Quadratic algebras for three dimensional non degenerate superintegrable systems with quadratic integrals of motion

Abstract

The three dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress and Miller have proved that in the case of non degenerate potentials there is 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. We show that in all the non degenerate cases (with one exception) there are at least two subalgebras of three integrals having a Poisson quadratic algebra structure, which is similar to the two dimensional case.

21 pages, Detailed version of the talk given at the XXVII Colloquium on Group Theoretical Methods in Physics, Yerevan, Armenia, Aug. 2008

References in corpus (2)

• Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
• Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds

Cited by in corpus (4)

• Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential
• Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• Ternary Poisson algebra for the non degenerate three dimensional Kepler Coulomb potential