The semiclassical limit from Hartree to Vlasov at positive density: strong uniform-in-time convergence and scattering
Year: 2026
Type: preprint
Abstract: Strong semiclassical convergence from the Hartree equation to the Vlasov equation is established in three dimensions near Penrose-stable homogeneous steady states at positive density. For sufficiently regular integrable interaction kernels, an $O(\hbar^2)$ convergence rate in weighted Sobolev spaces is proved on finite time intervals. Combining this estimate with earlier uniform-in-$\hbar$ phase-mixing and scattering bounds for the Hartree equation yields global existence and scattering for the corresponding Vlasov solution through the semiclassical limit. If the quantum and classical initial perturbations are aligned, the semiclassical convergence is uniform for all times, and the Wigner transforms of the quantum scattering profiles converge to the classical scattering profile at the explicit rate $O((\ln\hbar^{-1})^{-1/3})$.
Source: arXiv (Cornell University)
Author: Smith, Marnie
Institution: University of Cambridge
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171362501