The Hartree equation for infinitely many particles. I. Well-posedness theory

We show local and global well-posedness results for the Hartree equation $i\partial_tγ=[-Δ+w*ρ_γ,γ],$ where $γ$ 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 $γ_f=f(-Δ)$ 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 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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