On functions related to the spectral theory of Sturm--Liouville operators.

Stanfill, Jonathan, 1989-

On functions related to the spectral theory of Sturm--Liouville operators.

dc.contributor.advisor	Gesztesy, Fritz, 1953-
dc.creator	Stanfill, Jonathan, 1989-
dc.creator.orcid	0000-0002-4504-4942
dc.date.accessioned	2022-06-03T13:16:32Z
dc.date.available	2022-06-03T13:16:32Z
dc.date.created	2022-05
dc.date.issued	2022-04-01
dc.date.submitted	May 2022
dc.date.updated	2022-06-03T13:16:33Z
dc.description.abstract	Functions related to the spectral theory of differential operators have been extensively studied due to their many applications in mathematics and physics. In this dissertation, we will consider spectral ζ-functions, ζ-regularized functional determinants, and Donoghue m-functions associated with Sturm--Liouville operators. We apply our results to an array of examples, including regular Schrödinger operators as well as Jacobi and generalized Bessel operators in the singular context. We begin by employing 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. 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 a fundamental system of solutions of τy = zy. Next we turn to Donoghue m-functions. Assuming the standard local integrability hypotheses on the coefficients of the singular Sturm--Liouville differential equation τ, we study all corresponding self-adjoint realizations in L^2((a,b); rdx) and systematically construct the associated Donoghue m-functions in all cases where τ is in the limit circle case at least at one interval endpoint a or b. Finally, we construct Donoghue m-functions for the Jacobi differential operator in L^2 ((-1,1); (1-x)^alpha(1+x)^beta dx) associated with the differential expression τ alpha,beta = - (1-x)^-alpha(1+x)^-beta(d/dx)((1-x)^alpha+1((1+x)^beta+1)(d/dx), x E (-1,1), alpha,beta E R, whenever at least one endpoint, x = pm1, is in the limit circle case. In doing so, we provide a full treatment of the Jacobi operator's m-functions corresponding to coupled boundary conditions whenever both endpoints are in the limit circle case.
dc.format.mimetype	application/pdf
dc.identifier.uri	https://hdl.handle.net/2104/11939
dc.language.iso	en
dc.rights.accessrights	Worldwide access
dc.subject	Sturm--Liouville operators. Spectral zeta functions. Zeta regularized functional determinants. Jacobi equation. Donoghue m-function. Boundary conditions. Boundary values.
dc.title	On functions related to the spectral theory of Sturm--Liouville operators.
dc.type	Thesis
dc.type.material	text
thesis.degree.department	Baylor University. Dept. of Mathematics.
thesis.degree.grantor	Baylor University
thesis.degree.level	Doctoral
thesis.degree.name	Ph.D.