The exotic structure of the spectral $\zeta$-function for the Schr\"odinger operator with P\"oschl--Teller potential

TL;DR

This paper investigates the complex structure of the spectral ζ-function for Schrödinger operators with Pöschl--Teller potentials, revealing unusual branch points and the effects of potential perturbations on its analytic properties.

Contribution

It constructs and analyzes the spectral ζ-function for Pöschl--Teller potentials, uncovering its intricate branch point structure and the impact of potential perturbations.

Findings

Spectral ζ-function has a series of logarithmic branch points at nonpositive integers.

Additional branch points and simple poles depend on potential parameters.

Perturbing the potential significantly alters the ζ-function's meromorphic structure.

Abstract

This work focuses on the analysis of the spectral $\zeta$-function associated with a Schr\"{o}dinger operator endowed with a P\"oschl--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 P\"oschl--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.