Generalized five-dimensional Kepler system, Yang-Coulomb monopole and Hurwitz transformation

TL;DR

This paper introduces a new superintegrable 5D Kepler system with monopole interaction, relates it to an 8D oscillator via Hurwitz transformation, and derives its algebraic energy spectrum, expanding understanding of higher-dimensional integrable systems.

Contribution

It presents a novel superintegrable Hamiltonian combining 5D Kepler and Smorodinsky-Winternitz terms, and establishes a duality with an 8D singular oscillator through Hurwitz transformation.

Findings

New superintegrable 5D Kepler system with monopole interaction.

Dual 8D singular oscillator system with quadratic algebra.

Algebraic derivation of energy spectrum for the generalized system.

Abstract

The 5D Kepler system possesses many interesting properties. This system is superintegrable and also with a $su(2)$ nonAbelian monopole interaction (Yang-Coulomb monopole). This system is also related to a 8D isotropic harmonic oscillator by a Hurwitz transformation. We introduce a new superintegrable Hamiltonian that consists in a 5D Kepler system with new terms of Smorodinsky-Winternitz type. We obtain the integrals of motion of this systems. They generate a quadratic algebra with structure constants involving the Casimir operator of a $so(4)$ Lie algebra. We also show that this system remains superintegrable with a $su(2)$ nonAbelian monopole (generalized Yang-Coulomb monopole). We study this system using parabolic coordinates and obtain from Hurwitz transformation its dual that is a 8D singular oscillator. This 8D singular oscillator is also a new superintegrable system and multiseparable. We obtained its quadratic algebra that involves two Casimir operators of $so(4)$ Lie algebras. This correspondence is used to obtain algebraically the energy spectrum of the generalized Yang Coulomb monopole.