On superintegrable monopole systems

TL;DR

This paper reviews new superintegrable monopole systems in flat and curved spaces, highlighting their algebraic integrals of motion, symmetry algebras, and implications for energy spectra and wave functions.

Contribution

It introduces new families of superintegrable monopole systems with explicit higher-order integrals and algebraic structures, advancing understanding of their symmetries and spectral properties.

Findings

Identification of new superintegrable monopole models

Construction of algebraically independent integrals of motion

Analysis of symmetry polynomial algebras and spectral degeneracies

Abstract

Superintegrable systems with monopole interactions in flat and curved spaces have attracted much attention. For example, models in spaces with a Taub-NUT metric are well-known to admit the Kepler-type symmetries and provide non-trivial generalizations of the usual Kepler problems. In this paper, we overview new families of superintegrable Kepler, MIC-harmonic oscillator and deformed Kepler systems interacting with Yang-Coulomb monopoles in the flat and curved Taub-NUT spaces. We present their higher-order, algebraically independent integrals of motion via the direct and constructive approaches which prove the superintegrability of the models. The integrals form symmetry polynomial algebras of the systems with structure constants involving Casimir operators of certain Lie algebras. Such algebraic approaches provide a deeper understanding to the degeneracies of the energy spectra and connection between wave functions and differential equations and geometry.