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

Abstract

In this paper we introduce an index ${\ell }_c\in {\mathbb{N}}_0\cup \{\infty \}$ 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 Schrödinger 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 Schrödinger operators on the half-line with power potentials.

47 pages