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

TL;DR

This paper explores the algebraic structure of the Mie potential in a positively curved space, deriving spectrum, ladder operators, and Lie algebra generators, thus extending the understanding of quantum systems in curved geometries.

Contribution

It introduces a Lie algebraic extension of the Mie potential in positive curvature space, deriving spectrum and algebraic operators using polynomial solutions and factorization methods.

Findings

Derived the spectrum of the Mie potential in positive curvature.

Constructed ladder operators and Lie algebra generators.

Connected the algebraic structure to the Mie oscillator in curved space.

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.