An analogue of the P\"oschl-Teller anharmonic oscillator on an $N$-dimensional sphere

TL;DR

This paper derives analytical solutions for a Schr"odinger particle on an N-dimensional sphere under a P"oschl-Teller-like potential, generalizing known models and exploring the Euclidean limit.

Contribution

It introduces a hyperspherical analogue of the P"oschl-Teller anharmonic oscillator and provides closed-form energy eigenvalues and eigenfunctions for arbitrary dimensions.

Findings

Analytical energy eigenvalues and eigenfunctions derived.

Reproduces known 2D results by Kazaryan et al.

Recovers hyperspherical harmonic oscillator results when =0.

Abstract

A Schr\"odinger particle on an $N$-dimensional ($N\geqslant2$) hypersphere of radius $R$ is considered. The particle is subjected to the action of a force characterized by the potential $V(\theta)=2m\omega_{1}^{2}R^{2}\tan^{2}(\theta/2)+2m\omega_{2}^{2}R^{2}\cot^{2}(\theta/2)$, where $0\leqslant\theta\leqslant\pi$ is the hyperlatitude angular coordinate. In the general case when $\omega_{1}\neq\omega_{2}$, this is a model of a hyperspherical analogue of the P\"oschl-Teller anharmonic oscillator. Energy eigenvalues and normalized eigenfunctions for this system are found in closed analytical forms. For $N=2$, our results reproduce those obtained by Kazaryan et al. [Physica E 52 (2013) 122]. For $N\geqslant2$ arbitrary and for $\omega_{2}=0$, the results of Mardoyan and Petrosyan [J. Contemp. Phys. 48 (2013) 70] for their model of an isotropic hyperspherical harmonic oscillator are recovered. The Euclidean limit for the anharmonic oscillator in question is also discussed.