Completeness of multiseparable superintegrability on the complex 2-sphere

The possibility that Schrödinger's equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability. Here we demonstrate, for nondegenerate potentials, how to establish a complete list of such potentials that are superintegrable on the complex 2-sphere, using essentially algebraic means. We classify all such potentials that admit a pair of second-order constants of motion. Here 'nondegenerate' means that the potentials depend on four independent parameters. The method of proof generalizes to other spaces and dimensions. We show for the 2-sphere that all these superintegrable systems possess the remarkable property that they correspond to quadratic algebras, and we work out the detailed structure relations and their quantum analogues.