Content-Type: multipart/mixed; boundary="-------------9911191105240" This is a multi-part message in MIME format. ---------------9911191105240 Content-Type: text/plain; name="99-439.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-439.comments" For the Proceedings of `Quantum Theory and Symmetries' (Goslar, 18-22 July 1999) (World Scientific, 2000), edited by H.-D. Doebner, V.K. Dobrev, J.-D. Hennig and W. Luecke ---------------9911191105240 Content-Type: text/plain; name="99-439.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-439.keywords" Bose Gas, Gross-Pitaevskii equation ---------------9911191105240 Content-Type: application/x-tex; name="ASI_proc_1511.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ASI_proc_1511.tex" %% below is the ASI LATEX format for the Proceedings of %% `Quantum Theory and Symmetries' (Goslar, 18-22 July 1999) %% (World Scientific, 2000), %% edited by H.-D. Doebner, V.K. Dobrev, J.-D. Hennig and W. Luecke \documentclass[12pt]{article} \usepackage{amsmath,amsgen,amstext,amsbsy,amsopn,amsthm, amssymb} % THEOREM Environments --------------------------------------------------- \newtheorem{thm}{Theorem} \numberwithin{thm}{section} \newtheorem{cor}{Corollary} \numberwithin{cor}{section} \newtheorem{lem}{Lemma} \numberwithin{lem}{section} %\newtheorem{lema}{Lemma A} %\numberwithin{lem}{section} \newtheorem{prop}{Proposition} \numberwithin{prop}{section} \theoremstyle{definition} \newtheorem{defn}{Definition} \numberwithin{defn}{section} %\theoremstyle{remark} \newtheorem{rem}{Remark} \numberwithin{rem}{section} \newcommand{\PSI}{\vert\psi\rangle} \newcommand{\PHI}{\vert\phi\rangle} \newcommand{\D}{\mathcal{D}} \newcommand{\pd}{\partial} \newcommand{\eps}{\epsilon} \newcommand{\suli}{\sum\limits} \newcommand{\inli}{\int\limits} \newcommand{\bh}{{b/2}} \newcommand{\aob}{\left(\frac{a}{b}\right)} %\newcommand{\V}{\mathcal{V}} \newcommand{\V}{V} \newcommand{\al}{\alpha} \newcommand{\xij}{|x_i-x_j|} \newcommand{\half}{\mbox{$\frac{1}{2}$}} \newcommand{\E}{\mathcal{E}} \newcommand{\FF}{\mathcal{F}} \newcommand{\F}{F} \newcommand{\G}{G} %\newcommand{\G}{\mathcal{G}} \newcommand{\rmax}{\rho_{\al,\text{max}}} \newcommand{\rmin}{\rho_{\al,\text{min}}} \newcommand{\R}{\mathbb{R}} \newcommand{\real}{\text{Re}} \newcommand{\imag}{\text{Im}} \newcommand{\e}{\widetilde{e}} \newcommand{\vv}{\widetilde{v}} \newcommand{\RR}{\widetilde{R}} \newcommand{\as}{\widetilde{a}} \numberwithin{equation}{section} %\pagestyle{myheadings} \sloppy %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\fracone{\hbox{$\frac{1 }{ 2}$}} \def\p{{\vec p}} \def\A{{\bf A}} \def\B{{\bf B}} \def\R{{\bf R}} %\def\H{{\cal H}} \def\C{{\bf C}} \def\G{{\cal G}} \def\W{{\cal W}} \def\b{{\rm b}} \def\sc{{\cal C}} \def\E{{\cal E}} \def\a{{\alpha}} \def\x{{\vec{x}}} \def\const{{\rm const}} \def\lin{{\rm lin}} \def\loc{{\rm loc}} \def\spec{{\rm spec}} \def\supp{{\rm supp}} \def\mfr#1/#2{\hbox{$\frac{{#1} }{ {#2}}$}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hfuzz=10pt \pagestyle{empty} \textheight 8.5in \textwidth 6in %\textheight 22.5cm \textwidth 16cm \normalbaselineskip=12pt \normalbaselines \oddsidemargin 0.5cm \evensidemargin 0.5cm \topmargin -1cm \begin{document} \begin{center} \vspace*{1.0cm} {\LARGE{\bf The Ground State Energy and Density \\ of Interacting Bosons in a Trap}} \vskip 1.5cm {\large {\bf Elliott~H.~Lieb$^{1}$, Robert Seiringer$^2$ and Jakob Yngvason$^2$}} \vskip 0.5cm $^1$Departments of Physics and Mathematics, Princeton University, \\ P.~O.~Box 708, Princeton, New Jersey 08544-0708, USA\\ $^2$ Institut f\"ur Theoretische Physik, Universtit\"at Wien, \\ Boltzmanngasse 5, A-1090 Wien, Austria \end{center} \vspace{1 cm} \begin{abstract} In the theoretical description of recent experiments with dilute Bose gases confined in external potentials the Gross-Pitaevskii equation plays an important role. Its status as an approximation for the quantum mechanical many-body ground state problem has recently been rigorously clarified. A summary of this work is presented here. \end{abstract} \footnotetext[0]{\copyright 1999 by the authors. Reproduction of this work, in its entirety, by any means, is permitted for non-commercial purposes.} \vspace{1 cm} \section{Introduction} The Gross-Pitaevskii (GP) equation is a nonlinear Schr\"odinger equation that was introduced in the early sixties \cite{G1961}--\cite{G1963} as a phenomenological equation for the order parameter in superfluid ${\rm He}_{4}$. It has come into prominence again because of recent experiments on Bose-Einstein condensation of dilute gases in magnetic traps. The paper \cite{DGPS} brings an up to date review of these developments. One of the inputs needed for the justification of the GP equation starting from the many body Hamiltonian is the ground state energy of a a dilute, thermodynamically infinite, homogeneous Bose gas. The formula for this quantity is older than the GP equation but it has only very recently been derived rigorously for suitable interparticle potentials. See \cite{LY1998} and \cite{LY1999}. The paper \cite{LSY1999} goes one step further and derives the GP as a limit of the full quantum mechanical description. This derivation is summarized in the present contribution. The starting point of the investigation is the Hamiltonian for $N$ bosons that interact with each other via a spherically symmetric pair-potential $v(|\x_i - \x_j|)$ and are confined by an external potential $V(\x)$: \begin{equation} H = \sum_{i=1}^{N} \{- \Delta_i + V(\x_{i})\}+ \sum_{1 \leq i < j \leq N} v(|\x_i - \x_j|). \end{equation} The Hamiltonian acts on {\it symmetric} wave functions in $L^2(\R^{3N},d^{3N}x)$. The pair interaction $v$ is assumed to be {\it nonnegative} and of short range, more precisely, $v(r)\leq {\rm (const.)}\,r^{-(3+\varepsilon)}$ as $r\to\infty$, for some $\varepsilon>0$. The potential $V$ that represents the trap is continuous and $V(\x)\to\infty$ as $|\x|\to\infty$. By shifting the energy scale we can assume that $\min_{\x}V(\x)=0$. Units are chosen so that $\hbar=2m=1$, where $m$ is the particle mass. A natural energy unit is given by the ground state energy $\hbar\omega$ of the one particle Hamiltonian $-(\hbar^2/2m)\Delta+V$. The corresponding length unit, $\sqrt{\hbar/(m\omega)}$, measures the effective extension of the trap. The ground state wave function $\Psi_{0}(\x_{1},\dots,\x_{N})$ satisfies $H\Psi_{0}=E^{\rm QM}\Psi_{0}$, with the ground state energy $E^{\rm QM}=\inf{\rm \,spec\,}H$. If $v=0$, then %\begin{equation} $\Psi_{0}(\x_{1},\dots,\x_{N})= \prod_{i=1}^{N}\Phi_{0}(\x_{i}),$ %\end{equation} with $\Phi_{0}$ the ground state wave function of $- \Delta + V(\x)$. On the other hand, if $v\neq 0$ the ground state $\Psi_{0}(\x_{1},\dots,\x_{N})$ is, in general, far from being a product state if $N$ is large. In spite of this fact, recent experiments on BE condensation are usually interpreted in terms of a function $\Phi^{\rm GP}(\x)$ of a single $\x\in{\mathbb R}^3$, the solution of the {\it Gross-Pitaevskii equation} \begin{equation}\label{gpe}(- \Delta + V+8\pi a|\Phi|^2)\Phi=\lambda\Phi, \end{equation} together with the normalization condition \begin{equation}\label{norm} \int_{{\mathbb R}^3}|\Phi(\x)|^2=N. \end{equation} Here $a$ is the {\it scattering length} of the potential $v$: \begin{equation}a=\lim_{r\to\infty}\left(r-\frac{u_{0}(r)}{ u_{0}'(r)}\right)\end{equation} where $u_{0}$ satisfies the zero energy scattering equation, \begin{equation}\label{scatteq}- u^{\prime\prime}(r)+\mfr1/2 v(r) u(r)=0,\end{equation} and $u_{0}(0)=0$. (The factor $1/2$ is due to the reduced mass of the two-body problem.) The GP equation (\ref{gpe}) is the variational equation for the minimization of the GP {\it energy functional} \begin{equation}\label{gpf}\E^{\rm GP}[\Phi]=\int\left(|\nabla\Phi|^2+V|\Phi|^2+4\pi a|\Phi|^4\right)d^3\x\end{equation} with the subsidiary condition (\ref{norm}). The corresponding energy is \begin{equation}E^{\rm GP}(N,a)=\inf_{\int|\Phi|^2=N}\E^{\rm GP}[\Phi]= \E^{\rm GP}[\Phi^{\rm GP} ],\end{equation} with a unique, positive $\Phi^{\rm GP}$. The eigenvalue, $\lambda$, in (\ref{gpe}) is related to $E^{\rm GP}$ by \begin{equation} \lambda=dE^{\rm GP}(N,a)/dN=E^{\rm GP}(N,a)/N+4\pi a\bar \rho, \end{equation} where \begin{equation}\label{rhobar} \bar\rho=\frac 1N\int|\Phi^{\rm GP}(\x)|^4 d^3\x \end{equation} is the {\it mean density}. The minimizer $\Phi^{\rm GP}$ of (\ref{gpf}) with the condition (\ref{norm}) depends on $N$ and $a$, of course, and when this is important we denote it by $\Phi^{\rm GP}_{N,a}$. Mathematically, the GP equation is quite similar to the Thomas-Fermi-von Weizs\-\"acker equation \cite{lieb81} and its basic properties can be established by similar means. See \cite{LSY1999}, Sect.~2 and Appendix A. The idea is now that for {\it dilute} gases one should have \begin{equation}\label{approx}E^{\rm GP} \approx E^{\rm QM}\quad{\rm and}\quad \rho^{\rm QM}(\x)\approx \left|\Phi^{\rm GP}(\x)\right|^2\equiv \rho^{\rm GP}(\x),\end{equation} where the quantum mechanical particle density in the ground state is defined by \begin{equation} \rho^{\rm QM}(\x)=N\int|\Psi_{0}(\x,\x_{2},\dots,\x_{N})|^2d\x_{2}\cdots d\x_{N}. \end{equation} {\it Dilute} means here that \begin{equation}\bar\rho a^3\ll 1.\end{equation} The task is to make (\ref{approx}) precise and prove it! The first remark is that by scaling \begin{equation}E^{\rm GP}(N,a)=NE^{\rm GP}(1,Na)\quad \mbox{and}\quad \Phi^{\rm GP}_{N,a}=N^{1/2}\Phi^{\rm GP}_{1,Na}. \label{scaling} \end{equation} Hence $Na$ is the natural parameter in GP theory. It should also be noted that $E^{\rm QM}$ depends on $N$ and $v$ and not only on the scattering length $a$. However, in the limit we are about to define, it is really only $a$ that matters. To bring this out we write \begin{equation}v(r)=(a_1/a)^2v_1(a_1r/a), \end{equation} where $v_{1}$ has scattering length $a_{1}$ and regard $v_{1}$ as {\it fixed}. Then $E^{\rm QM}$ is a function $E^{\rm QM}(N,a)$ of $N$ and $a$ and we can state our main result. \begin{thm}[Dilute limit of the QM ground state energy and density] \begin{equation}\label{econv} \lim_{N\to\infty}\frac{{E^{\rm QM}(N,a)}}{ {E^{\rm GP}(N,a)}}=1\end{equation} and \begin{equation}\label{dconv} \lim_{N\to\infty}\frac{1}{ N}\rho^{\rm QM}_{N,a}(\x)= \left |{\Phi^{\rm GP}_{1,Na}}(\x)\right|^2\end{equation} if $Na$ is fixed. The convergence in (\ref{dconv}) is in the weak $L_1$-sense. \end{thm} \noindent In particular, the limits depend only on the scattering length $a_{1}=Na$ of $v_{1}$ and not on details of the potential. \medskip \noindent{\it Remark.} Since $\bar\rho\sim N$ the attribute `dilute' may seem a bit strange for these limits. However, what matters is that the scattering length $a$ is small compared to the mean particle distance, $\sim\bar\rho^{1/3}$, and if $Na$ is kept constant, then $\bar\rho a^3\sim N^{-2}$. It is also important to remember that the unit of length, $\sqrt{\hbar/(m\omega)}\equiv a_{\rm trap}$, depends on the external potential, and $a$ really stands for $a/a_{\rm trap}$. If the external potential is scaled with $N$, the both $a_{\rm trap}$ and the energy unit $\hbar\omega$ depend on $N$. For instance, in order to achieve a finite transition temperature for the BE condensation of a noninteracting gas in a parabolic trap it is necessary to keep $N\omega^3$ fixed in the thermodynamic limit, and hence $a_{\rm trap}\sim N^{1/6}$ and $\hbar\omega\sim N^{-1/3}$. (See \cite{DGPS}, Eq.\ (14).) However, since the energy unit cancels in (\ref{econv}) a dependence of $V$ on $N$ does not affect the validity of (\ref{econv}), and (\ref{dconv}) also remains valid taking into account that both sides really contain the factor $a_{\rm trap}^{-3}$. \section{The dilute homogeneous Bose gas} The motivation for the last term in the GP energy functional (\ref{gpf}) is an asymptotic formula for the quantum mechanical ground state energy $E_{0}(N,L)$ of $N$ bosons in a rectangular {\it box} of side length $L$ (i.e., the {\it homogeneous} case), that was put forward by several authors many decades ago. Consider the energy per particle in the thermodynamic limit with $\rho=N/L^3$ fixed: \begin{equation}e_{0}(\rho)= \lim_{L\to\infty}E_{0}(\rho L^3,L)/(\rho L^3).\end{equation} According to the pioneering work of Bogoliubov \cite{BO} the leading term in the {\it low density asymptotics} of $e_{0}(\rho)$ is given by \begin{equation}e_{0}(\rho)\approx4\pi\rho a\end{equation} for $\rho a^3\ll 1$. In the 50's and early 60's several derivations of this formula were presented \cite{Lee-Huang-YangEtc}.% \cite{Lieb63}. They all depended on some special assumptions about the ground state that have never been proved or the selection of some special terms from a perturbation expansion that most likely diverges. The only rigorous estimates of this period were obtained by Dyson \cite{dyson} for hard spheres: \begin{equation}\frac{1+2 Y^{1/3}}{ (1-Y^{1/3})^2}\geq \frac{e_{0}(\rho)}{ 4\pi\rho a}\geq\frac{1}{ 10\sqrt 2} \end{equation} with $Y\equiv 4\pi\rho a^3/3$. The upper bound has the right asymptotic form, and it is not very difficult to generalize it to other potentials than hard spheres. The lower bound on the other hand is too low by a factor of about $1/14$ and a remedy for this situation was obtained only 40 years after Dyson's paper: \begin{thm}[Lower bound for a homogeneous gas] If $v$ is nonnegative and of finite range, then \begin{equation}\label{lbd}\frac{e_{0}(\rho)}{ 4\pi\rho a}\geq (1-C\, Y^{1/17})\end{equation} \medskip with some constant $C$. \end{thm} The proof of this theorem is given in \cite{LY1998}, see also the exposition in \cite{LY1999}. For the application to the proof of Theorem 1.1 we need a version for finite boxes that is implicitly contained in \cite{LY1998} and explicitly stated in \cite{LY1999}: \begin{thm}[Lower bound in a finite box] \label{lbthm2} For a positive potential $v$ with finite range there is a $\delta>0$ such that \begin{equation}\label{lbd2}E_{0}(N,L)/N\geq 4\pi\mu\rho a \left(1-C\, Y^{1/17}\right) \end{equation} for all $N$ and $L$ with $Y<\delta$ and $L/a>C'Y^{-6/17}$. Here $C$ and $C'$ are constants, independent of $N$ and $L$. (Note that the condition on $L/a$ requires in particular that $N$ must be large enough, $N>\hbox{\rm (const.)}Y^{-1/17}$.) \end{thm} These results are stated for interactions of finite range. An extension to potentials $v$ of infinite range decreasing faster than $1/r^3$ at infinity is obtained by approximating $v$ by finite range potentials, controlling the change of the scattering length as the cut-off is removed. In this case the estimate holds also, but possibly with an exponent different from $1/17$ and a different constant. It should be noted, however, that the form of the error term in (\ref{lbd}) is dictated by the method of proof. The true error term presumably does not have a negative sign for sufficiently small $Y$. \section{An upper bound for the QM energy} We now turn to the inhomogeneous gas. In order to prove Eq.\ (\ref{econv}) one has to establish upper and lower bounds for $E^{\rm QM}$ in terms of $E^{\rm GP}$ with errors that vanish in the limit considered. As usual, the upper bound is easier. It is based on test wave functions of the form \begin{equation}\label{ansatz}\Psi(\x_{1},\dots,\x_{N}) =\prod_{i=1}^N\Phi^{\rm GP}(\x_{i})F(\x_{1},\dots,\x_{N}).\end{equation} where $F$ is constructed in the following way: \begin{equation}F(\x_1,\dots,\x_N)=\prod_{i=1}^N %F_i(\x_1,\dots,\x_i) \quad %{\rm with} \quad %F_i(\x_1,\dots,\x_i)= f(t_i(\x_1,\dots,\x_i)),\end{equation} where $t_i = \min\{|\x_i-\x_j|, 1\leq j\leq i-1\}$ is the distance of $\x_{i}$ to its {\it nearest neighbor} among the points $\x_1,\dots,\x_{i-1}$ and $f$ is a function of $t\geq 0$. With $u_{0}$ the zero energy scattering solution and $f_{0}(r)=u_{0}(r)/r$ the function $f$ can be taken as \begin{equation}\label{eff}f(r)=f_{0}(r)/f_{0}(b)\end{equation} for $r0$ so that \begin{equation}N_{0}>cN\end{equation} for all $N$. \medskip This definition applies also if $\langle\cdot\rangle_0$ is replaced by a thermal equilibrium state at nonzero temperature. Note that the density in momentum space is \begin{equation}\langle \tilde a^*(\p)\tilde a(\p)\rangle_0=\int\exp(i\p\cdot(\x-\x^\prime))\gamma{}_{N}(\x,\x^\prime)d\x d\x^\prime \ , \end{equation} and this differs from $\left|\int \exp(i\p\cdot\x)\sqrt{\rho(\x)}d\x\right|^2$ unless $\gamma{}_{N}$ has rank 1. \medskip It is often claimed that $\Phi^{\rm GP}$ is (approximately) the eigenfunction to the highest eigenvalue of $\gamma{}_{N}(\x,\x^\prime)$ and hence that $|\tilde\Phi^{\rm GP}(\vec p)|^2$ gives the momentum distribution of the condensate, but this is not proved yet. In fact, so far the only cases with genuine interaction where BE condensation has been rigorously established in the ground state are lattice gases at precisely half filling. (The hard core lattice Bose gas corresponds to the $XY$ spin $1/2$ model and BE condensation was proved in dimension $\geq3$ \cite{DLS} and dimension $2$ \cite{LKS1988a}. The hard core lattice gas with nearest neighbor repulsion corresponds to the Heisenberg antiferromagnet and condensation was proved for high dimension in \cite{DLS} and dimension $\geq3$ in \cite{LKS1988}. Dimension $2$ is still open.) \begin{thebibliography}{99} \bibitem{G1961}E.P. Gross, {\it Structure of a Quantized Vortex in Boson Systems,} Nuovo Cimento {\bf 20}, 454--466 (1961). \bibitem{P1961} L.P. Pitaevskii, {\it Vortex lines in an imperfect Bose gas,} Sov. Phys. JETP, {\bf 13}, 451--454 (1961). \bibitem{G1963} E.P. Gross, {\it Hydrodynamics of a superfluid condensate,} J. Math. Phys. {\bf 4}, 195--207 (1963). \bibitem{DGPS} F. Dalfovo, S.\ Giorgini, L.P.\ Pitaevskii, S.\ Stringari, {\it Theory of Bose-Einstein condensation in trapped gases}, Rev. Mod. Phys. \textbf{71}, 463--512 (1999). \bibitem{LY1998} E.H. Lieb, J. Yngvason, {\it Ground state energy of the low density Bose gas}, Phys. Rev. Lett. \textbf{80}, 2504--2507 (1998). \bibitem{LY1999} E.H. Lieb, J. Yngvason, {\it Ground State Energy a Dilute Bose Gas}, in {Differential Equations and mathematical Physics}, Proceedings of an International Conference held at the University of Alabama at Birmingham, March 16--20 1999, pp.\ 271--282 (1999). \bibitem{LSY1999} E.H. Lieb, R.\ Seiringer, and J. Yngvason, {\it Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional}, Phys.\ Rev.\ A, in press (1999). \bibitem{lieb81}E.H. Lieb, {\it Thomas-fermi and related theories of atoms and molecules}, Rev. Mod. Phys. {\bf 53}, 603--641 (1981). \bibitem{BO} N.N. Bogoliubov, J. Phys. (U.S.S.R.) {\bf 11}, 23 (1947); N.N. Bogoliubov and D.N. Zubarev, Sov. Phys.-JETP {\bf 1}, 83 (1955). \bibitem{Lee-Huang-YangEtc}K.~Huang, C.N.~Yang, Phys. Rev. {\bf 105}, 767-775 (1957); T.D.~Lee, K.~Huang, and C.N.~Yang, Phys. Rev. {\bf 106}, 1135-1145 (1957); K.A. Brueckner, K. Sawada, Phys. Rev. {\bf 106}, 1117-1127, 1128-1135 (1957); S.T. Beliaev, Sov. Phys.-JETP {\bf 7}, 299-307 (1958); T.T. Wu, Phys. Rev. {\bf 115}, 1390 (1959); N. Hugenholtz, D. Pines, Phys. Rev. {\bf 116}, 489 (1959); M. Girardeau, R. Arnowitt, Phys. Rev. {\bf 113}, 755 (1959); T.D. Lee, C.N. Yang, Phys. Rev. {\bf 117}, 12 (1960); E.H. Lieb, Phys. Rev. {\bf 130}, 2518--2528 (1963). \bibitem{dyson} F.J. Dyson, {\it Ground-State Energy of a Hard-Sphere Gas}, Phys. Rev. \textbf{106}, 20--24 (1957). \bibitem{DLS} E.H. Lieb, F.J. Dyson and B. Simon, {\it Phase Transitions in Quantum Spin Systems with Isotropic and Non-Isotropic Interactions}, J. Stat. Phys. {\bf 18}, 335-383 (1978). \bibitem{LKS1988a} E.H. Lieb, T. Kennedy and S. Shastry, {\it The $XY$ Model has Long-Range Order for all Spins and all Dimensions Greater than One}, Phys. Rev. Lett. {\bf 61}, 2582-2584 (1988). \bibitem {LKS1988} E.H. Lieb, T. Kennedy and S. Shastry, {\it Existence of N\'eel Order in Some Spin 1/2 Heisenberg Antiferromagnets}, J. Stat. Phys. {\bf 53}, 1019 (1988). \end{thebibliography} \end{document} ---------------9911191105240--