Speaker
Description
Superintegrable systems provide a natural laboratory in which to study symmetries, hidden structures and exactly solvable quantum models. A classical example is the hydrogen atom: the high degeneracy of its bound-state spectrum is explained by an o(4) symmetry, while the full collection of bound states can be organized using the larger dynamical algebra o(4,2), the Lie algebra of the conformal group of space-time.
I will use this familiar example as motivation for the generic superintegrable system on the two-sphere. For this model, the symmetry algebra is the rank-one Racah algebra. I will explain how it sits inside a larger algebraic structure, namely the rank-two Jacobi algebra, which is the dynamical algebra of the system. This identification gives an algebraic solution of the quantum model and explains the natural appearance of two-variable Jacobi polynomials and univariate Racah polynomials.