3 FORMULAS FOR EIGENFUNCTIONS OF INTEGRABLE SCHRODINGER-OPERATORS

We give three formulae for meromorphic eigenfunctions (scattering star es) of Sutherland's integrable N-body Schrodinger operators and their generalizations. The first is an explicit computation of the Etingof-K irillov traces of intertwining operators, the second an integral repre sentation of hypergeometric type, and the third is a formula of Bethe ansatz type. The last two formulas are degenerations of elliptic formu las obtained previously in connection with the Knizhnik-Zamolodchikov- Bernard equation. The Bethe ansatz formulas in the elliptic case are r eviewed and discussed in more detail here: Eigenfunctions are parametr ized by a 'Hermite-Bethe' variety, a generalization of the spectral va riety of the Lame operator, We also give the q-deformed version of our first formula. In the scalar sin case, this gives common eigenfunctio ns of the commuting Macdonald-Ruijsenaars difference operators.