New spin Calogero–Sutherland models related to BN-type Dunkl operators

Abstract

We construct several new families of exactly and quasi-exactly solvable BC N -type Calogero–Sutherland models with internal degrees of freedom. Our approach is based on the introduction of a new family of Dunkl operators of B N type which, together with the original B N -type Dunkl operators, are shown to preserve certain polynomial subspaces of finite dimension. We prove that a wide class of quadratic combinations involving these three sets of Dunkl operators always yields a spin Calogero–Sutherland model, which is (quasi-)exactly solvable by construction. We show that all the spin Calogero–Sutherland models obtainable within this framework can be expressed in a unified way in terms of a Weierstrass ℘ function with suitable half-periods. This provides a natural spin counterpart of the well-known general formula for a scalar completely integrable potential of BC N type due to Olshanetsky and Perelomov. As an illustration of our method, we exactly compute several energy levels and their corresponding wavefunctions of an elliptic quasi-exactly solvable potential for two and three particles of spin 1/2.