The classical Darboux III oscillator: factorization, Spectrum Generating Algebra and solution to the equations of motion

TL;DR

This paper applies the Spectrum Generating Algebra technique to the Darboux III oscillator, a maximally superintegrable system, to analyze its symmetries and solve its equations of motion, extending methods used for other curved space systems.

Contribution

The paper extends the SGA method to the Darboux III oscillator, providing a new approach to analyze its symmetry algebra and solve its equations of motion in non-constant curvature space.

Findings

SGA technique successfully applied to Darboux III oscillator

Derived the symmetry algebra of the system

Obtained explicit solutions to the equations of motion

Abstract

In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be interpreted as a one-parameter deformation of the Kepler-Coulomb system. Such a Hamiltonian is associated to a specific Bertrand space of non-constant curvature. The SGA procedure unveils the symmetry algebra underlying the Hamiltonian system and, moreover, enables one to solve the equations of motion. Here we will follow the same path to tackle the Darboux III system, another maximally superintegrable system, which can indeed be viewed as a natural deformation of the isotropic harmonic oscillator where the flat Euclidean space is again replaced by another space of non-constant curvature.