Models of quadratic algebras generated by second order superintegrable systems in 2 dimensions

Sarah Post

A Hamiltonian, or energy function, is a central object in the study of classical and quantum mechanics. We can exploit the higher order symmetry of a Hamiltonian by studying its ``constants of the motion``, classically, or commuting differential operators for quantum systems. Systems with a maximal number of independent symmetries are called superintegrable and can be used to find eigenvalues and eigenfunctions for the Hamiltonian by algebraic means. Specifically, if we require the symmetry operators to be 2nd order, we obtain a quadratic algebra generated by the operators. Models of such algebras can be analyzed to find spectral information as well as many connections with special function theory. In many cases, it is possible to construct Bargmann-type Hilbert spaces from the algebras; these include both a functional inner product and a delta function belonging to the space.

In this talk, I hope to give a brief introduction to superintegrable systems in two dimensions and give examples of one variable models of the quadratic algebras including both differential and difference operator models. We see the appearance of hypergeometric and Bessel functions as well as Wilson polynomials from the models. I also give an example of the analysis of a variable mass Hamiltonian using models of the quadratic algebra.

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