Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd $k$

@article{Quesne2009SuperintegrabilityOT,
  title={Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd \$k\$},
  author={C Quesne},
  journal={arXiv: Mathematical Physics},
  year={2009}
}
  • C. Quesne
  • Published 23 November 2009
  • Mathematics
  • arXiv: Mathematical Physics
In a recent FTC by Tremblay {\sl et al} (2009 {\sl J. Phys. A: Math. Theor.} {\bf 42} 205206), it has been conjectured that for any integer value of $k$, some novel exactly solvable and integrable quantum Hamiltonian $H_k$ on a plane is superintegrable and that the additional integral of motion is a $2k$th-order differential operator $Y_{2k}$. Here we demonstrate the conjecture for the infinite family of Hamiltonians $H_k$ with odd $k \ge 3$, whose first member corresponds to the three-body… 
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We extend recent work by Tremblay, Turbiner, and Winternitz which analyzes an infinite family of solvable and integrable quantum systems in the plane, indexed by the positive parameter k. Key
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We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrodinger eigenvalue equation HΨ ≡ (Δ2 + V)Ψ = EΨ on any 2D Riemannian manifold, real or complex,
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