Semiclassical evolution with low regularity

Abstract

We prove semiclassical estimates for the Schrödinger-von Neumann evolution with C1,1 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 and to concentrating pure and mixed states without any regularity assumption. In a appendix, we finally estimate the dependence in the dimension of the constant appearing on the Calderón-Vaillancourt Theorem.