Sharp semiclassical spectral asymptotics for local magnetic Schr\"odinger operators on $\mathbb{R}^d$ without full regularity

TL;DR

This paper derives precise spectral asymptotics for magnetic Schr"odinger operators in high dimensions, assuming smooth magnetic potentials and highly differentiable electric potentials with Hölder continuity.

Contribution

It provides the first sharp semiclassical spectral asymptotics for such operators under minimal regularity assumptions on the potentials.

Findings

Established sharp spectral asymptotics for localised counting functions.

Derived asymptotics for Riesz means of the spectral distribution.

Extended semiclassical analysis to operators with less regular potentials.

Abstract

We consider operators acting in $L^2(\mathbb{R}^d)$ with $d\geq3$ that locally behave as a magnetic Schr\"odinger operator. For the magnetic Schr\"odinger operators we suppose the magnetic potentials are smooth and the electric potential is five times differentiable and the fifth derivatives are H\"older continuous. Under these assumptions, we establish sharp spectral asymptotics for localised counting functions and Riesz means.