Sharp semiclassical spectral asymptotics for Schr\"odinger operators with non-smooth potentials

TL;DR

This paper establishes sharp semiclassical spectral asymptotics for Schr"odinger operators with non-smooth potentials in higher dimensions, relaxing regularity assumptions while maintaining precise asymptotic estimates.

Contribution

It provides the first sharp spectral asymptotics for Schr"odinger operators with potentials lacking full regularity, extending classical results to less smooth cases.

Findings

Derived sharp asymptotics for the spectral counting function.

Extended asymptotic results to Riesz means with order b3 in (0,1].

Achieved results under minimal regularity assumptions on the potential.

Abstract

We consider semiclassical Schr\"odinger operators acting in $L^2(\mathbb{R}^d)$ with $d\geq3$. For these operators we establish a sharp spectral asymptotics without full regularity. For the counting function we assume the potential is locally integrable and that the negative part of the potential minus a constant is one time differentiable and the derivative is H\"older continues with parameter $\mu\geq1/2$. Moreover we also consider sharp Riesz means of order $\gamma$ with $\gamma\in(0,1]$. Here we assume the potential is locally integrable and that the negative part of the potential minus a constant is two time differentiable and the second derivative is H\"older continues with parameter $\mu$ that depends on $\gamma$.