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NON-LOCAL CONVOLUTION TYPE OPERATORS WITH POTENTIAL: ESSENTIAL AND INFINITE DISCRETE SPECTRUM
  • Andrey Piatnitski,
  • Denis Borisov,
  • Elena Zhizhina
Andrey Piatnitski
UiT Norges arktiske universitet - Campus Narvik

Corresponding Author:apiatnitski@gmail.com

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Denis Borisov
Institut matematiki s vycislitel'nym centrom FGBUN Ufimskogo federal'nogo issledovatel'skogo centra Rossijskoj akademii nauk
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Elena Zhizhina
UiT Norges arktiske universitet - Campus Narvik
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Abstract

The goal of this note is to study the spectrum of a self-adjoint convolution operator in L 2 ( R d ) with an integrable kernel that is perturbed by an essentially bounded real-valued potential tending to zero at infinity. We show that the essential spectrum of such operator is the union of the spectrum of the convolution operator and of the essential range of the potential. Then we provide several sufficient conditions for the existence of a countable sequence of discrete eigenvalues. For operators having non-connected essential spectrum we give sufficient conditions for the existence of discrete eigenvalues in the corresponding spectral gaps.
13 Mar 2024Submitted to Mathematical Methods in the Applied Sciences
14 Mar 2024Submission Checks Completed
14 Mar 2024Assigned to Editor
20 Mar 2024Review(s) Completed, Editorial Evaluation Pending
18 Apr 2024Reviewer(s) Assigned
04 Sep 2024Editorial Decision: Revise Major
12 Sep 20241st Revision Received
17 Sep 2024Submission Checks Completed
17 Sep 2024Assigned to Editor
17 Sep 2024Review(s) Completed, Editorial Evaluation Pending
05 Oct 2024Reviewer(s) Assigned