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

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 $SL(2,\mathbb{R})$ 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 $\zeta$-regularized functional determinant is also explicitly computed.