Absence of eigenvalues for integro-differential operators with periodic coefficients

TL;DR

This paper investigates the spectral properties of certain nonselfadjoint integro-differential operators, demonstrating the absence of eigenvalues under perturbations, with applications to the Hill operator.

Contribution

It introduces new spectral results showing the absence of eigenvalues for a class of nonselfadjoint integro-differential operators using perturbation theory.

Findings

No eigenvalues for specific nonselfadjoint integro-differential operators

Spectral properties of the perturbed Hill operator are characterized

Results apply to operators acting in Lp spaces on positive real line or entire real line

Abstract

Applying perturbation theory methods, the absence of the point spectrum for some nonselfadjoint integro-differential operators is investigated. The considered differential operators are of arbitrary order and act in either $\mathbf{L}_{p}(\mathbb{R}_{+})$ or $\mathbf{L}_{p}(\mathbb{R}) (1\leq p<\infty)$. As an application of general results, new spectral properties of the perturbed Hill operator are derived.