Regular singular Sturm-Liouville operators and their zeta-determinants

TL;DR

This paper derives a general formula for the zeta-determinant of Sturm-Liouville operators with regular singular potentials on [0,1], extending previous results to broader boundary conditions and potentials.

Contribution

It provides a unified approach to compute zeta-determinants for Sturm-Liouville operators with general regular singular potentials and separated boundary conditions.

Findings

Established existence of zeta-determinants for the operators.

Derived a formula involving the Wronski-determinant of solutions.

Extended previous results to more general potentials and boundary conditions.

Abstract

We consider Sturm-Liouville operators on the line segment [0, 1] with general regular singular potentials and separated boundary conditions. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski- determinant of a fundamental system of solutions adapted to the boundary conditions. This generalizes the earlier work of the first author, treating general regular singular potentials but only the Dirichlet boundary conditions at the singular end, and the recent results by Kirsten-Loya-Park for general separated boundary conditions but only special regular singular potentials.