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

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.