Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials

Abstract

The structure theory for the quadratic algebra generated by first and second order constants of the motion for 2D second order superintegrable systems with nondegenerate (3-parameter) and or 2-parameter potentials is well understood, but the results for the strictly 1-parameter case have been incomplete. Here we work out this structure theory and prove that the quadratic algebra generated by first and second order constants of the motion for systems with 4 second order constants of the motion must close at order three with the functional relationship between the 4 generators of order four. We also show that every 1-parameter superintegrable system is Stäckel equivalent to a system on a constant curvature space.

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Cited by in corpus (13)

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• Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials
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• Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
• Tools for Verifying Classical and Quantum Superintegrability
• Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
• Quadratic algebra contractions and 2nd order superintegrable systems
• An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
• Superintegrable systems with spin and second-order (pseudo)tensor integrals of motion
• Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems
• Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces
• Doubly Exotic $N$th-order Order Superintegrable Classical Systems Separating in Cartesian Coordinates
• Generalized quadratic commutator algebras of PBW-type