Atlantis Press Paper Details: Separable Potentials and a Triality in Two-Dimensional Spaces of Constant Curvature

We characterize and completely describe some types of separable potentials in twdimensional spaces, S2 [1]2 , of any (positive, zero or negative) constant curvature and either definite or indefinite signature type. The results are formulated in a way which applies at once for the two-dimensional sphere S2 , hyperbolic plane H2 , AntiDeSiter / DeSitter two-dimensional spaces AdS1+1 / dS1+1 as well as for their flat analogues E2 and M1+1 . This is achieved through an approach of Cayley-Klein type with two parameters, 1 and 2, to encompass all curvatures and signature types. We dicuss six coordinate systems allowing separation of the Hamilton-Jacobi equation for natural Hamiltonians in S2 [1]2 and relate them by a formal triality transformation, which seems to be a clue to introduce general "elliptic coordinates" for any CK space concisely. As an application we give, in any S2 [1]2 , the explicit expressions for the Fradkin tensor and for the Runge-Lenz vector, i.e., the constants of motion for the harmonic oscillator and Kepler potential on any S2 [1]2 .

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title:		Separable Potentials and a Triality in Two-Dimensional Spaces of Constant Curvature
publication:		JNMP
volume-issue:		12 - 2
pages:		230 - 252
ISSN:		1402-9251
DOI:		doi:10.2991/jnmp.2005.12.2.6 (how to use a DOI)
author(s):		José F. CARINENA, Manuel F RAÑADA, Mariano SANTANDER
publication date:		May 2005
abstract:		We characterize and completely describe some types of separable potentials in twdimensional spaces, S2 [1]2 , of any (positive, zero or negative) constant curvature and either definite or indefinite signature type. The results are formulated in a way which applies at once for the two-dimensional sphere S2 , hyperbolic plane H2 , AntiDeSiter / DeSitter two-dimensional spaces AdS1+1 / dS1+1 as well as for their flat analogues E2 and M1+1 . This is achieved through an approach of Cayley-Klein type with two parameters, 1 and 2, to encompass all curvatures and signature types. We dicuss six coordinate systems allowing separation of the Hamilton-Jacobi equation for natural Hamiltonians in S2 [1]2 and relate them by a formal triality transformation, which seems to be a clue to introduce general "elliptic coordinates" for any CK space concisely. As an application we give, in any S2 [1]2 , the explicit expressions for the Fradkin tensor and for the Runge-Lenz vector, i.e., the constants of motion for the harmonic oscillator and Kepler potential on any S2 [1]2 .
copyright:		© Atlantis Press. This article is distributed under the terms of the Creative Commons Attribution License, which permits non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited.
full text:		122art6.pdf (259 K)