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

Abstract

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 (−∆), describing an homogeneous Fermi 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 (−∆) in a Schatten space, the system weakly converges to the stationary state for large times.

The Hartree Equation for Infinitely Many Particles I. Well-Posedness Theory

“…It is an interesting question to investigate the asymptotic stability of these states, that is, the weak limit of γ (t) when t → ∞. This is studied in [34].…”

On stability of ground states for finite crystals in the Schrödinger–Poisson model

“…These results were extended by Catto, Le Bris, Lions and others to Thomas-Fermi and Hartree-Fock models [7,8,9]. Further results in this direction were established by Cancés, Lahbabi, Lewin, Sabin, Stoltz, and others [5,6,22,25,26]. All these results concern either the convergence of the ground state of finite particle systems in the thermodynamic limit or the existence of the ground state for infinite particle systems.…”

The Hartree equation for infinite quantum systems

“…We explained the results contained in [21] which prove the asymptotic stability of translation-invariant quantum density matrices under the nonlinear Hartree flow (Theorem 4). A key element of the proof is a generalization of Strichartz estimates to density matrices (Theorem 1), which was discovered in [12] and extended later in [13].…”

“…In [19,21] we addressed the question of the well-posedness of (1.1) and of the large time behaviour of its solutions, in the case where Tr |γ 0 | = +∞, that is for quantum systems with an infinite number of particles. This context was strongly motivated from the study of the dynamics of (1.1) around a class of stationary solutions which are translation-invariant.…”

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AbstractWe review some recent results obtained with Mathieu Lewin [21] concerning the nonlinear Hartree equation for density matrices of infinite trace, describing the time evolution of quantum systems with infinitely many particles. Our main result is the asymptotic stability of a large class of translationinvariant density matrices which are stationary solutions to the Hartree equation.

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