Content-Type: multipart/mixed; boundary="-------------0102070250702" This is a multi-part message in MIME format. ---------------0102070250702 Content-Type: text/plain; name="01-64.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-64.keywords" exchange energy, lowest Landau band, one-dimensional quantum systems ---------------0102070250702 Content-Type: application/x-tex; name="corr0602.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="corr0602.tex" \newcommand{\version}{February 7, 2001} \documentclass[12pt]{article} \usepackage{amsmath,amsgen,amstext,amsbsy,amsopn,amsthm,amssymb} \pagestyle{myheadings} \swapnumbers \setlength{\voffset}{-.75truein} \setlength{\textheight}{9.25truein} \setlength{\textwidth}{6.5truein} \setlength{\hoffset}{-.7truein} \theoremstyle{plain} \newtheorem{thm}{THEOREM} \newtheorem{cl}[thm]{COROLLARY} \newtheorem{lem}[thm]{LEMMA} \newtheorem{proposition}[thm]{PROPOSITION} \theoremstyle{definition} \newtheorem{rem}[thm]{Remark} \newcommand{\infspec}{{\rm inf\ spec\ }} \newcommand{\nab}{\left(-i\nabla+\eta\A(\x)\right)} \newcommand{\R}{{\mathbb R}} \newcommand{\C}{{\mathbb C}} \newcommand{\N}{{\mathbb N}} \newcommand{\D}{{\mathcal D}} \newcommand{\Ll}{{\mathcal L}} \newcommand{\Hh}{{\mathcal H}} \newcommand{\Cc}{{\mathcal C}} \newcommand{\eps}{\varepsilon} \newcommand{\A}{{\bf A}} \newcommand{\abf}{{\bf a}} \newcommand{\B}{{\bf b}} \newcommand{\Aa}{{\bf A}} \newcommand{\x}{{\bf x}} \newcommand{\y}{{\bf y}} \newcommand{\X}{{\bf X}} \newcommand{\0}{{\bf 0}} \newcommand{\rr}{{\bf r}} \newcommand{\bfeta}{{\bf z}} \newcommand{\xpp}{\x^\perp} \newcommand{\ypp}{\y^\perp} \newcommand{\yperp}{\y_\perp} \newcommand{\rperp}{{\bf r}_\perp} \newcommand{\xpar}{x^\parallel} \newcommand{\ypar}{y^\parallel} \newcommand{\ppar}{p^\parallel} \newcommand{\Lpar}{L^\parallel} \newcommand{\Tr}{{\rm Tr}} \newcommand{\half}{\mbox{$\frac{1}{2}$}} \newcommand{\third}{\mbox{$\frac{1}{3}$}} \newcommand{\rg}{\rho_\Gamma} \newcommand{\rddm}{\rho^{\rm DDM}} \newcommand{\rdm}{\rho^{\rm DM}} \newcommand{\gddm}{\Gamma^{\rm DDM}} \newcommand{\hddm}{h^{\rm DDM}} \newcommand{\Eddm}{E^{\rm DDM}} \newcommand{\Pddm}{\Phi^{\rm DDM}} \newcommand{\dx}{\frac\partial{\partial x}} \newcommand{\dy}{\frac\partial{\partial y}} \newcommand{\ED}{{\mathcal E}^{\rm DDM}_{B,Z}} \newcommand{\Elin}{{\mathcal E}^{\rm DDM}_{\rm lin}} \newcommand{\Econf}{E_{\rm conf}} \newcommand{\tE}{{\mathcal E}^{\rm MH}_{\eta , \rm lin}} \newcommand{\Edens}{{\hat{\mathcal E}}^{\rm MH}} \newcommand{\uw}{\underline w} \newcommand{\rhs}{\rho^{\rm HS}} \newcommand{\bsigma}{\mathord{\hbox{\boldmath $\sigma$}}} \newcommand{\rmd}{{d}} \newcommand{\al}{{\alpha}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \date{\small\version} \begin{document} \markboth{\scriptsize{HS \version}}{\scriptsize{HS \version}} \title{\bf{Bounds on one-dimensional exchange energies with application to lowest Landau band quantum mechanics}} \author{\vspace{5pt} Christian Hainzl$^1$ and Robert Seiringer$^{2}$\\ \vspace{-4pt}\small{ Institut f\"ur Theoretische Physik, Universit\"at Wien}\\ \small{Boltzmanngasse 5, A-1090 Vienna, Austria}} \date{\small\version} \maketitle \begin{abstract} By means of a generalization of the Fefferman-de la Llave decomposition we derive a general lower bound on the interaction energy of one-dimensional quantum systems. We apply this result to a specific class of lowest Landau band wave functions. \end{abstract} \footnotetext[1]{E-Mail: \texttt{hainzl@thp.univie.ac.at}} \footnotetext[2]{E-Mail: \texttt{rseiring@ap.univie.ac.at}} \bigskip An important issue in the quantum mechanics of many interacting particles is the description of the energy of the system in terms of the particle density. In particular, a lower bound to the difference of the interaction energy and its \lq\lq direct\rq\rq\ part is of interest. For the three-dimensional Coulomb potential it is known that for any $N$-particle wave function $\Psi$ the so-called Lieb-Oxford inequality \cite{LO81} \begin{equation}\label{LO} \langle\Psi|\sum_{i0} \frac 1{2r}\int_{x-r}^{x+r} |f(y)| dy. \end{equation} For $p>1$ the inequality \begin{equation}\label{hlineq} \|f^*\|_p\leq 2\left(\frac {2p}{p-1}\right)^{1/p} \|f\|_p \end{equation} holds for all $f\in\Ll^p(\R)$ \cite{SW71}. \begin{lem}[General bound on the exchange energy]\label{corr} Let $V:\R_+\to \R_+$ be a convex function, with $\lim_{r\to\infty}V(r)=0$. Let $\psi\in \Ll^2(\R^N)$. Then, for all $\beta(z) \geq 0$, \begin{equation}\label{corin} {\rm Ex}_\psi\geq -\half \int_{-\infty}^\infty dz \left( \rho_\psi^{*2}(z)\int_0^{\beta(z)} V''(r)r^2 dr+\rho_\psi^*(z) \int_{\beta(z)}^\infty V''(r) r dr\right). \end{equation} \end{lem} \begin{proof} With the aid of Lemma \ref{l1} we can write \begin{equation}\label{10} \sum_{i < j} V(|x_i - x_j|) = 2 \int_{-\infty}^\infty dz \int_0^\infty dr V''(2r) \sum_{i0$ is the magnetic field strength. We are in particular interested in the strong field case, where $B\gg \bar\rho_\psi^2$. Using the relation \begin{equation}\label{defwm} B^{-1/2}V_{m,m}(B^{-1/2}z)=\int_0^\infty dq e^{-q |z|} e^{-q^2} L_m(q^2/2)^2\equiv W_m(z) \end{equation} (see \cite{V76}), where $L_m$ are the Laguerre polynomials, one easily sees that the potentials $V_{m,m}$ are smooth convex functions away from $z=0$, for all $m\in \N_0$. Hence we can use Lemma \ref{corr} to get a lower bound on the exchange energy for these potentials. In the following, we will use the estimate \begin{equation}\label{w0} W_m(0)\leq W_0(0)=\sqrt\frac\pi 4<1. \end{equation} We do not try to give the best possible constants in the bound stated below, but concentrate on the asymptotic behavior of ${\rm Ex}_\psi$ for large $B/\bar\rho_\psi^2$. \begin{thm}[Exchange energy for $V_{m,m}$]\label{corrthm} Let $V=V_{m,m}$ be given by (\ref{defvmn}), for some $m\in \N_0$ and $B>0$. Then, for all $\psi\in\Ll^2(\R^N)$ with $\rho_\psi\in\Ll^2(\R)$, \begin{equation}\label{corvmn} {\rm Ex}_\psi\geq - 16 N \bar\rho_\psi \left( \ln\left[e^3+\frac{B^{1/2}}{\bar\rho_\psi}\right]+2\right). \end{equation} \end{thm} \begin{proof} {}From (\ref{defvmn}) and (\ref{defwm}) one easily verifies the estimates \begin{equation} W_m''(r)\leq \frac 2{r^2} W_m(r)\quad{\rm and}\quad W_m(r)\leq \min\left\{\frac 1 r, W_m(0)\right\}. \end{equation} Using this and (\ref{w0}) we get \begin{equation} \int_\beta^\infty V''(r)r dr \leq \frac 2\beta \end{equation} and \begin{equation} \int_0^\beta V''(r)r^2 dr \leq 2\left(1+\left[\ln \beta B^{1/2}\right]_+\right), \end{equation} where $[t]_+=\max\{t,0\}$. Choosing $\beta=\beta(z)=\rho_\psi^*(z)^{-1}$ Lemma \ref{corr} implies that \begin{equation} {\rm Ex}_\psi\geq -\int_{-\infty}^\infty \rho_\psi^*(z)^2\left(2+\left[\ln B^{1/2}/\rho_\psi^*(z)\right]_+\right) dz. \end{equation} Next we use that for $a>0$ \begin{equation} [\ln a]_+=\inf_{s>0}\frac 1{se} a^s\leq \inf_{00$, the exchange energy is easily calculated to be \begin{equation} {\rm Ex}_\psi=-N\bar\rho_\psi \int_0^{B^{1/2}/ \bar\rho_\psi}W_m(r)\left(1-\frac{\bar\rho_\psi} {B^{1/2}}r\right)dr, \end{equation} which is precisely of the order $N\bar\rho_\psi \ln[ B^{1/2}/\bar\rho_\psi]$ for $B\gg \bar\rho_\psi^2$. The same holds for the bosonic case, i.e., for $\psi$ the totally symmetric product of the $\varphi_i$'s. We now apply our results to a model described by the Hamiltonian \begin{equation} H_m=\sum_{i=1}^N\left(-\hbar^2\frac{\partial^2} {\partial x_i^2}-Z V_m(x_i)\right)+\sum_{i