An analogue of the Pöschl–Teller anharmonic oscillator on an N-dimensional sphere

Radosław Szmytkowski

doi:10.1140/epjp/s13360-025-06252-w

2023 Impact factor 2.8

EPJ PLUS - Condensed Matter and Complex Systems

Open Access

Eur. Phys. J. Plus (2025) 140: 361
https://doi.org/10.1140/epjp/s13360-025-06252-w

Regular Article

An analogue of the Pöschl–Teller anharmonic oscillator on an N-dimensional sphere

Radosław Szmytkowski^a

Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, ul. Gabriela Narutowicza 11/12, 80-233, Gdańsk, Poland

^a radoslaw.szmytkowski@pg.edu.pl

Received: 3 January 2025
Accepted: 23 March 2025
Published online: 5 May 2025

Abstract

A Schrödinger particle on an N-dimensional ( $N ⩾ 2$ $N\geqslant 2$ ) hypersphere of radius R is considered. The particle is subjected to the action of a force characterized by the potential $V (θ) = 2 m ω_{1}^{2} R^{2} {tan}^{2} (θ / 2) + 2 m ω_{2}^{2} R^{2} {cot}^{2} (θ / 2)$ $V(\theta )=2m\omega _{1}^{2}R^{2}\tan ^{2}(\theta /2) +2m\omega _{2}^{2}R^{2}\cot ^{2}(\theta /2)$ , where $0 ⩽ θ ⩽ π$ $0\leqslant \theta \leqslant \pi$ is the hyperlatitude angular coordinate. In the general case when $ω_{1} \neq ω_{2}$ $\omega _{1}\ne \omega _{2}$ , this is a model of a hyperspherical analogue of the Pöschl–Teller anharmonic oscillator. Energy eigenvalues and normalized eigenfunctions for this system are found in closed analytical forms. For $N = 2$ $$N=2$$ , our results reproduce those obtained by Kazaryan et al. (Physica E 52:122, 2013). For $N ⩾ 2$ $N\geqslant 2$ arbitrary and for $ω_{2} = 0$ $\omega _{2}=0$ , the results of Mardoyan and Petrosyan (J. Contemp. Phys. 48:70, 2013) for their model of an isotropic hyperspherical harmonic oscillator are recovered. The Euclidean limit for the anharmonic oscillator in question is also discussed.

© The Author(s) 2025

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.