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Abstract
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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.
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Keywords
Weyl law, Schrödinger operator
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Mathematical Subject Classification
Primary: 35P20
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Milestones
Received: 20 July 2021
Revised: 5 April 2022
Accepted: 15 June 2022
Published: 24 April 2023
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© 2023 The Author(s), under
exclusive license to MSP (Mathematical Sciences
Publishers). |
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