标题： 无限粒子的Hartree方程。 I. 适定性理论 显示英文标题

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

摘要： 我们展示了Hartree方程$$i\partial_t\gamma=[-\Delta+w*\rho_\gamma,\gamma],$$的局部和全局适定性结果，其中$\gamma$是$L^2(\R^d)$上的有界自伴算子，$\rho_\gamma(x)=\gamma(x,x)$以及$w$是一个光滑的短程相互作用势。初始数据$\gamma(0)$被假定为平移不变态$\gamma_f=f(-\Delta)$的扰动，该态描述了一个具有无限多个粒子的量子系统，例如零温度下的费米海，或者正温度下的费米-狄拉克和玻色-爱因斯坦气体。 全局适定性由状态$\gamma(t)$的相对（自由）能量的守恒得出，该能量是相对于平稳状态$\gamma_f$计算的。 我们确实使用了相对熵的一般概念，这使得能够处理一大类平稳状态$f(-\Delta)$。 我们的结果基于正密度下的 Lieb-Thirring 不等式以及针对正交函数的最近 Strichartz 不等式，这两者均由 Frank、Lieb、Seiringer 和本文的第一作者提出。 显示英文摘要

摘要： We show local and global well-posedness results for the Hartree equation $$i\partial_t\gamma=[-\Delta+w*\rho_\gamma,\gamma],$$ where $\gamma$ is a bounded self-adjoint operator on $L^2(\R^d)$, $\rho_\gamma(x)=\gamma(x,x)$ and $w$ is a smooth short-range interaction potential. The initial datum $\gamma(0)$ is assumed to be a perturbation of a translation-invariant state $\gamma_f=f(-\Delta)$ 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 $\gamma(t)$, counted relatively to the stationary state $\gamma_f$. We indeed use a general notion of relative entropy, which allows to treat a wide class of stationary states $f(-\Delta)$. 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.