Polynomial Associative Algebras for Quantum Superintegrable Systems with a Third Order Integral of Motion

  • Ian Marquette
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 144)


We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integral of motion. We construct the most general associative cubic algebra and we present specific realizations. We use them to calculate the energy spectrum. All classical and quantum superintegrable potentials separable in cartesian coordinates with a third order integral are known. The general formalism is applied to one of the quantum potentials.


Associative Algebra Casimir Operator Polynomial Algebra Quantum Potential Superintegrable System 
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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Ian Marquette
    • 1
  1. 1.Département de physique et Centre de recherche mathématiqueUniversité de MontréalSuccursale Centre-Ville, MontréalCanada

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