Brief communications

On infinite discrete spectrum of convolution operators with potentials

Denis Borisov^abc, Elena Zhizhina^d, Andrey Piatnitski<sup>de

a</sup> Institute of Mathematics with Computing Centre, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa, Russia
^b Peoples' Friendship University of Russia, Moscow, Russia
^c Bashkir State Pedagogical University n. a. M. Akmulla, Ufa, Russia
^d Moscow Institute of Physics and Technology (National Research University), Higher School of Contemporary Mathematics, Moscow, Russia
^e UiT The Arctic University of Norway, Campus Narvik, Narvik, Norway

Abstract: In $L_2(\mathbb{R}^d)$, we consider a self-adjoint operator which is the sum of a convolution operator and a potential. With minimal assumptions on the convolution kernel and the potential, we describe the location of its essential spectrum and give sufficient conditions for the existence of infinite series of discrete eigenvalues accumulating at the edges of the essential spectrum. We also discuss the case where a non-empty discrete spectrum appears in gaps of the essential spectrum.

Keywords: convolution operators with potentials, infinite series of eigenvalues, gaps in the essential spectrum, eigenvalues in gaps.

MSC: 47A10

Received: 04.11.2024
Revised: 16.01.2025
Accepted: 21.01.2025

DOI: 10.4213/faa4263

 English version: Functional Analysis and Its Applications, 2025, 59:4, 457–461

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