Hyperspherical Harmonics, Separation of Variables and the Bethe Ansatz

TL;DR

This paper explores the connection between harmonic functions on spheres, the Bethe ansatz method, and integrable systems, providing explicit solutions and clarifying their interrelations through algebraic and analytical techniques.

Contribution

It establishes a link between solutions to Helmholtz's equation on spheres and the Bethe ansatz approach in integrable systems, introducing a new method for constructing harmonic functions.

Findings

Explicit joint eigenfunctions expressed as homogeneous polynomials.

Clarification of the relation between harmonic analysis and the Bethe ansatz.

Connection of the R-matrix approach to classical harmonic functions.

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.