Convolution Type Operators With Potential: Essential and Infinite Discrete Spectrum

ABSTRACT The goal of this note is to study the spectrum of a self‐adjoint convolution operator in 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 nonconnected essential spectrum, we give sufficient conditions for the existence of discrete eigenvalues in the corresponding gaps of the essential spectrum.