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

2009/02/01 by Y. Tanoudis, Tanoudis, Y., C. Daskaloyannis +1
Mathematics · Physics and Astronomy · #17B63 #70H06 #Advanced Fiber Laser Technologies #FOS: Physical sciences #Mathematical Physics (math-ph) #Photorefractive and Nonlinear Optics #Quantum Mechanics and Non-Hermitian Physics #math-ph #math.MP #msc:17B63 #msc:70H06

paper · pdf · doi:10.48550/arxiv.0902.0130

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

arxiv created 2009/02/01 · openalex publication_date 2009/02/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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