This work focuses on the analysis of the spectral \(\zeta \) -function associated with a Schrödinger operator endowed with a Pöschl–Teller potential. We construct the spectral \(\zeta \) -function using a contour integral representation and, for particular self-adjoint extensions, we perform its analytic continuation to a larger region of the complex plane. We show that the spectral \(\zeta \) -function in these cases can possess a very unusual and remarkable structure consisting of a series of logarithmic branch points located at every nonpositive integer value of s along with infinitely many additional branch points (and finitely many simple poles) whose locations depend on the parameters of the problem. By comparing the Pöschl–Teller potential to the classic Bessel potential, we further illustrate that perturbing a given potential by a smooth potential on a finite interval can greatly affect the meromorphic structure and branch points of the spectral \(\zeta \) -function in surprising ways.
