Superintegrability and higher order integrals for quantum systems

We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrodinger eigenvalue equation II Psi (Delta(2) + V) Psi = E Psi on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. The flat space equations with potentials V = alpha(x + iy)(k-1)/(x - iy)(k+1) in Cartesian coordinates, and V = ar(2) + beta/r(2) cos(2) k theta + gamma/r(2) sin(2) k theta (the Tremblay, Turbiner and Winternitz system) in polar coordinates, have each been shown to be classically superintegrable for all rational numbers k. We apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k. We develop the classical analog of the quantum canonical form for a symmetry. It is clear that our methods will generalize to other Hamiltonian systems.