Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 031, 24 pages      arXiv:1011.6548      https://doi.org/10.3842/SIGMA.2011.031
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

A Recurrence Relation Approach to Higher Order Quantum Superintegrability

Ernie G. Kalnins a, Jonathan M. Kress b and Willard Miller Jr. c
a) Department of Mathematics, University of Waikato, Hamilton, New Zealand
b) School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia
c) School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA

Received January 27, 2011, in final form March 20, 2011; Published online March 28, 2011

Abstract
We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates. We show that the method provides rigorous proofs of superintegrability and explicit constructions of higher order generators for the symmetry algebra. We apply the method to 5 families of systems, each depending on a parameter k, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system, and we give proofs of quantum superintegrability for all rational values of k, new for 4 of these systems. In addition, we show that the explicit information supplied by the special function recurrence relations allows us to prove, for the first time in 4 cases, that the symmetry algebra generated by our lowest order symmetries closes and to determine the associated structure equations of the algebras for each k. We have no proof that our generating symmetries are of lowest possible order, but we have no counterexamples, and we are confident we can can always find any missing generators from our raising and lowering operator recurrences. We also get for free, one variable models of the action of the symmetry algebra in terms of difference operators. We describe how the Stäckel transform acts and show that it preserves the structure equations.

Key words: superintegrability; quadratic algebras; special functions.

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