Weyl laws for Schr\"odinger operators on compact manifolds with boundary

TL;DR

This paper establishes Weyl laws for Schrödinger operators with singular potentials on compact manifolds with boundary, improving remainder estimates under certain geometric conditions, extending classical spectral asymptotics results.

Contribution

It extends classical Weyl law results to Schrödinger operators with singular potentials on manifolds with boundary, using heat kernel bounds and wave equation perturbation techniques.

Findings

Weyl laws proven for Schrödinger operators with critical singular potentials.

Improved Weyl remainder estimates when periodic geodesic billiards have measure zero.

Extension of classical results by Seeley, Ivrii, and Melrose.

Abstract

We prove Weyl laws for Schr\"odinger operators with critically singular potentials on compact manifolds with boundary. We also improve the Weyl remainder estimates under the condition that the set of all periodic geodesic billiards has measure 0. These extend the classical results by Seeley, Ivrii and Melrose. The proof uses the Gaussian heat kernel bounds for short times and a perturbation argument involving the wave equation.