Content-Type: multipart/mixed; boundary="-------------0010030527736" This is a multi-part message in MIME format. ---------------0010030527736 Content-Type: text/plain; name="00-392.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-392.comments" For the proceedings of the International Conference on Partial Differential Equations, Clausthal, July 24-28, 2000 ---------------0010030527736 Content-Type: text/plain; name="00-392.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-392.keywords" interacting Bose gas, ground state, Gross-Pitaevskii functional, scattering length ---------------0010030527736 Content-Type: application/x-tex; name="claus.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="claus.tex" %------------------------------------------------------------ % % PARTIAL DIFFERENTIAL EQUATIONS 2000 PRCOCEEDINGS % %------------------------------------------------------------ \documentclass{amsart} %\documentclass[runningheads,envcountsect]{pde2000} %------------------------------------------------------------ % Additional Packages (Optional) %------------------------------------------------------------ %\usepackage{amsmath} %\usepackage{amssymb} %\usepackage{amsthm} %\usepackage{latexsym} %------------------------------------------------------------ % Your own definitions. % Please reduce your own definitions to an absolute % minimum. (You are not allowed to use \renewcommand) %------------------------------------------------------------ \newcommand{\R}{\mathbb{R}} \def\x{{\vec{x}}} \def\E{{\mathcal E}} \newcommand{\al}{\alpha} \newcommand{\suli}{\sum\limits} \newcommand{\rmax}{\rho_{\al,{\rm max}}} \newcommand{\rmin}{\rho_{\al,{\rm min}}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} %------------------------------------------------------------ \begin{document} %------------------------------------------------------------ % Titlepage %------------------------------------------------------------ \title[Bosons in a Trap]{Bosons in a Trap: Asymptotic Exactness of the Gross-Pitaevskii Ground State Energy Formula} % The title of your article %\titlerunning{Bosons in a Trap} % allows abbreviation of title, if the full title is too long % to fit in the running head \author{Robert Seiringer} \address{Institut f\"ur Theoretische Physik, Universit\"at Wien, Boltzmanngasse 5, A-1090 Vienna, Austria} \email{rseiring@ap.univie.ac.at} %------------------------------------------------------------ % Subject classifications %------------------------------------------------------------ \subjclass{Primary 81V70; Secondary 35Q55, 46N50} %------------------------------------------------------------ % End of document %------------------------------------------------------------ %\authorrunning{Robert Seiringer} % if there are more than two authors, % please abbreviate author list for running head %------------------------------------------------------------ % Abstract. %------------------------------------------------------------ \begin{abstract} Recent experimental breakthroughs in the treatment of dilute Bose gases have renewed interest in their quantum mechanical description, respectively in approximations to it. The ground state properties of dilute Bose gases confined in external potentials and interacting via repulsive short range forces are usually described by means of the Gross-Pitaevskii energy functional. In joint work with Elliott H. Lieb and Jakob Yngvason its status as an approximation for the quantum mechanical many-body ground state problem has recently been rigorously clarified. We present a summary of this work, for both the two- and three-dimensional case. \end{abstract} %------------------------------------------------------------ \maketitle %------------------------------------------------------------ \section{Introduction} The Gross-Pitaevskii (GP) functional was introduced in the early sixties as a phenomenological description of the order parameter in superfluid ${\rm He}_{4}$ \cite{G1961,P1961,G1963}. 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. The present contribution is based on the joint work \cite{LSY1999,LSY2000} with Elliott H. Lieb and Jakob Yngvason (see also \cite{LSY1999a}). The starting point of our investigation is the Hamiltonian for $N$ identical bosons moving in $\R^D$, $D=2$ or $3$, that interact with each other via a radially 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 $\otimes^N L^2(\R^{D},d\x)$. The pair interaction $v$ is assumed to be {\it nonnegative} and of short range, more precisely, we demand it to have a finite scattering length. (For a definition of the scattering length in arbitrary dimension see \cite{LY2000}.) The potential $V$ that represents the trap is locally bounded 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. We are interested in the ground state energy $E^{\rm QM}=\inf{\rm \,spec\,}H$. Besides $N$ it depends on the potentials $V$ and $v$, but with $V$ fixed and \begin{equation}\label{scalv} v(r)=(a_1/a)^2v_1(a_1r/a), \end{equation} where $v_{1}$ has scattering length $a_{1}$ and is regarded as {\it fixed}, $E^{\rm QM}$ is a function of $N$ and $a$ only. The corresponding ground state density is given by \begin{equation} \rho^{\rm QM}(\x)=N\int|\Psi_{0}(\x,\x_{2},\dots,\x_{N})|^2d\x_{2}\dots d\x_{N}, \end{equation} where $\Psi_0$ is a ground state wave function of $H$. Note that $v$ given in ({\ref{scalv}) has scattering length $a$. Here $a$ is dimensionless and really stands for $a\sqrt{m\omega/\hbar}$. Hence a scaling of $v$ like (\ref{scalv}) is equivalent to scaling the external potential $V$ at fixed $v$. In particlar the limit $a\to 0$ with fixed $V$ is equivalent to the limit $\omega\to 0$ with fixed $v$, if one introduces the scaling $V(\x)=\omega V_1(\omega ^{1/2} \x)$ for some fixed $V_1$. Recent experiments on Bose-Einstein condensation are usually interpreted in terms of a function $\Phi^{\rm GP}(\x)$ of $\x\in{\R}^D$, which minimizes the {\it Gross-Pitaevskii energy functional} \begin{equation}\label{gpf} \E^{\rm GP}[\Phi]=\int_{\R^D}\left(|\nabla\Phi|^2+V|\Phi|^2+4\pi g|\Phi|^4\right)d\x \end{equation} under the subsidiary condition $\int|\Phi|^2=N$. The corresponding energy is \begin{equation} E^{\rm GP}(N,g)=\inf_{\int|\Phi|^2=N}\E^{\rm GP}[\Phi]=\E^{\rm GP}[\Phi^{\rm GP}]. \end{equation} The parameter $g$ is different in dimensions 2 and 3. However, for any value of $g>0$ and $N>0$ it can be shown that a unique, strictly positive $\Phi^{\rm GP}$ exists \cite{LSY1999}. It depends on these parameters, of course, and when this is important we denote it by $\Phi^{\rm GP}_{N,g}$. The motivation of the term $4\pi g|\Phi|^4$ in the GP functional comes from the ground state energy density, $\varepsilon_0(\rho)$, of a a dilute, thermodynamically infinite, homogeneous Bose gas of density $\rho$, interacting via a repulsive potential with scattering length $a$. The formulas for this quantity are older than the GP functional \cite{BO,Lee,Schick}, at least for $D=3$, but they have only very recently been derived rigorously for suitable interparticle potentials. See \cite{LY1998} and \cite{LY2000}. They are given by \begin{eqnarray}\nonumber \varepsilon_0(\rho)&\approx& 4\pi a \rho^2\qquad {\rm for\ }D=3, \\ \varepsilon_0(\rho)&\approx& 4\pi \rho^2 |\ln(a^2\rho)|^{-1}\qquad {\rm for\ }D=2, \end{eqnarray} where $\approx$ means that the formulas are valid for {\it dilute} gases, where $a^D\rho\ll 1$. Hence the natural choice of the parameter $g$ is \begin{eqnarray} g &=& a \qquad {\rm for\ }D=3, \\ \label{g2} g &=& |\ln(a^2\bar\rho)|^{-1}\qquad {\rm for\ }D=2, \end{eqnarray} where $\bar\rho$ is the {\it mean GP density} \begin{equation}\label{rhobar} \bar\rho=\frac 1N\int|\Phi^{\rm GP}(\x)|^4 d\x. \end{equation} Note that $\Phi^{\rm GP}$ depends on $g$, so (\ref{g2}) together with (\ref{rhobar}) are non-linear equations for $g$. Alternatively, one could define $g$ using the minimizer for $g=1$ in the definition of $\bar\rho$. Since $\bar\rho$ appears only under a logarithm, this would not effect our leading order calculations. For the same reason one could use the TF minimizer (see below) instead of the GP minimizer to define $g$. Note also that unlike in the three-dimensional case $g$ depends on $N$ in the two-dimensional case. The idea is now that with this choice of $g$ one should, for {\it dilute} gases, have that \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} This is made precise in the following theorems. Note that by scaling \begin{equation} E^{\rm GP}(N,g)=NE^{\rm GP}(1,Ng)\quad\mbox{and}\quad \Phi^{\rm GP}_{N,g}(\x)=N^{1/2}\Phi^{\rm GP}_{1,Ng}(\x). \label{scaling} \end{equation} Hence $Ng$ is the natural parameter in GP theory. With this in mind we can state our first main result. \begin{theorem}[The GP limit of the QM ground state energy and density]\label{thm1} ~ \\ If $N\to\infty$ with $Ng$ fixed, then \begin{equation}\label{econv} \lim_{N\to\infty}\frac{{E^{\rm QM}(N,a)}}{ {E^{\rm GP}(N,g)}}=1, \end{equation} and \begin{equation}\label{dconv} \lim_{N\to\infty}\frac{1}{ N}\rho^{\rm QM}(\x)= \left |{\Phi^{\rm GP}_{1,Ng}}(\x)\right|^2 \end{equation} in the weak $L_1$-sense. \end{theorem} Note that by hypothesis of the theorem above it really applies to dilute gases, since for fixed $Ng$ (which we refer to as the GP case) the mean density $\bar\rho$ is of order $N$ and \begin{equation} a^3\bar\rho\sim N^{-2}\quad\mbox{for}\quad D=3,\quad a^2\bar\rho\sim \exp(-N)\quad\mbox{for}\quad D=2. \end{equation} Especially for $D=2$ this is an unsatisfactory restriction, since $a$ has to decrease exponentially with $N$. For a slower decrease $Ng$ tends to infinity with $N$, and the same holds for $D=3$ if $a$ does not decrease at least as $N^{-1}$. In this case, the gradient term in the GP functional becomes negligible compared to the other terms and the so-called {\it Thomas-Fermi (TF) functional} \begin{equation}\label{gtf} \E^{\rm TF}[\rho]=\int_{\R^D}\left(V\rho+4\pi g\rho^2\right)d\x \end{equation} arises. It is defined for nonnegative functions $\rho$ on $\R^D$. Its ground state energy $E^{\rm TF}$ and density $\rho^{\rm TF}$ are defined analogously to the GP case. Our second main result is that minimization of (\ref{gtf}) reproduces correctly the ground state energy and density of the many-body Hamiltonian in the limit when $N\to\infty$, $a^D\bar \rho\to 0$, but $Ng\to \infty$ (which we refer to as the TF case), provided the external potential is reasonably well behaved. We will assume that $V$ is asymptotically equal to some function $W$ that is homogeneous of some order $s>0$ and locally H\"older continuous (see \cite{LSY2000} for a precise definition). This condition can be relaxed, but it seems adequate for most practical applications and simplifies things considerably. \begin{theorem}[The TF limit of the QM ground state energy and density]\label{thm2} ~ \\ Assume that $V$ satisfies the conditions stated above. If $\gamma\equiv Ng\to\infty$ as $N\to\infty$, but still $a^D\bar\rho\to 0$, then \begin{equation}\label{econftf} \lim_{N\to\infty}\frac{E^{\rm QM}(N,a)} {E^{\rm TF}(N,g)}=1, \end{equation} and \begin{equation}\label{dconvtf} \lim_{N\to\infty}\frac{\gamma^{D/(s+D)}}{N}\rho^{\rm QM}(\gamma^{1/(s+D)}\x)= \tilde\rho^{\rm TF}(\x) \end{equation} in the weak $L_1$-sense, where $\tilde\rho^{\rm TF}$ is the minimizer of the TF functional under the condition $\int\rho=1$, $g=1$, and with $V$ replaced by $W$. \end{theorem} \noindent{\it Remark.} The theorems are independent of the interaction potential $v_1$ in (\ref{scalv}). This means that in the limit we consider only the scattering length effects the ground state properties, and not the details of the potential. Note also that the particular limit we consider is {\it not} a mean field limit, since the interaction potential is very hard in this limit; in fact the term $4\pi g|\Phi|^4$ is mostly kinetic energy. \medskip In the following, we will present a brief sketch of the proof of Theorems \ref{thm1} and \ref{thm2}. We will derive appropriate upper and lower bounds on the ground state energy $E^{\rm QM}$. The convergence of the densities follows from the convergence of the energies in the usual way by variation with respect to the external potential. We refer to \cite{LSY1999} and \cite{LSY2000} for details. \section{Upper bound to the QM energy} To derive an upper bound on $E^{\rm QM}$ we use a generalization of a trial wave function of Dyson \cite{dyson}, who used this function to give an upper bound on the ground state energy of the homogeneous hard core Bose gas. It is 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(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 $r\geq 0$. We choose it to be \begin{equation} f(r)=\left\{\begin{array}{cl} f_{0}(r)/f_0(b) \quad &\mbox{for}\quad r