On Infinite Discrete Spectrum of Convolution Operators with Potentials

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.