Quadratic algebra structure in the 5D Kepler system with non-central potentials and Yang-Coulomb monopole interaction

TL;DR

This paper analyzes the algebraic structure of a 5D Kepler system with non-central potentials under a Yang-Coulomb monopole, deriving its spectrum and separability properties using quadratic algebra techniques.

Contribution

It constructs the quadratic algebra of integrals of motion for the 5D system and derives its energy spectrum algebraically, also exploring its separability in various coordinates.

Findings

Quadratic algebra $Q(3; L^{so(4)}, T^{su(2)})\oplus so(4)$ constructed

Energy spectrum obtained algebraically via algebraic methods

System is multiseparable in hyperspherical and parabolic coordinates

Abstract

We construct the integrals of motion for the 5D deformed Kepler system with non-central potentials in $su(2)$ Yang-Coulomb monopole field. We show that these integrals form a higher rank quadratic algebra $Q(3; L^{so(4)}, T^{su(2)})\oplus so(4)$, with structure constants involving the Casimir operators of $so(4)$ and $su(2)$ Lie algebras. We realize the quadratic algebra in terms of the deformed oscillator and construct its finite-dimensional unitary representations. This enable us to derive the energy spectrum of the system algebraically. Furthermore we show that the model is multiseparable and allows separation of variables in the hyperspherical and parabolic coordinates. We also show the separability of its 8D dual system (i.e. the 8D singular harmonic oscillator) in the Euler and cylindrical coordinates.