Spectral zeta-Functions and zeta-Regularized Functional Determinants for Regular Sturm-Liouville Operators

TL;DR

This paper develops a unified method to compute spectral zeta-functions and regularized determinants for regular Sturm-Liouville operators, providing explicit formulas and examples for various boundary conditions and potentials.

Contribution

It introduces a new approach to evaluate spectral zeta-functions and determinants for Sturm-Liouville problems using fundamental solutions and analytic continuation techniques.

Findings

Explicit formulas for zeta-function values at positive integers.

Full analytic continuation of the zeta-function achieved.

Application to Schrödinger operators with various potentials.

Abstract

The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $\zeta$-functions to efficiently compute values of spectral $\zeta$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $\tau$. Depending on the underlying boundary conditions, we express the $\zeta$-function values in terms of a fundamental system of solutions of $\tau y = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $\zeta$-function through a Liouville transformation and provide an explicit expression for the $\zeta$-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schr\"{o}dinger operators with zero, piecewise constant, and a linear potential on a compact interval.