Generalized MICZ-Kepler system, duality, polynomial and deformed oscillator algebras

TL;DR

This paper explores the algebraic structures of the generalized MICZ-Kepler system and its dual, deriving their quadratic algebras and energy spectra using deformed oscillator algebra, with implications for higher-dimensional superintegrable systems.

Contribution

It introduces a new algebraic derivation of the MICZ-Kepler energy spectrum on S^3 and extends polynomial algebra results to higher-dimensional superintegrable systems.

Findings

Quadratic algebra of the MICZ-Kepler system in E_3 is constructed.

Realization of these algebras via deformed oscillator algebra is achieved.

New algebraic derivation of the MICZ-Kepler energy spectrum on S^3 is provided.

Abstract

We present the quadratic algebra of the generalized MICZ-Kepler system in three-dimensional Euclidean space $E_{3}$ and its dual the four dimensional singular oscillator in four-dimensional Euclidean space $E_{4}$. We present their realization in terms of a deformed oscillator algebra using the Daskaloyannis construction. The structure constants are in these cases function not only of the Hamiltonian but also of other integrals commuting with all generators of the quadratic algebra. We also present a new algebraic derivation of the energy spectrum of the MICZ-Kepler system on the three sphere $S^{3}$ using a quadratic algebra. These results point out also that results and explicit formula for structure functions obtained for quadratic, cubic and higher order polynomial algebras in context of two-dimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles.