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

Abstract

A Schrödinger particle on an N-dimensional (\(N\geqslant 2\)) hypersphere of radius R is considered. The particle is subjected to the action of a force characterized by the potential \(V(\theta )=2m\omega _{1}^{2}R^{2}\tan ^{2}(\theta /2) +2m\omega _{2}^{2}R^{2}\cot ^{2}(\theta /2)\), where \(0\leqslant \theta \leqslant \pi\) is the hyperlatitude angular coordinate. In the general case when \(\omega _{1}\ne \omega _{2}\), 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 52:122, 2013). For \(N\geqslant 2\) arbitrary and for \(\omega _{2}=0\), 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.