An optimal semiclassical bound on commutators of spectral projections with position and momentum operators

Abstract

We prove an optimal semiclassical bound on the trace norm of the following commutators $$[{\varvec{1}}_{(-\infty ,0]}(H_\hbar ),x]$$ , $$[{\varvec{1}}_{(-\infty ,0]}(H_\hbar ),-i\hbar \nabla ]$$ and $$[{\varvec{1}}_{(-\infty ,0]}(H_\hbar ),e^{i\langle t, x\rangle }]$$ , where $$H_\hbar $$ is a Schrodinger operator with a semiclassical parameter $$\hbar $$ , x is the position operator, $$-\,i\hbar \nabla $$ is the momentum operator, and t in $${\mathbb {R}}^d$$ is a parameter. These bounds are in the non-interacting setting the ones introduced as an assumption by N. Benedikter, M. Porta and B. Schlein in a study of the mean-field evolution of a fermionic system.