Indefinite Sturm-Liouville operators in polar form

TL;DR

This paper studies indefinite Sturm-Liouville operators in polar form, establishing conditions for their spectral properties and similarity to self-adjoint operators using Krein space theory.

Contribution

It introduces new criteria for the existence of Riesz bases and similarity to self-adjoint operators for indefinite Sturm-Liouville operators in a Krein space setting.

Findings

Conditions for Riesz basis of eigenfunctions

Criteria for operator similarity to self-adjoint operators

Abstract results on regularity of critical points in Krein spaces

Abstract

We consider the indefinite Sturm-Liouville differential expression \[\mathfrak{a}(f) := - \frac{1}{w}\left( \frac{1}{r} f' \right)',\] where $\mathfrak{a}$ is defined on a finite or infinite open interval $I$ with $0\in I$ and the coefficients $r$ and $w$ are locally summable and such that $r(x)$ and $(\operatorname{sgn} x) w(x)$ are positive a.e. on $I$. With the differential expression $\mathfrak{a}$ we associate a nonnegative self-adjoint operator $A$ in the Krein space $L^2_w(I)$, which is viewed as a coupling of symmetric operators in Hilbert spaces related to the intersections of $I$ with the positive and the negative semi-axis. For the operator $A$ we derive conditions in terms of the coefficients $w$ and $r$ for the existence of a Riesz basis consisting of generalized eigenfunctions of $A$ and for the similarity of $A$ to a self-adjoint operator in a Hilbert space $L^2_{|w|}(I)$. These results are obtained as consequences of abstract results about the regularity of critical points of nonnegative self-adjoint operators in Krein spaces, which are couplings of two symmetric operators acting in Hilbert spaces.