Superintegrability and higher order integrals for quantum systems

E. G. Kalnins, J. M. Kress, W. Miller

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45 Scopus citations


We 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 2k + γ/r 2sin 2k (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 languageEnglish (US)
Article number265205
JournalJournal of Physics A: Mathematical and Theoretical
Issue number26
StatePublished - Jun 16 2010


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