Pointwise Weyl Laws for Schr\"odinger operators with singular potentials

TL;DR

This paper establishes that pointwise Weyl laws remain valid for Schrödinger operators with certain singular potentials on Riemannian manifolds, extending classical spectral asymptotics to broader potential classes.

Contribution

It proves the validity of pointwise Weyl laws for Schrödinger operators with potentials in the Kato class and in $L^n(M)$, broadening the scope of spectral asymptotics under singular perturbations.

Findings

Weyl law holds for potentials in the Kato class.

Standard sharp error term $O(\lambda^{n-1})$ is valid for potentials in $L^n(M)$.

Results extend classical Weyl law to operators with singular potentials.

Abstract

We consider the Schr\"odinger operators $H_V=-\Delta_g+V$ with singular potentials $V$ on general $n$-dimensional Riemannian manifolds and study whether various forms of pointwise Weyl law remain valid under this pertubation. We prove that the pointwise Weyl law holds for potentials in the Kato class, which is the minimal assumption to ensure that $H_V$ is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. Moreover, we show that the pointwise Weyl law with the standard sharp error term $O(\lambda^{n-1})$ holds for potentials in $L^n(M)$.