Jonathan Stanfill
The Ohio State University
Title
A finer limit circle/limit point classification for Sturm-Liouville operators
Abstract
We introduce a new finer limit point/limit circle classification for Sturm-Liouville equations by defining the regularization index. The results rely on the integrability of the product of the principal and nonprincipal solutions near singular endpoints and constructing a spectral parameter power series (i.e., a Taylor series in the spectral parameter z) for solutions of the Sturm-Liouville problem. The regularization index at the singular endpoint is then defined by comparing the growth in x of the coefficients of the power series. Implications include classifying when these series are well-behaved asymptotic series (in a precise sense), quantifying how far certain limit point endpoints are away from being Darboux transformed to a limit circle endpoint, and extending Weyl eigenvalue asymptotics to singular problems. These results will be motivated by multiple examples.
This talk is based on joint work with Mateusz Piorkowski.