Finer limit circle/limit point classification for Sturm-Liouville operators

TL;DR

This paper introduces a new index called the regularization index for classifying endpoints of Sturm-Liouville operators, extending the classical limit circle/limit point theory and analyzing stability under perturbations.

Contribution

It defines the regularization index, extends the limit circle/limit point classification, and studies its stability under perturbations for general Sturm-Liouville operators.

Findings

The regularization index generalizes limit circle/limit point classification.

The index is stable under certain perturbations.

Applications include Bessel, Jacobi, and Schrödinger operators.

Abstract

In this paper we introduce an index $\ell_c \in \mathbb{N}_0 \cup \lbrace \infty \rbrace$ which we call the `regularization index' associated to the endpoints, $c\in\{a,b\}$, of nonoscillatory Sturm-Liouville differential expressions with trace class resolvents. This notion extends the limit circle/limit point dichotomy in the sense that $\ell_c~=~0$ at some endpoint if and only if the expression is in the limit circle case. In the limit point case $\ell_c>0$, a natural interpretation in terms of iterated Darboux transforms is provided. We also show stability of the index $\ell_c$ for a suitable class of perturbations, extending earlier work on perturbations of spherical Schr\"odinger operators to the case of general three-coefficient Sturm-Liouville operators. We demonstrate our results by considering a variety of examples including generalized Bessel operators, Jacobi differential operators, and Schr\"odinger operators on the half-line with power potentials.