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The Hartree equation for infinitely many particles. II: Dispersion and scattering in 2D. (English) Zbl 1301.35122

Summary: 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 \(f(-\Delta)\), 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 \(f(-\Delta)\) in a Schatten space, the system weakly converges to the stationary state for large times.
For part I, see [the authors, arXiv:1310.0603, to appear in Commun Math. Phys., doi:10.1007/s00220-014-2098-6].

MSC:

35Q40 PDEs in connection with quantum mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
35B35 Stability in context of PDEs
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