A classical (or quantum) second order superintegrable system is an integrable -dimensional Hamiltonian system with potential that admits functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multiseparability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions, and with quasiexactly solvable systems. Here, we announce a complete classification of nondegenerate (i.e., four-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly ten nondegenerate potentials.
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November 2007
Research Article|
November 28 2007
Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties
E. G. Kalnins;
E. G. Kalnins
Department of Mathematics,
University of Waikato
, Hamilton 3240, New Zealand
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J. M. Kress;
J. M. Kress
a)
School of Mathematics,
The University of New South Wales
, Sydney, New South Wales 2052, Australia
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W. Miller, Jr.
W. Miller, Jr.
b)
School of Mathematics,
University of Minnesota
, Minneapolis, Minnesota 55455, USA
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a)
Electronic mail: [email protected].
b)
Electronic mail: [email protected].
J. Math. Phys. 48, 113518 (2007)
Article history
Received:
June 04 2007
Accepted:
November 02 2007
Citation
E. G. Kalnins, J. M. Kress, W. Miller; Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties. J. Math. Phys. 1 November 2007; 48 (11): 113518. https://doi.org/10.1063/1.2817821
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