The classical Darboux III oscillator: factorization, Spectrum Generating Algebra and solution 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.

14 pages, 6 figures. Based on the contribution presented at "The XXIII International Conference on Integrable Systems and Quantum Symmetries" (ISQS-23), June 23-27, 2015, Prague, Czech Republic

References in corpus (12)

• Superintegrable Systems in Darboux spaces
• Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability
• Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles
• Factorizations of one dimensional classical systems
• A maximally superintegrable system on an n-dimensional space of nonconstant curvature
• Bertrand spacetimes as Kepler/oscillator potentials
• Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds
• A new exactly solvable quantum model in N dimensions
• N-dimensional sl(2)-coalgebra spaces with non-constant curvature
• The classical Taub-Nut System: factorization, spectrum generating algebra and solution to the equations of motion
• On two superintegrable nonlinear oscillators in N dimensions
• Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: II. Darboux Spaces DIII and DIV

Cited by in corpus (4)

• Higher Order Quantum Superintegrability: a new "Painlevé conjecture"
• The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories
• Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras
• Explicit isomorphisms for the symmetry algebras of continuous and discrete isotropic oscillators