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Sharp Weyl laws with singular potentials

Rupert L. Frank and Julien Sabin

Vol. 5 (2023), No. 1, 85–144

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, for any Kato class potential, a pointwise Weyl law holds with the standard sharp remainder term, provided a certain, explicit additional term is taken into account. This 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 Avakumović to the case of Schrödinger operators with singular potentials.

Weyl law, Schrödinger operator
Mathematical Subject Classification
Primary: 35P20
Received: 20 July 2021
Revised: 5 April 2022
Accepted: 15 June 2022
Published: 24 April 2023
Rupert L. Frank
Mathematisches Institut
Ludwig-Maximilians-Universitat München
Munich Center for Quantum Science and Technology (MCQST)
Department of Mathematics
Pasadena, CA
United States
Julien Sabin
Centre de Mathematiques Laurent Schwartz
École Polytechnique