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Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials

J. Math. Phys. 16, 512 (1975); doi:10.1063/1.522574

Issue Date: March 1975

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C. P. Boyer
Centro de Investigacion en Matematicas Aplicadas y en Sistemas, Universidad Nacional Autonoma de México, México 20, D.F., Mexico

E. G. Kalnins and W. Miller, Jr.
Centre de recherches mathématiques, Université de Montréal, Montréal 101, P.Q., Canada
As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of SU (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
Permalink: http://link.aip.org/link/?JMAPAQ/16/512/1
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PACS

  • 03.65.Ge
    Classical and quantum physics; mechanics and fields Quantum mechanics Solutions of wave equations; bound states
  • 02.20.Sv
    Mathematical methods in physics Group theory Lie algebras of Lie groups
  • YEAR: 1975

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
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REFERENCES (10)
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  1. C. P. Boyer, E. G. Kalnins, and W. Miller Jr., J. Math. Phys. 16, 499 (1975).
  2. V. Bargmann, Commun. Pure Appl. Math. 14, 187 (1961).
  3. K. B. Wolf, J. Math. Phys. 15, 2102 (1974).
  4. J. Patera and P. Winternitz, J. Math. Phys. 14, 1130 (1970).
  5. F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, 1964).
  6. It is easy to see from our mapping that the subspace of polynomials in [script J]2 map onto polynomial solutions of (*) in L2(R2). This is closely related to a recent article on generalized heat polynomials, G. G. Bilodeau, SIAM J. Math. Anal, 5, 43 (1974).
  7. P. Kramer, M. Moshinsky, and T. H. Seligman, to appear in Group Theory and Its Applications III, edited by E. M. Loebl (Academic, New York, to be published);
    8. A. O. Barut and L. Girardello, Commun. Math. Phys. 21, 41 (1971).
  8. N. Ja. Vilenkin, Special Functions and the Theory of Group Representations (Amer. Math. Soc. Translation, Providence, R.I., 1968).
  9. E. Chacon and M. de Llano, Rev. Méx. Fis. 12, 57 (1963);
    10. Z. Pluhař and J. Tolar, Czech. J. Phys. 14, 287 (1964).
  10. W. Miller Jr., J. Math. Phys. 13, 648 (1972).

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