Spectral zetaFunctions and zetaRegularized Functional Determinants for Regular SturmLiouville Operators
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 (threecoefficient) selfadjoint SturmLiouville 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 Schrödinger operators with zero, piecewise constant, and a linear potential on a compact interval.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2101.12295
 Bibcode:
 2021arXiv210112295F
 Keywords:

 Mathematics  Spectral Theory;
 Mathematical Physics;
 Primary: 47A10;
 47B10;
 47G10. Secondary: 34B27;
 34L40
 EPrint:
 44 pages