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

Abstract

We show local and global well-posedness results for the Hartree equationwhere γ is a bounded self-adjoint operator on L 2 (R d ), ρ γ (x) = γ (x, x) and w is a smooth short-range interaction potential. The initial datum γ (0) is assumed to be a perturbation of a translation-invariant state γ f = f (− ) which describes a quantum system with an infinite number of particles, such as the Fermi sea at zero temperature, or the Fermi-Dirac and Bose-Einstein gases at positive temperature. Global well-posedness follows from the conservation of the relative (free) energy of the state γ (t), counted relatively to the stationary state γ f . We indeed use a general notion of relative entropy, which allows us to treat a wide class of stationary states f (− ). Our results are based on a Lieb-Thirring inequality at positive density and on a recent Strichartz inequality for orthonormal functions, which are both due to Frank, Lieb, Seiringer and the first author of this article.

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.…”

“…In [25], Lewin and Sabin have established the well-posedness for the reduced von Neumann equation, describing the Fermi gas, with density matrices of infinite trace and pair-wise interaction potentials w ∈ L 1 (R 3 ). Moreover, the authors prove the asymptotic stability of translation-invariant stationary states for 2D Fermi gas [26].…”

Trace Class Conditions for Functions of Schrödinger Operators

“…Beyond the applications to minimization problems mentioned above, we note that Lieb-Thirring inequalities with μ > 0 recently also proved useful in time-dependent problems [24].…”

The Hartree equation for infinite quantum systems

“…In a first work [19], we showed the global existence of solutions to (1.1) around a large class of stationary states γ ref , in the adequate energy space. The operator γ ref has not only an infinite trace, it also has an infinite (total) energy, which is defined as…”

“…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.…”