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 -