Semiclassical Evolution With Low Regularity

TL;DR

This paper establishes semiclassical estimates for Schrödinger-von Neumann evolution with low regularity potentials and density matrices, applicable across different weak topologies and uniformly in the number of particles.

Contribution

It provides new semiclassical estimates for quantum dynamics with low regularity potentials, extending previous results to broader classes of initial states and high-dimensional systems.

Findings

Estimates hold for potentials with $C^{1,1}$ regularity.

Results are valid for initial states with Wigner functions 7 times differentiable.

Uniform estimates are obtained for $N$-body quantum dynamics.

Abstract

We prove semiclassical estimates for the Schr\''odinger-von Neumann evolution with $C^{1,1}$ potentials and density matrices whose square root have either Wigner functions with low regularity independent of the dimension, or matrix elements between Hermite functions having long range decay. The estimates are settled in different weak topologies and apply to initial density operators whose square root have Wigner functions $7$ times differentiable, independently of the dimension. They also apply to the $N$ body quantum dynamics uniformly in $N$. In a appendix, we finally estimate the dependence in the dimension of the constant appearing on the Calderon-Vaillancourt Theorem.