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Schrödinger operators with complex sparse potentials

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posted on 2022-04-14, 08:47 authored by Jean-Claude CueninJean-Claude Cuenin

We establish quantitative upper and lower bounds for Schrödinger operators with complex potentials that satisfy some weak form of sparsity. Our first result is a quantitative version of an example, due to S. Bögli (Commun Math Phys 352:629–639, 2017), of a Schrödinger operator with eigenvalues accumulating to every point of the essential spectrum. The second result shows that the eigenvalue bounds of Frank (Bull Lond Math Soc 43:745–750, 2011 and Trans Am Math Soc 370:219–240, 2018) can be improved for sparse potentials. The third result generalizes a theorem of Klaus (Ann Inst H Poincaré Sect A (N.S.) 38:7–13, 1983) on the characterization of the essential spectrum to the multidimensional non-selfadjoint case. The fourth result shows that, in one dimension, the purely imaginary (non-sparse) step potential has unexpectedly many eigenvalues, comparable to the number of resonances. Our examples show that several known upper bounds are sharp.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Communications in Mathematical Physics

Volume

392

Issue

3

Pages

951 - 992

Publisher

Springer

Version

  • VoR (Version of Record)

Rights holder

© The Author

Publisher statement

This is an Open Access Article. It is published by Springer under the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Full details of this licence are available at: https://creativecommons.org/licenses/by/4.0/

Acceptance date

2022-02-22

Publication date

2022-04-05

Copyright date

2022

ISSN

0010-3616

eISSN

1432-0916

Language

  • en

Depositor

Dr Jean-Claude Cuenin. Deposit date: 13 April 2022

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