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Letters in Mathematical Physics

, Volume 33, Issue 1, pp 61–74 | Cite as

Harmonics on hyperspheres, separation of variables and the bethe ansatz

  • J. Harnad
  • P. Winternitz
Article

Abstract

The relation between solutions to Helmholtz's equation on the sphereSn-1 and the\([\mathfrak{s}\mathfrak{l}(2)]^n \) n Gaudin spin chain is clarified. The joint eigenfunctions of the Laplacian and a complete set of commuting second-order operators suggested by theR-matrix approach to integrable systems, based on the loop algebra\(\widetilde{\mathfrak{s}\mathfrak{l}}(2)_R \), are found in terms of homogeneous polynomials in the ambient space. The relation of this method of determining a basis of harmonic functions onSn-1 to the Bethe ansatz approach to integrable systems is explained.

Mathematics Subject Classifications (1991)

81R99 82B23 17B65 35J05 34A26 

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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • J. Harnad
    • 1
    • 2
  • P. Winternitz
    • 3
  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontréalCanada
  2. 2.Centre de recherches mathématiquesUniversité de Montréal C.P.MontréalCanada
  3. 3.Centre de recherches mathématiquesUniversité de Montréal C. P.MontréalCanada

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