Browsing by Subject "quadratic algebras"

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    Models for the 3D singular isotropic oscillator quadratic algebra
    (2008-11-06) Kalnins, E.G.; Miller Jr., W.; Post, Sarah
    We give the first explicit construction of the quadratic algebra for a 3D quantum superintegrable system with nondegenerate (4-parameter) potential together with realizations of irreducible representations of the quadratic algebra in terms of differential-differential or differential-difference and difference-difference operators in two variables. The example is the singular isotropic oscillator. We point out that the quantum models arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras for superintegrable systems in $n$ dimensions and are closely related to Hecke algebras and multivariable orthogonal polynomials.
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    Structure theory for 2D superintegrable systems with 1-parameter potentials
    (2008-11-06) Kalnins, E.G.; Kress, Jonathan M.; Miller Jr., W.; Post, Sarah
    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 St\"ackel equivalent to a system on a constant curvature space.