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

TL;DR

This paper introduces new families of BC_N-type Calogero-Sutherland models with internal spins, constructed via novel Dunkl operators, and demonstrates their (quasi-)exact solvability and explicit solutions for specific cases.

Contribution

The authors develop new Dunkl operators of B_N type that preserve polynomial subspaces, enabling the construction of a unified class of spin Calogero-Sutherland models with explicit solutions.

Findings

New Dunkl operators of B_N type introduced.

Construction of (quasi-)exactly solvable spin models.

Explicit energy levels and wavefunctions for specific cases.

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.