Weyl formulae for Schr\"odinger operators with critically singular potentials

TL;DR

This paper generalizes classical Weyl formulae for Schrödinger operators on compact manifolds with critically singular potentials, achieving optimal error bounds under minimal assumptions on the potential and geometric conditions.

Contribution

It extends Weyl law error estimates to Schrödinger operators with minimal potential regularity, including the Kato class, and under various geometric conditions.

Findings

Achieved $O(\lambda^{n-1})$ error bounds with minimal potential assumptions.

Extended Duistermaat-Guillemin theorem to singular potentials with $o(\lambda^{n-1})$ bounds.

Improved error bounds for tori with stronger potential regularity.

Abstract

We obtain generalizations of classical versions of the Weyl formula involving Schr\"odinger operators $H_V=-\Delta_g+V(x)$ on compact boundaryless Riemannian manifolds with critically singular potentials $V$. In particular, we extend the classical results of Avakumovi\'{c} , Levitan and H\"ormander by obtaining $O(\lambda^{n-1})$ bounds for the error term in the Weyl formula in the universal case when we merely assume that $V$ belongs to the Kato class, ${\mathcal K}(M)$, which is the minimal assumption to ensure that $H_V$ is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. In this case, we can also obtain extensions of the Duistermaat-Guillemin theorem yielding $o(\lambda^{n-1})$ bounds for the error term under generic conditions on the geodesic flow, and we can also extend B\'erard's theorem yielding $O(\lambda^{n-1}/\log \lambda)$ error bounds under the assumption that the principal curvatures are non-positive everywhere. We can obtain further improvements for tori, which are essentially optimal, if we strengthen the assumption on the potential to $V\in L^p(M)\cap {\mathcal K}(M)$ for appropriate exponents $p=p_n$.