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Recursion operators, higher-order symmetries and superintegrability in quantum mechanics

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Czechoslovak Journal of Physics Aims and scope

Abstract

A connection between the theory of superintegrable quantum-mechanical systems, which admit a maximal number of integrals of motion, and the standard Lie group theory is established. It is shown that the flows generated by first- and second-order Lie symmetries of the bidimensional Schrödinger equation can be classified and interpreted as quantum-mechanical operators which commute with integrable or superintegrable Hamiltonians. In this way, all known superintegrable potentials in the plane are naturally obtained and slightly more general integrals of motion are found.

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Authors and Affiliations

  1. Feza Gürsey Institute, PO Box 6, Cengelkoy, 81220, Istanbul, Turkey

    M. B. Sheftel

  2. Department of Higher Mathematics, North Western Polytechnical Institute, Millionnaya Str. 5, 191186, St. Petersburg, Russia

    M. B. Sheftel

  3. Università di Lecce and INFN sez. di Lecce, via per Arnesano, 73100, Lecce, Italy

    P. Tempesta

  4. Centre de Recherches Mathématiques and Dép. de Mathématiques et Statistique, Université de Montréal, C.P. 6128-CV, H3C3J7, Montréal, QC, Canada

    P. Winternitz

Authors
  1. M. B. Sheftel
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  2. P. Tempesta
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  3. P. Winternitz
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Additional information

This study was started while M.S. and P.T. were visiting the Centre de Recherches Mathématiques, Université de Montréal and finished while M.S. and P.W. were visiting the Dipartimento di Fisica, Università di Lecce.

The research of P.W. was partly supported by research grants from NSERC of Canada and FCAR du Québec.

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Sheftel, M.B., Tempesta, P. & Winternitz, P. Recursion operators, higher-order symmetries and superintegrability in quantum mechanics. Czech J Phys 51, 392–399 (2001). https://doi.org/10.1023/A:1017553909398

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  • DOI: https://doi.org/10.1023/A:1017553909398

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