Recurrence approach and higher rank polynomial algebras for superintegrable monopole systems

TL;DR

This paper explores superintegrable monopole systems, deriving polynomial algebras and energy spectra using recurrence methods, and introduces a new model in generalized Taub-NUT space with explicit solutions.

Contribution

It presents a novel superintegrable monopole system in Taub-NUT space and develops recurrence-based methods to derive its polynomial algebra and energy spectrum.

Findings

Polynomial algebra satisfied by integrals of motion

Explicit wave functions in terms of Laguerre and Jacobi polynomials

Degenerate energy spectra derived from algebraic structure

Abstract

We revisit the MIC-harmonic oscillator in flat space with monopole interaction and derive the polynomial algebra satisfied by the integrals of motion and its energy spectrum using the ad hoc recurrence approach. We introduce a superintegrable monopole system in generalized Taub-NUT space. The Schr\"{o}dinger equation of this model is solved in spherical coordinates in the framework of St\"{a}ckel transformation. It is shown that wave functions of the quantum system can be expressed in terms of the product of Laguerre and Jacobi polynomials. We construct ladder and shift operators based on the corresponding wave functions and obtain the recurrence formulas. By applying these recurrence relations, we construct higher order algebraically independent integrals of motion. We show the integrals form a polynomial algebra. We construct the structure functions of the polynomial algebra and obtain the degenerate energy spectra of the model.