$su(1,1)\simeq so(2,1)$ Lie Algebraic Extensions of the Mie-type Interactions with Positive Constant Curvature

Abstract

The Schr\"{o}dinger equation in three dimensional space with constant positive curvature is studied for the Mie potential. Using analytic polynomial solutions, we have obtained whole spectrum of the corresponding system. With the aid of factorization method, ladder operators are obtained within the variable and function transformations. Using ladder operators, we have given the generators of $so(2,1)$ algebra and the Casimir operator which are related to the Mie Oscillator on the positive curvature.