Atlantis Press Paper Details: Jacobi, Ellipsoidal Coordinates and Superintegrable Systems

We describe Jacobi's method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems to which we then turn. A Schrödinger operator with potential on a Riemannian space is second-order sperintegrable if there are 2n - 1 (classically) functionally independent second-order symmetry operators. (The 2n - 1 is the maximum possible number of such symmtries.) These systems are of considerable interest in the theory of special functions because they are multiseparable, i.e., variables separate in several coordinate sets and are explicitly solvable in terms of special functions. The interrelationships between separable solutions provides much additional information about the systems. We give an example of a superintegrable system and then present very recent results exhibiting the general structure of superintegrable systems in all real or complex two-dimensional spaces and three-dimensional conformally flat spaces and a complete list of such spaces and potentials in two dimensions.


title:		Jacobi, Ellipsoidal Coordinates and Superintegrable Systems
publication:		JNMP
volume-issue:		12 - 2
pages:		209 - 229
ISSN:		1402-9251
DOI:		doi:10.2991/jnmp.2005.12.2.5 (how to use a DOI)
author(s):		E G KALNINS, J M KRESS, W MILLER §
publication date:		May 2005
abstract:		We describe Jacobi's method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems to which we then turn. A Schrödinger operator with potential on a Riemannian space is second-order sperintegrable if there are 2n - 1 (classically) functionally independent second-order symmetry operators. (The 2n - 1 is the maximum possible number of such symmtries.) These systems are of considerable interest in the theory of special functions because they are multiseparable, i.e., variables separate in several coordinate sets and are explicitly solvable in terms of special functions. The interrelationships between separable solutions provides much additional information about the systems. We give an example of a superintegrable system and then present very recent results exhibiting the general structure of superintegrable systems in all real or complex two-dimensional spaces and three-dimensional conformally flat spaces and a complete list of such spaces and potentials in two dimensions.
copyright:		© Atlantis Press. This is an open-access article 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:		122art5.pdf (207 K)