Superintegrable potentials on 3D Riemannian and Lorentzian spaces with non-constant curvature

TL;DR

This paper constructs a broad class of superintegrable potentials on 3D curved spaces with non-constant curvature using a quantum sl(2,R) coalgebra, analyzing their geometric and dynamical properties.

Contribution

It introduces a novel coalgebra-based framework for superintegrable potentials on 3D curved spaces, including non-constant curvature analogues of classical geometries.

Findings

Hamiltonians have at least three independent constants of motion.

Intrinsic oscillator and Kepler potentials are identified on these spaces.

Explicit examples of superintegrable potentials are provided.

Abstract

A quantum sl(2,R) coalgebra is shown to underly the construction of a large class of superintegrable potentials on 3D curved spaces, that include the non-constant curvature analogues of the spherical, hyperbolic and (anti-)de Sitter spaces. The connection and curvature tensors for these "deformed" spaces are fully studied by working on two different phase spaces. The former directly comes from a 3D symplectic realization of the deformed coalgebra, while the latter is obtained through a map leading to a spherical-type phase space. In this framework, the non-deformed limit is identified with the flat contraction leading to the Euclidean and Minkowskian spaces/potentials. The resulting Hamiltonians always admit, at least, three functionally independent constants of motion coming from the coalgebra structure. Furthermore, the intrinsic oscillator and Kepler potentials on such Riemannian and Lorentzian spaces of non-constant curvature are identified, and several examples of them are explicitly presented.