 01188 W.H. Aschbacher, J. Froehlich, G.M. Graf, K. Schnee, M. Troyer
 Symmetry Breaking Regime in the Nonlinear Hartree Equation
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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 twobody 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 GrossPitaevskii functional of BoseEinstein
condensation for {\it attractive} interatomic forces that overcomes the breakdown of
this theory at the collapse point of the condensate.
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