Integrable potentials on spaces with curvature from quantum groups

TL;DR

This paper constructs integrable and superintegrable Hamiltonian systems on curved spaces derived from quantum group deformations, generalizing classical potentials like harmonic oscillator and Kepler on non-constant curvature spaces.

Contribution

It introduces a family of integrable systems on deformed spaces with variable curvature using quantum group techniques, including new superintegrable potentials.

Findings

Constructed integrable systems on deformed spheres, hyperbolic, and (anti-)de Sitter spaces.

Proposed analogues of harmonic oscillator and Kepler potentials on these curved spaces.

Identified superintegrable systems with constant curvature related to the deformation parameter.

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.