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New spin Calogero-Sutherland models related to B_N-type Dunkl operators

F. Finkel·D. Gomez-Ullate·A. Gonzalez-Lopez·M. A. Rodriguez·R. Zhdanov
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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 two new families 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 P 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.

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F. Finkel, D. Gomez-Ullate, A. Gonzalez-Lopez, M. A. Rodriguez, R. Zhdanov. 2001-04-18. New spin Calogero-Sutherland models related to B_N-type Dunkl operators. https://doi.org/10.1016/s0550-3213(01)00378-9

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