The exotic structure of the spectral $ζ$-function for the Schrödinger operator with Pöschl--Teller potential
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912641475149824 |
|---|---|
| author | Fucci, Guglielmo Stanfill, Jonathan |
| author_facet | Fucci, Guglielmo Stanfill, Jonathan |
| contents | This work focuses on the analysis of the spectral $ζ$-function associated with a Schrödinger operator endowed with a Pöschl--Teller potential. We construct the spectral $ζ$-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 $ζ$-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 $ζ$-function in surprising ways. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17860 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The exotic structure of the spectral $ζ$-function for the Schrödinger operator with Pöschl--Teller potential Fucci, Guglielmo Stanfill, Jonathan Mathematical Physics Spectral Theory This work focuses on the analysis of the spectral $ζ$-function associated with a Schrödinger operator endowed with a Pöschl--Teller potential. We construct the spectral $ζ$-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 $ζ$-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 $ζ$-function in surprising ways. |
| title | The exotic structure of the spectral $ζ$-function for the Schrödinger operator with Pöschl--Teller potential |
| topic | Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2411.17860 |