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

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.