Vol. 7, No. 6, 2014

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ISSN: 1948-206X (e-only)
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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
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