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

A Schrödinger particle on an $N$-dimensional ($N⩾2$) hypersphere of radius $R$ is considered. The particle is subjected to the action of a force characterized by the potential $V(\theta )=2m{\omega }_1^2{R}^2{\tan }^2(\theta \mathrm{/}2)+2m{\omega }_2^2{R}^2{\cot }^2(\theta \mathrm{/}2)$, where $0⩽\theta ⩽\pi$ is the hyperlatitude angular coordinate. In the general case when ${\omega }_1\mathrm{\ne }{\omega }_2$, 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 $N=2$, our results reproduce those obtained by Kazaryan et al. [Physica E 52 (2013) 122]. For $N⩾2$ 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.