Perturbation and spectral theory for singular indefinite Sturm-Liouville operators

TL;DR

This paper explores the spectral properties of singular indefinite Sturm-Liouville operators with sign-changing weights, analyzing how perturbations affect their spectra and eigenvalues in an indefinite inner product setting.

Contribution

It introduces a spectral analysis framework for indefinite Sturm-Liouville operators with sign-changing weights, including perturbation effects and Kneser type results.

Findings

Spectral properties differ from classical Hilbert space cases.

Perturbations of indefinite weights influence eigenvalue distribution.

Established results on essential spectra and eigenvalue accumulation.

Abstract

We study singular Sturm-Liouville operators of the form \[ \frac{1}{r_j}\left(-\frac{\mathrm d}{\mathrm dx}p_j\frac{\mathrm d}{\mathrm dx}+q_j\right),\qquad j=0,1, \] in $L^2((a,b);r_j)$, where, in contrast to the usual assumptions, the weight functions $r_j$ have different signs near the singular endpoints $a$ and $b$. In this situation the associated maximal operators become self-adjoint with respect to indefinite inner products and their spectral properties differ essentially from the Hilbert space situation. We investigate the essential spectra and accumulation properties of nonreal and real discrete eigenvalues; we emphasize that here also perturbations of the indefinite weights $r_j$ are allowed. Special attention is paid to Kneser type results in the indefinite setting and to $L^1$ perturbations of periodic operators.