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Communications in Mathematical Physics

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

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

  • Mathieu Lewin
  • Julien Sabin
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.

Keywords

Relative Entropy Lewin Rank Operator Strichartz Estimate Relative Free Energy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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