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

The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $ζ$-functions to efficiently compute values of spectral $ζ$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $τ$. Depending on the underlying boundary conditions, we express the $ζ$-function values in terms of a fundamental system of solutions of $τy = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $ζ$-function through a Liouville transformation and provide an explicit expression for the $ζ$-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 Schrödinger operators with zero, piecewise constant, and a linear potential on a compact interval.

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