On spectra of convolution operators with potentials

TL;DR

This paper investigates the spectral characteristics of a class of self-adjoint operators in $L_2(\\mathds R^d)$, combining convolution operators with potentials, and provides conditions for the existence and bounds of discrete spectra.

Contribution

It offers new insights into the spectral analysis of convolution operators with potentials, including criteria for discrete spectrum existence and bounds, extending classical Schrödinger operator results.

Findings

Essential spectrum is the union of convolution spectrum and potential image.

Conditions for discrete eigenvalues are established.

Comparison with classical Schrödinger operators highlights similarities and differences.

Abstract

This paper focuses on the spectral properties of a bounded self-adjoint operator in $L_2(\mathds R^d)$ being the sum of a convolution operator with an integrable convolution kernel and an operator of multiplication by a continuous potential converging to zero at infinity. We study both the essential and the discrete spectra of this operator. It is shown that the essential spectrum of the sum is the union of the essential spectrum of the convolution operator and the image of the potential. We then provide a number of sufficient conditions for the existence of discrete spectrum and obtain lower and upper bounds for the number of discrete eigenvalues. Special attention is paid to the case of operators possessing countably many points of the discrete spectrum. We also compare the spectral properties of the operators considered in this work with those of classical Schr\"odinger operators.