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

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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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Authors and Affiliations

  1. Mathematics Department (UMR 8088), CNRS & Université de Cergy-Pontoise, 95000, Cergy-Pontoise, France

    Mathieu Lewin & Julien Sabin

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  1. Mathieu Lewin
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  2. Julien Sabin
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Correspondence to Mathieu Lewin.

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Communicated by L. Erdös

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Lewin, M., Sabin, J. The Hartree Equation for Infinitely Many Particles I. Well-Posedness Theory. Commun. Math. Phys. 334, 117–170 (2015). https://doi.org/10.1007/s00220-014-2098-6

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  • DOI: https://doi.org/10.1007/s00220-014-2098-6

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