Perturbations of periodic Sturm--Liouville operators

TL;DR

This paper investigates how perturbations in periodic Sturm--Liouville operators affect their spectral properties, showing stability of the essential spectrum and conditions for eigenvalues in spectral gaps under various assumptions.

Contribution

It extends Rofe-Beketov's classical results by establishing spectral stability and eigenvalue conditions under $L^1$, first, and second moment assumptions on perturbations.

Findings

Essential spectrum remains unchanged under $L^1$ perturbations.

Finitely many eigenvalues can appear in spectral gaps with first moment conditions.

Band edges are not eigenvalues under second moment conditions.

Abstract

We study perturbations of the self-adjoint periodic Sturm--Liouville operator \[ A_0 = \frac{1}{r_0}\left(-\frac{\mathrm d}{\mathrm dx} p_0 \frac{\mathrm d}{\mathrm dx} + q_0\right) \] and conclude under $L^1$-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.