01-188 W.H. Aschbacher, J. Froehlich, G.M. Graf, K. Schnee, M. Troyer

Symmetry Breaking Regime in the Nonlinear Hartree Equation (2468K, Latex2e with 2 eps figures) May 21, 01

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Abstract. The present paper is concerned with minimizers $\Phi$ of the Hartree energy functional ${\mathcal H}_g [{\bar \psi},\psi]=\frac{1}{2}\|\nabla\psi\|_2^2+(\psi,v\psi)_2+g(\psi,V\ast|\psi|^2\psi)_2$ with $g<0$ on the configuration space $\R^d$, $d \ge 2$, for a general class of external potentials $v$ and two-body potentials $V$. We prove that a minimizer $\Phi$ does not have the symmetry properties of the potential $v$ for $|g|$ strictly larger than a critical $g_\ast > 0$. A numerical investigation visualizes this symmetry breaking regime in the simple case of an external double well potential. As a particular application of these results and as a motivation for the investigation of the Hartree functional, we propose a generalization of the Gross-Pitaevskii functional of Bose-Einstein condensation for {\it attractive} interatomic forces that overcomes the break-down of this theory at the collapse point of the condensate.

Files: 01-188.src( 01-188.keywords , symbrk.tex , symbrk1.eps , symbrk2.eps )