On the Plancherel Formula for the (Discrete) Laplacian in a Weyl Chamber with Repulsive Boundary Conditions at the Walls

Abstract

It is known from early work of Gaudin that the quantum system of n Bosonic particles on the line with a pairwise delta-potential interaction admits a natural generalization in terms of the root systems of simple Lie algebras. The corresponding quantum eigenvalue problem amounts to that of a Laplacian in a convex cone, the Weyl chamber, with linear homogeneous boundary conditions at the walls. In this paper we study a discretization of this eigenvalue problem, which is characterized by a discrete Laplacian on the dominant cone of the weight lattice endowed with suitable linear homogeneous conditions at the boundary. The eigenfunctions of this discrete model are computed by the Bethe Ansatz method. The orthogonality and completeness of the resulting Bethe wave functions (i.e., the Plancherel formula) turn out to follow from an elementary computation performed by Macdonald in his study of the zonal spherical functions on p-adic simple Lie groups. Through a continuum limit, the Plancherel formula for the ordinary Laplacian in the Weyl chamber with linear homogeneous boundary conditions is recovered. Throughout this paper we restrict ourselves to the case of repulsive boundary conditions.