The exotic structure of the spectral $\zeta$-function for the Schrödinger operator with Pöschl--Teller potential

Abstract

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.

arXiv comment33 pages, LaTex

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