Dynamical symmetries for superintegrable quantum systems

Abstract

We study the dynamical symmetries of a class of two-dimensional superintegrable systems on a 2-sphere, obtained by a procedure based on the Marsden-Weinstein reduction, by considering its shape-invariant intertwining operators. These are obtained by generalizing the techniques of factorization of one-dimensional systems. We firstly obtain a pair of noncommuting Lie algebras $su(2)$ that originate the algebra $so(4)$. By considering three spherical coordinate systems we get the algebra $u(3)$ that can be enlarged by `reflexions' to $so(6)$. The bounded eigenstates of the Hamiltonian hierarchies are associated to the irreducible unitary representations of these dynamical algebras.