Spectral $$\varvec{\zeta }$$-functions and $$\varvec{\zeta }$$-regularized functional determinants for regular Sturm–Liouville operators

Abstract

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 with 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 τ y = zy 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.

On a Nonlocal Integral Operator Commuting with the Laplacian and the Sturm-Liouville Problem I: Low Rank Perturbations of the Operator

“…This is an expansion on old ideas due to Kato [31]. The spectral questions explored here connect in a fundamental way with the works [9,8,22,21].…”

“…Remark 1.1 The matrix B is sometimes written in terms of the sines and cosines to reflect the self-adjointness; see, e.g. [21,22,24]. To highlight this angular dependence we note that one can indeed recast (2) in the form (3), using the KAN decomposition of B , viz., , where r > 0, and θ, n ∈ R. (This is a unique decomposition called "the Iwasawa decomposition" in the Lie theory literature; it is a special case of QR factorization; see [14]).…”