### Abstract

We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (Δ_{2} + 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 = αr^{2} + β/r^{2}cos ^{2}k + γ/r ^{2}sin ^{2}k (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.

Original language | English (US) |
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Article number | 265205 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 43 |

Issue number | 26 |

DOIs | |

State | Published - Jun 16 2010 |

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## Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*43*(26), [265205]. https://doi.org/10.1088/1751-8113/43/26/265205