标题： 非线性Hartree方程的量子散射态的半经典极限 显示英文标题

标题： Semi-classical limit of quantum scattering states for the nonlinear Hartree equation

摘要： 本文研究了半经典条件下量子粒子的长时间动力学。 首先，我们证明对于具有短程相互作用势的非线性Hartree方程，小数据解满足色散界限并且它们发生散射，其中小性条件和界限与表示约化普朗克常数的小参数$\hbar\in(0,1]$无关。 然后，取半经典极限$\hbar\to0$，我们证明此类量子散射态的Wigner变换弱-*收敛到Vlasov方程对应的经典散射态。 作为直接结果，我们在不假设初始数据正则性的前提下，建立了Vlasov方程的小数据散射。 我们的分析基于自由薛定谔流的新统一色散估计，该估计简单但至关重要，可用于包含奇异相互作用势，如逆幂律势$\frac{1}{|x|^a}$其中$1<a<\frac{5}{3}$。 显示英文摘要

摘要： This article concerns the long-time dynamics of quantum particles in the semi-classical regime. First, we show that for the nonlinear Hartree equation with short-range interaction potential, small-data solutions obey dispersion bounds and they scatter, where the smallness conditions and the bounds are independent of the small parameter $\hbar\in(0,1]$ representing the reduced Planck constant. Then, taking the semi-classical limit $\hbar\to0$, we prove that the Wigner transforms of such quantum scattering states converge weakly-* to the corresponding classical scattering states for the Vlasov equation. As a direct consequence, we establish small-data scattering for the Vlasov equation without assuming regularity on initial data. Our analysis is based on a new uniform dispersion estimate for the free Schr\"odinger flow, which is simple but crucial to include singular interaction potentials such as inverse power-law potential $\frac{1}{|x|^a}$ with $1<a<\frac{5}{3}$.