The spectral $ζ$-function of Sturm--Liouville operators with $N$ generalized point potentials

Abstract

This work analyzes the spectral zeta function associated with regular and singular Sturm--Liouville operators endowed with $N$ generalized point potentials. The point interactions are characterized as a particular class of self-adjoint extensions of Sturm--Liouville operators defined on a chain of $N+1$ adjacent intervals. Each interaction is described by $SL(2,\R)$ matching conditions relating generalized boundary values across neighboring intervals. We construct the spectral zeta function associated with Sturm--Liouville operators endowed with $N$ generalized point potentials in terms of a contour integral involving an appropriate characteristic function. The process of analytic continuation of the spectral zeta function to a neighborhood of the origin is illustrated by means of specific examples where the $ζ$-regularized functional determinant is also explicitly computed.