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arXiv:0907.4466 · Journal of Statistical Physics

Derivation of the Time Dependent Gross-Pitaevskii Equation Without Positivity Condition on the Interaction

Peter Pickl

Published May 10, 2010• 53 citations
arXivHTMLPDF
DOI
10.1007/s10955-010-9981-0

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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].