An optimal semiclassical bound on certain commutators

Abstract

We prove an optimal semiclassical bound on the trace norm of the following commutators $[1_{(-\infty ,0]}(H_{\hslash }),x]$, $[1_{(-\infty ,0]}(H_{\hslash }),-i\hslash \nabla ]$ and $[1_{(-\infty ,0]}(H_{\hslash }),e^{itx}]$, where $H_{\hslash }$ is a Schrödinger operator with a semiclassical parameter $\hslash$, $x$ is the position operator and $-i\hslash \nabla$ is the momentum operator. These bounds corresponds to a mean-field version of bounds introduced as an assumption by N. Benedikter, M. Porta and B. Schlein in a study of the mean-field evolution of a fermionic system.