An optimal semiclassical bound on certain commutators

TL;DR

This paper establishes optimal semiclassical bounds on the trace norms of certain commutators involving Schrödinger operators, advancing the understanding of mean-field quantum dynamics for fermionic systems.

Contribution

It provides the first optimal semiclassical bounds on specific commutators related to Schrödinger operators, extending prior assumptions in mean-field fermionic system analysis.

Findings

Proves optimal bounds on commutators involving spectral projections of Schrödinger operators.

Extends mean-field bounds to a semiclassical setting with precise trace norm estimates.

Supports the theoretical framework for analyzing fermionic quantum systems.

Abstract

We prove an optimal semiclassical bound on the trace norm of the following commutators $[\boldsymbol{1}_{(-\infty,0]}(H_\hbar),x]$, $[\boldsymbol{1}_{(-\infty,0]}(H_\hbar),-i\hbar\nabla]$ and $[\boldsymbol{1}_{(-\infty,0]}(H_\hbar),e^{itx}]$, where $H_\hbar$ is a Schr\"odinger operator with a semiclassical parameter $\hbar$, $x$ is the position operator and $-i\hbar\nabla$ 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.