Kalnins, Ernie G.Kress, Jonathan M.Miller, W., Jr.2010-06-282010-06-282010Kalnins, E.G., Kress, J.M. & Miller, W., Jr. (2010). Superintegrability and higher order integrals for quantum systems. Journal of Physics A: Mathematical and Theoretical, 43(26), Art. No. 265205.https://hdl.handle.net/10289/4050We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (Δ2 + V)Ψ = EΨ 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 = α(x + iy)k − 1/(x − iy)k + 1 in Cartesian coordinates, and V = αr² + β/r²cos ²kθ + γ/r²sin ²kθ (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.enmathematicsSchrödinger eigenvalue equationJournal Article10.1088/1751-8113/43/26/265205