Classical and quantum higher order superintegrable systems from coalgebra symmetry

Abstract

The N-dimensional generalization of Bertrand spaces as families of maximally superintegrable (M.S.) systems on spaces with a nonconstant curvature is analyzed. Considering the classification of two-dimensional radial systems admitting three constants of motion at most quadratic in momenta, we will be able to generate a new class of spherically symmetric M.S. systems by using a technique based on coalgebra. The three-dimensional realization of these systems provides the entire classification of classical spherically symmetric M.S. systems admitting periodic trajectories. We show that in dimension N > 2, these systems (classical and quantum) admit, in general, higher order constants of motion and turn out to be exactly solvable. Furthermore, it is possible to obtain non-radial M.S. systems by introducing the projection of the original radial system to a suitable lower dimensional space.