Derivation of the time dependent Gross-Pitaevskii equation for a class of non purely positive potentials

Abstract

We present a microscopic derivation of the time-dependent Gross-Pitaevskii equation starting from an interacting N-particle system of Bosons. We prove convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the respective Gross-Pitaevskii equation. Our work extends a previous result by one of us (P.P.[44]) to interaction potentials which need not to be nonnegative, but may have a sufficiently small negative part. One key estimate in our proof is an operator inequality which was first proven by Jun Yin, see [49].

arXiv comment34 pages

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Cited by in corpus (6)

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• Derivation of the time-dependent Gross-Pitaevskii equation for the dipolar gases
• Universal low-energy behavior in a quantum Lorentz gas with Gross-Pitaevskii potentials