The spectral \(\zeta \)-function for quasi-regular Sturm–Liouville operators

Abstract

In this work, we analyze the spectral \(\zeta \)-function associated with the self-adjoint extensions, \(T_{A,B}\), of quasi-regular Sturm–Liouville operators that are bounded from below. By utilizing the Green’s function formalism, we find the characteristic function, which implicitly provides the eigenvalues associated with a given self-adjoint extension \(T_{A,B}\). The characteristic function is then employed to construct a contour integral representation for the spectral \(\zeta \)-function of \(T_{A,B}\). By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the \(\zeta \)-function to a larger region of the complex plane. We also present a method for computing the value of the spectral \(\zeta \)-function of \(T_{A,B}\) at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral \(\zeta \)-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of s.