TY - JOUR
T1 - Fine structure for 3D second-order superintegrable systems
T2 - Three-parameter potentials
AU - Kalnins, E. G.
AU - Kress, J. M.
AU - Miller, W.
PY - 2007/6/1
Y1 - 2007/6/1
N2 - A classical (or quantum) superintegrable system of second order is an integrable n-dimensional Hamiltonian system with potential that admits 2n - 1 functionally independent constants of the motion quadratic in the momenta, the maximum possible. For n ≤ 3 on conformally flat spaces with nondegenerate, i.e., four-parameter potentials (the extreme case), we have worked out the structure and classified most of the possible spaces and potentials. Here, we extend the analysis to a more degenerate class of functionally linearly independent superintegrable systems, the three-parameter potential case. We show that for 'true' three-parameter potentials the algebra of constants of the motion no longer closes at order 6 but still all such systems are Stäckel transforms of systems on complex Euclidean space or the complex 3-sphere. This is a significant step towards the complete structure analysis of all types of second-order superintegrable systems.
AB - A classical (or quantum) superintegrable system of second order is an integrable n-dimensional Hamiltonian system with potential that admits 2n - 1 functionally independent constants of the motion quadratic in the momenta, the maximum possible. For n ≤ 3 on conformally flat spaces with nondegenerate, i.e., four-parameter potentials (the extreme case), we have worked out the structure and classified most of the possible spaces and potentials. Here, we extend the analysis to a more degenerate class of functionally linearly independent superintegrable systems, the three-parameter potential case. We show that for 'true' three-parameter potentials the algebra of constants of the motion no longer closes at order 6 but still all such systems are Stäckel transforms of systems on complex Euclidean space or the complex 3-sphere. This is a significant step towards the complete structure analysis of all types of second-order superintegrable systems.
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U2 - 10.1088/1751-8113/40/22/008
DO - 10.1088/1751-8113/40/22/008
M3 - Article
AN - SCOPUS:34249313359
SN - 1751-8113
VL - 40
SP - 5875
EP - 5892
JO - Journal of Physics A: Mathematical and Theoretical
JF - Journal of Physics A: Mathematical and Theoretical
IS - 22
M1 - 008
ER -