New spin Calogero-Sutherland models related to 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 appr oach 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 tha t a wide class of quadratic combinations involving these three sets of Dunk l operators always yields a spin Calogero-Sutherland model, which is (quasi -)exactly solvable by construction. We show that all the spin Calogero-Suth erland models obtainable within this framework can be expressed in a unifie d way in terms of a Weierstrass rho function with suitable half-periods. Th is provides a natural spin counterpart of the well-known general formula fo r a scalar completely integrable potential of BCN type due to Olshanetsky a nd Perelomov. As an illustration of our method, we exactly compute several energy levels and their corresponding wavefunctions of an elliptic quasi-ex actly solvable potential for two and three particles of spin 1/2. (C) 2001 Elsevier Science B.V. All rights, reserved.