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

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 $1<a<\frac{5}{3}$.