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 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 once differentiable, with its derivative being Hölder continuous with parameter $\mu \ge 1/2$. 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 twice differentiable, with its second derivative being Hölder continuous with parameter $\mu$ that depends on $\gamma$.