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Superintegrability and higher-order constants for classical and quantum systems

  • Elementary Particles and Fields
  • Theory
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Abstract

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 components of their analysis were to demonstrate that there are closed orbits in the corresponding classical system if k is rational, and for a number of examples there are generating quantum symmetries that are higher order differential operators than two. Indeed they conjectured that for a general class of potentials of this type, quantum constants of higher order should exist. We give credence to this conjecture by showing that for an even more general class of potentials in classicalmechanics, there are higher-order constants of the motion as polynomials in the momenta. Thus these systems are all superintegrable.

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Authors and Affiliations

  1. Department of Mathematics, University of Waikato, Hamilton, New Zealand

    E. G. Kalnins

  2. School of Mathematics, University of Minnesota, Minneapolis, USA

    W. Miller Jr.

  3. Laboratory of Theoretical Physics, Joint Institute of Nuclear Research, Dubna, Russia

    G. S. Pogosyan

Authors
  1. E. G. Kalnins
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  2. W. Miller Jr.
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  3. G. S. Pogosyan
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The text was submitted by the authors in English.

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Kalnins, E.G., Miller, W. & Pogosyan, G.S. Superintegrability and higher-order constants for classical and quantum systems. Phys. Atom. Nuclei 74, 914–918 (2011). https://doi.org/10.1134/S1063778811060159

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  • DOI: https://doi.org/10.1134/S1063778811060159

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