A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators

TL;DR

This paper investigates the spectral properties of indefinite Sturm-Liouville operators with a focus on eigenvalues in the essential spectrum, using a functional model to analyze algebraic multiplicities and critical points.

Contribution

It introduces an abstract functional model for indefinite Sturm-Liouville operators and characterizes eigenvalue multiplicities and critical points in this context.

Findings

Eigenvalues in the essential spectrum are characterized.

Algebraic multiplicities of eigenvalues are determined.

Operators with finite singular critical points are identified.

Abstract

Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered.