Communications in Mathematical Physics

, Volume 334, Issue 1, pp 117–170 | Cite as

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

Article

Abstract

We show local and global well-posedness results for the Hartree equation
$$i\partial_t\gamma = [-\Delta + w * \rho_\gamma, \gamma],$$
where γ is a bounded self-adjoint operator on \({L^2(\mathbb{R}^d)}\) , 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(−Δ) 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(−Δ). 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.

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Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Mathematics Department (UMR 8088)CNRS & Université de Cergy-PontoiseCergy-PontoiseFrance

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