Vol. 12, No. 6, 2019

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Gross–Pitaevskii dynamics for Bose–Einstein condensates

Christian Brennecke and Benjamin Schlein

Vol. 12 (2019), No. 6, 1513–1596
Abstract

We study the time-evolution of initially trapped Bose–Einstein condensates in the Gross–Pitaevskii regime. We show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross–Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e., on the number of excitations of the Bose–Einstein condensate). To reach this goal, we combine the method of Lewin, Nam and Schlein (2015), who analyzed fluctuations around the Hartree dynamics for N-particle initial data in the mean-field regime, with ideas of Benedikter, de Oliveira and Schlein (2015), who considered the evolution of Fock-space initial data in the Gross–Pitaevskii regime.

Keywords
Bose–Einstein condensates, quantum dynamics, Gross–Pitaevskii equation
Mathematical Subject Classification 2010
Primary: 35Q40, 81V70
Milestones
Received: 15 March 2018
Revised: 26 June 2018
Accepted: 25 October 2018
Published: 7 February 2019
Authors
Christian Brennecke
Institute of Mathematics
University of Zurich
Zurich
Switzerland
Benjamin Schlein
Institute of Mathematics
University of Zurich
Zurich
Switzerland