#### Vol. 7, No. 6, 2014

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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 $f\left(-\Delta \right)$, 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 $f\left(-\Delta \right)$ 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
Primary: 35Q40