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

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

2008-12-22

arxiv.org/abs/0812.4124v1

Phys.At.Nucl.73: 255-263,2010

Physics

Mathematical Physics

14 pages. Based in the contribution presented at the Group 27 conference, Yerevan, Armenia, August 13-19, 2008

Scientific paper

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.

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