B N - type Dunkl operators

We construct several new families of exactly and quasi-exactly solvable BCN -type Calogero– Sutherland models with internal degrees of freedom. Our approach is based on the introduction of a new family of Dunkl operators of BN type which, together with the original BN -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 BCN 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.