The Hartree equation for infinitely many particles, II: Dispersion and scattering in 2D

Lewin, Mathieu; Sabin, Julien

doi:10.2140/apde.2014.7.1339

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The Hartree equation for infinitely many particles, II: Dispersion and scattering in 2D

Mathieu Lewin and Julien Sabin

Vol. 7 (2014), No. 6, 1339–1363

DOI: 10.2140/apde.2014.7.1339

Abstract

We consider the nonlinear Hartree equation for an interacting gas containing infinitely many particles and we investigate the large-time stability of the stationary states of the form $f (- Δ)$ , describing a homogeneous quantum gas. Under suitable assumptions on the interaction potential and on the momentum distribution $f$ , we prove that the stationary state is asymptotically stable in dimension 2. More precisely, for any initial datum which is a small perturbation of $f (- Δ)$ in a Schatten space, the system weakly converges to the stationary state for large times.

Keywords

Hartree equation, infinite quantum systems, Strichartz inequality, scattering, Lindhard function

Mathematical Subject Classification 2010

Primary: 35Q40

Milestones

Received: 2 October 2013

Accepted: 9 June 2014

Published: 18 October 2014

Authors

Mathieu Lewin
CNRS & Université de Cergy-Pontoise Mathematics Department UMR 8088 F-95000 Cergy-Pontoise France

Julien Sabin
Université de Cergy-Pontoise Mathematics Department UMR 8088 F-95000 Cergy-Pontoise France

Vol. 7, No. 6, 2014