Sharp Weyl laws with singular potentials

TL;DR

This paper investigates how singular potentials in the Schr"odinger operator on a 3D manifold affect Weyl's law, showing a modified law with an additional lower-order term and providing counterexamples where the classical law fails.

Contribution

It extends the Weyl law analysis to Schr"odinger operators with singular potentials, revealing conditions under which the classical law is preserved or violated.

Findings

Pointwise Weyl law holds with a lower-order correction term.

Examples show singular potentials can violate the classical Weyl law.

Method extended from Avakumović to singular Schr"odinger operators.

Abstract

We consider the Laplace--Beltrami operator on a three-dimensional Riemannian manifold perturbed by a potential from the Kato class and study whether various forms of Weyl's law remain valid under this perturbation. We show that a pointwise Weyl law holds, modified by an additional term, for any Kato class potential with the standard sharp remainder term. The additional term is always of lower order than the leading term, but it may or may not be of lower order than the sharp remainder term. In particular, we provide examples of singular potentials for which this additional term violates the sharp pointwise Weyl law of the standard Laplace-Beltrami operator. For the proof we extend the method of Avakumovi\'c to the case of Schr\"odinger operators with singular potentials.