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arXiv · hep-th/9405085

Hyperspherical Harmonics, Separation of Variables and the Bethe Ansatz

J. Harnad·P. Winternitz
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Abstract

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

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J. Harnad, P. Winternitz. 1994-05-12. Hyperspherical Harmonics, Separation of Variables and the Bethe Ansatz. https://doi.org/10.1007/bf00750812

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