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Sharp Weyl laws with singular potentials

Frank, Rupert L. and Sabin, Julien (2020) Sharp Weyl laws with singular potentials. . (Unpublished)

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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 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 Avakumović to the case of Schrödinger operators with singular potentials.

Item Type:Report or Paper (Discussion Paper)
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Frank, Rupert L.0000-0001-7973-4688
Additional Information:© 2020 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
Record Number:CaltechAUTHORS:20200819-151324737
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:105037
Deposited By: Tony Diaz
Deposited On:19 Aug 2020 22:22
Last Modified:02 Jun 2023 01:08

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