{j = 1}^{N} |u_j (x)|^2 \ . \end{eqnarray*} $u_j$ is a {\bf Riesz potential} ($m=0$) or {\bf Bessel potential} ($m > 0$) of $f_j$. If $n = 1$ and $m > 0$ then, $\rho \in C^{0, 1/2}(\R^n)$ and $\parallel \rho \parallel_{L^{\infty}(\R)} \leq L/m$. If $n = 2$ and $m > 0$ then for all $1 \leq p < \infty \\ \noindent \parallel\rho \parallel_{L^p(\R^2)} \leq B_p m^{-2/p} N^{1/p}.$ If $n \geq 3, p = n/(n-2)$ and $m \geq 0$ (including $m = 0$) then $\parallel \rho \parallel_{L^p(\R^n)} \leq A_n N^{1/p}.$ Here, $L, B_p, A_n$ are universal constants. Without the orthogonality, $N^{1/p}$ would have to be replaced by $N$. Further generalizations are possible~\cite{EL4}. \begin{thebibliography}{99} \bibitem[1]{AL} AIZENMAN, M.A. AND LIEB, E.H.: `On semiclassical bounds for eigenvalues of Schr\"odinger operators', {\it Phys. Lett.} {\bf 66A} (1978), 427-429. \bibitem[2]{BL} BENGURIA, R. AND LOSS, M.: `A simple proof of a theorem of Laptev and Weidl', preprint, 1999. \bibitem[3]{BS} BLANCHARD, PH. AND STUBBE, J.: `Bound states for Schr\"odinger Hamiltonians: phase space methods and applications', {\it Rev. Math. Phys.} {\bf 8} (1996), 503-547. \bibitem[4]{MC} CWIKEL, M.: `Weak type estimates for singular values and the number of bound states of Schr\"odinger operators', {\it Ann. Math.} {\bf 106} (1977), 93-100. \bibitem[5]{ID} DAUBECHIES, I.: `An uncertainty principle for fermions with generalized kinetic energy', {\it Commun. Math. Phys.} {\bf 90} (1983), 511-520. \bibitem[6]{HR} HELFFER, B. AND ROBERT, D.: `Riesz means of bound states and semi-classical limit connected with a Lieb-Thirring conjecture, II', {\it Ann. Inst. Henri Poincar\'e, Sect. Physique Th\'eorique} {\bf 53} (1990), 139-147. \bibitem[7]{HLT} HUNDERTMARK, D., LIEB, E.H. AND THOMAS, L.E.: `A sharp bound for an eigenvalue moment of the one-dimensional Schr\"odinger operator', {\it Adv. Theor. Math. Phys.} {\bf 2} (1998), 719-731. \bibitem[8]{LW} LAPTEV, A. AND WEIDL, T.: `Sharp Lieb-Thirring inequalities in high dimensions', Acta Math., in press, 1999. \bibitem[9]{EL} LIEB, E.H.: `The numbers of bound states of one-body Schr\"odinger operators and the Weyl problem', Vol. 36 of {\it Proc. Symp. Pure Math.,} Amer. Math. Soc., 1980, pp. 241-251. (cf. [3] and [10] for improvements.) \bibitem[10]{EL4} LIEB, E.H.: `An $L^p$ bound for the Riesz and Bessel potentials of orthonormal functions', {\it J. Funct. Anal.} {\bf 51} (1983), 159-165. \bibitem[11]{EL2} LIEB, E.H.: `On characteristic exponents in turbulence', {\it Commun. Math. Phys.} {\bf 92} (1984), 473-480. \bibitem[12]{EL3} LIEB, E.H.: `Kinetic energy bounds and their applications to the stability of matter', in H. HOLDEN and A. JENSEN (eds.): {\it Springer Lecture Notes in Physics} {\bf 345} (1989), 371-382. \bibitem[13]{LT} LIEB, E.H. AND THIRRING W.: `Inequalities for the moments of the eigenvalues of the Schr\"odinger Hamiltonian and their relation to Sobolev inequalities', in E. LIEB, B. SIMON, A. WIGHTMAN (eds.): {\it `Studies in Mathematical Physics}', Princeton University Press, 1976, pp. 269-303. See also W. Thirring (ed.) {\it The Stability of Matter: From Atoms to Stars, Selecta of E.H. Lieb}, Springer, 1997. \bibitem[14]{GR} ROSENBLJUM, G.V.: `Distribution of the discrete spectrum of singular differential operators', {\it Dok. Akad. Nauk SSSR} {\bf 202} (1972), 1012-1015. The details are given in {\it Izv. Vyss. Ucebn. Zaved. Matem.} {\bf 164} (1976), 75-86. (English trans. {\it Sov. Math. (Iz VUZ)} {\bf 20} (1976), 63-71. \bibitem[15]{BSi} SIMON, B.: `Functional integration and quantum physics': Vol. 86 of {\it Pure and Applied Mathematics}, Academic Press, 1979. \bibitem[16]{TW} WEIDL, T.: `On the Lieb-Thirring constants $L_{\gamma, 1}$ for $\gamma \geq 1/2$'. {\it Commun. Math. Phys.} {\bf 178} no. 1 (1996), 135-146. \end{thebibliography} \bigskip \noindent \hfill{\it Elliott H. Lieb}\\ \noindent \rightline{\it Departments of Mathematics and Physics} \noindent \rightline{\it Princeton University}\\ \bigskip \footnoterule \bigskip \noindent {\copyright 1998 by Elliott H. Lieb} \end{document} ---------------0003281509240--