2024/09/11 by Fucci, Guglielmo, Piorkowski, Mateusz, Stanfill, Jonathan
#34L40 #47B10 #47G10. Secondary: 34B27 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary: 47A10 #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2409.06922
In this work we analyze the spectral ζ-function associated with the self-adjoint extensions, TA,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 TA,B. The characteristic function is then employed to construct a contour integral representation for the spectral ζ-function of TA,B. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the ζ-function to a larger region of the complex plane. We also present a method for computing the value of the spectral ζ-function of TA,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 ζ-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.