Integrable potentials on spaces with curvature from quantum groups

Abstract

A family of classical integrable systems defined on a deformation of the two-dimensional sphere, hyperbolic and (anti-)de Sitter spaces is constructed through Hamiltonians defined on the non-standard quantum deformation of a sl(2) Poisson coalgebra. All these spaces have a non-constant curvature that depends on the deformation parameter z. As particular cases, the analogues of the harmonic oscillator and Kepler--Coulomb potentials on such spaces are proposed. Another deformed Hamiltonian is also shown to provide superintegrable systems on the usual sphere, hyperbolic and (anti-)de Sitter spaces with a constant curvature that exactly coincides with z. According to each specific space, the resulting potential is interpreted as the superposition of a central harmonic oscillator with either two more oscillators or centrifugal barriers. The non-deformed limit z=0 of all these Hamiltonians can then be regarded as the zero-curvature limit (contraction) which leads to the corresponding (super)integrable systems on the flat Euclidean and Minkowskian spaces.

19 pages, 1 figure. Two references added

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Cited by in corpus (25)

• The quantum harmonic oscillator on the sphere and the hyperbolic plane
• Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature
• Cuboctahedric Higgs oscillator from the Calogero model
• Position-dependent mass, finite-gap systems, and supersymmetry
• The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach
• (Super)integrability from coalgebra symmetry: formalism and applications
• Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature
• The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach II
• Anisotropic inharmonic Higgs oscillator and related (MICZ-)Kepler-like systems
• Second Hopf map and Yang-Coulomb system on 5d (pseudo)sphere
• A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog
• (Anti)de Sitter/Poincare symmetries and representations from Poincare/Galilei through a classical deformation approach
• Hidden symmetries of the Higgs oscillator and the conformal algebra
• Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system
• Superintegrability on sl(2)-coalgebra spaces
• Superintegrability on N-dimensional spaces of constant curvature from so(N+1) and its contractions
• Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature
• A unified approach to Poisson-Hopf deformations of Lie-Hamilton systems based on sl(2)
• Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations
• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
• Curvature as an integrable deformation
• Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum
• Non-coboundary Poisson-Lie structures on the book group
• Superintegrable potentials on 3D Riemannian and Lorentzian spaces with non-constant curvature
• Integrable potentials on Cayley-Klein spaces from quantum groups