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Higher Order Quantum Superintegrability: A New “Painlevé Conjecture”

Higher Order Quantum Superintegrability
  • Ian MarquetteEmail author
  • Pavel Winternitz
Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order N > 2. Two types of such superintegrable potentials exist. The first type consists of “standard potentials” that satisfy linear differential equations. The second type consists of “exotic potentials” that satisfy nonlinear equations. For N = 3, 4, and 5 these equations have the Painlevé property. We conjecture that this is true for all N ≥ 3. The two integrals X and Y  commute with the Hamiltonian, but not with each other. Together they generate a polynomial algebra (for any N) of integrals of motion. We show how this algebra can be used to calculate the energy spectrum and the wave functions.

Keywords

Superintegrable systems Painlevé transcendents Polynomial algebras Exact solvability Higher order integrals Chazy class 

PACS

03.65.Fd 

Notes

Acknowledgements

The research of I. M. was supported by the Australian Research Council through Discovery Early Career Researcher Award DE130101067 and Australian Research Council Discovery Project DP 160101376. The research of P.W. was partially supported by an NSERC discovery research grant.

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Authors and Affiliations

  1. 1.School of Mathematics and PhysicsThe University of QueenslandBrisbane, St-LuciaAustralia
  2. 2.Centre de recherches mathématiques et Département de Mathématiques et de StatistiqueUniversité de MontréalMontréalCanada

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