## Analysis & PDE

• Anal. PDE
• Volume 12, Number 6 (2019), 1513-1596.

### Gross–Pitaevskii dynamics for Bose–Einstein condensates

#### 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.

#### Article information

Source
Anal. PDE, Volume 12, Number 6 (2019), 1513-1596.

Dates
Revised: 26 June 2018
Accepted: 25 October 2018
First available in Project Euclid: 12 March 2019

https://projecteuclid.org/euclid.apde/1552356129

Digital Object Identifier
doi:10.2140/apde.2019.12.1513

Mathematical Reviews number (MathSciNet)
MR3921312

Zentralblatt MATH identifier
07061133

#### Citation

Brennecke, Christian; Schlein, Benjamin. Gross–Pitaevskii dynamics for Bose–Einstein condensates. Anal. PDE 12 (2019), no. 6, 1513--1596. doi:10.2140/apde.2019.12.1513. https://projecteuclid.org/euclid.apde/1552356129

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