We prove a quantitative and global in time semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension $d\ge 3$, including the case of a Coulomb singularity in dimension $d=3$. This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small. As an intermediate result, we also obtain quantitative bounds on the space and velocity moments of even order and the asymptotic behavior of the spatial density due to dispersion effects, uniform in the Planck constant ħ.

中文翻译：

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集中初始数据的从 Hartree 到 Vlasov 方程的全局半经典极限

我们证明了在维度上奇异相互作用势的情况下从 Hartree 到 Vlasov 方程的定量和全局时间半经典极限$d\ge 3$，包括维数库仑奇点的情况 $d=3$. 这个结果适用于足够集中的初始数据，因为一些空间矩最初足够小。作为中间结果，我们还获得了偶数阶空间和速度矩的定量界限以及由于色散效应引起的空间密度的渐近行为，在普朗克常数ħ中是均匀的。