Algebraic integrability for the Schr�dinger equation and finite reflection groups

Abstract

Algebraic integrability of ann-dimensional Schrodinger equation means that it has more thann independent quantum integrals. Forn=1, the problem of describing such equations arose in the theory of finite-gap potentials. The present paper gives a construction which associates finite reflection groups (in particular, Weyl groups of simple Lie algebras) with algebraically integrable multidimensional Schrodinger equations. These equations correspond to special values of the parameters in the generalization of the Calogero—Sutherland system proposed by Olshanetsky and Perelomov. The analytic properties of a joint eigenfunction of the corresponding commutative rings of differential operators are described. Explicit expressions are obtained for the solution of the quantum Calogero—Sutherland problem for a special value of the coupling constant.