Browsing by Author "Post, Sarah"

Now showing 1 - 4 of 4
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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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    Models of quadratic quantum algebras and their relation to classical superintegrable systems
    (2008-09-12) Kalnins, E.G.; Miller Jr., W.; Post, Sarah
    We show how to construct realizations (models) of quadratic algebras for 2 dimensional second order superintegrable systems in terms of differential or difference operators in one variable. We demonstrate how various models of the quantum algebras arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras related to superintegrable systems in n dimensions and are intimately related to multivariable orthogonal polynomials.
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    Models of second-order superintegrable systems.
    (2009-08) Post, Sarah
    The study of superintegrable systems has progressed far beyond analysis of specific examples, especially in the case where the constants of the motion are quadratic in the momenta. In this thesis, I begin with a brief overview of the structure analysis for second order superintegrable systems both in classical and quantum mechanics. In 2d and 3d conformally at spaces, the algebra generated by the constants of the motion has been proven to be a finitely generated quadratic algebra with closure at finite order. Models are exhibited of the quadratic algebras for each equivalence class of 2d second order quantum superintegrable systems. I also describe some classical models of the algebras and their role in determining the quantum systems. Finally, a model for the 3d singular isotropic oscillator quadratic algebra is given.
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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.