Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

Abstract

A classical ͑or quantum͒ second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n − 1 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-parameter͒ 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.

Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere

“…Systems of 2nd order have been well studied and there is now a structure and classification theory [8,9,10,11,12,13], especially for the cases n = 2, 3. For 3rd and higher order superintegrable systems there have been recent dramatic advances but no structure and classification theory as yet [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29].…”

Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential

“…According to Kalnins-Kress-Miller "5 to 6" theorem [1], in any three-dimensional Hamiltonian system with nondegenerate potential with quadratic integrals of motion, there is always exist a 6th integral that is functionally depended with the other integrals. This sixth integral of motion is linearly independent with the 5 functionally independent integrals.…”

The algebra of the quantum nondegenerate three-dimensional Kepler-Coulomb potential

“…The Coulomb potential differs from the other non degenerate potentials that described in [1] since posses one integral of fourth order in addition to above, quadratic one which denoted by B 1 and have the following form:…”

“…By studying all the known non degenerate potentials given by Kalnins, Kress, Miller [1], we can show that: Proposition: In the case of the non degenerate with quadratic integrals of motion, on a conformally flat manifold, the integrals of motion satisfy a parafermionic-like quadratic Poisson Algebra with 5 generators which described from the following:…”

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In the three dimensional flat space any classical Hamiltonian, which has five functionally independent integrals of motion, including the Hamiltonian, is characterized as superintegrable. Kalnins, Kress and Miller [1] have proved that, in the case of non degenerate potentials, i.e potentials depending linearly on four parameters, with quadratic symmetries, posses a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral imply that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators.

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