Commutator Estimates and Quantitative Local Weyl's Law for Schr\"odinger Operators with Non-Smooth Potentials

TL;DR

This paper develops commutator estimates for semi-classical Schrödinger operators with non-smooth potentials and applies them to establish quantitative local Weyl laws in various function spaces, including complex interacting particle systems.

Contribution

It introduces new commutator estimates for projection operators with non-smooth potentials and applies them to derive quantitative Weyl laws for both non-interacting and interacting particles.

Findings

Established commutator estimates in Schatten norms for potentials with limited regularity.

Proved quantitative local Weyl laws in L^p spaces for Schrödinger operators.

Extended analysis to systems with singular interactions like Coulomb potential.

Abstract

We analyze semi-classical Schr\"odinger operators with potentials of class $C^{1,1/2}$ and establish commutator estimates for the associated projection operators in Schatten norms. These are then applied to prove quantitative versions of the local and phase space Weyl laws in $L^p$ spaces. We study both non-interacting, and interacting particle systems. In particular, we are able to treat the case of the minimizers of the Hartree energy in the case of repulsive singular pair interactions such as the Coulomb potential.