Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x

TL;DR

This paper investigates the spectral properties of singular Sturm-Liouville operators with indefinite weights, focusing on local definitizability, critical points, and spectral behavior near infinity, with applications to various operator classes.

Contribution

It characterizes local definitizability and critical points for indefinite Sturm-Liouville operators, and constructs operators with non-real spectrum accumulating to real points.

Findings

Infinity is a regular critical point for the operator.

Constructed operators with non-real spectrum accumulating to real points.

Applied results to multiple classes of Sturm-Liouville operators.

Abstract

We consider a singular Sturm-Liouville expression with the indefinite weight sgn x. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of $\infty$. Moreover, in this situation, the point $\infty$ is a regular critical point. We construct an operator $A=(\sgn x)(-d^2/dx^2+q)$ with non-real spectrum accumulating to a real point. The obtained results are applied to several classes of Sturm-Liouville operators.