Gross–Pitaevskii dynamics for Bose–Einstein condensates

Brennecke, Christian; Schlein, Benjamin

doi:10.2140/apde.2019.12.1513

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Gross–Pitaevskii dynamics for Bose–Einstein condensates

Christian Brennecke and Benjamin Schlein

Vol. 12 (2019), No. 6, 1513–1596

DOI: 10.2140/apde.2019.12.1513

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

Vol. 12, No. 6, 2019