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Spectral \(\zeta\)-functions and \(\zeta\)-regularized functional determinants for regular Sturm-Liouville operators. (English) Zbl 07423435

Summary: 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 with 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 \(\tau y=zy\) 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 Schrödinger operators with zero, piecewise constant, and a linear potential on a compact interval.

MSC:

47A10 Spectrum, resolvent
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47G10 Integral operators
34B27 Green’s functions for ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B24 Sturm-Liouville theory
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