Sharp semiclassical spectral asymptotics for Schrödinger operators with non-smooth potentials

Abstract

We consider semiclassical Schrödinger operators acting in $L^2({\mathbb{R}}^d)$ with $d\ge 3$. 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 Hölder continues with parameter $\hat{I}¼\ge 1/2$. Moreover we also consider sharp Riesz means of order ${\hat{I}}^3$ with ${\hat{I}}^3\in (0,1]$. 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 Hölder continues with parameter $\hat{I}¼$ that depends on ${\hat{I}}^3$.

arXiv admin note: text overlap with arXiv:2309.03716. Some references have been updated and non-smooth results have been added