On the point spectrum of some perturbed differential operators with periodic coefficients

TL;DR

This paper investigates the finiteness of the point spectrum of certain perturbed differential operators with periodic coefficients, using perturbation theory and resolvent analysis.

Contribution

It introduces an abstract analytical framework for studying point spectrum finiteness and applies it to differential operators with periodic coefficients under perturbations.

Findings

Finiteness of the point spectrum is established for a class of perturbed differential operators.

Analytical continuation of the resolvent function is used to analyze spectral properties.

The results extend understanding of spectral behavior in perturbed periodic differential operators.

Abstract

Finiteness of the point spectrum of linear operators acting in a Banach space is investigated from point of view of perturbation theory. In the first part of the paper we present an abstract result based on analytical continuation of the resolvent function through continuous spectrum. In the second part, the abstract result is applied to differential operators which can be represented as a differential operator with periodic coefficients perturbed by an arbitrary subordinated differential operator.