Hyperspherical Harmonics, Separation of Variables and the Bethe Ansatz

Details Hyperspherical Harmonics, Separation of Variables and the Bethe Ansatz Hyperspherical Harmonics, Separation of Variables and the Bethe Ansatz

1994-05-12

arxiv.org/abs/hep-th/9405085v1

Lett.Math.Phys. 33 (1995) 61-74

Physics

High Energy Physics

High Energy Physics - Theory

14 pgs, Plain Tex, preprint CRM--2174 (May, 1994)

Scientific paper

10.1007/BF00750812

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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