Three-body generalization of the Sutherland model with internal degrees of freedom

A generalized spin Sutherland model including a three-body potential is proposed. The problem is analyzed in terms of three first-order differential-difference operators, obtained by combining SUSYQM supercharges with the elements of the dihedral group D6. Three alternative commuting operators are also introduced. PACS: 03.65.Fd, 02.20.Df, 11.30.Pb Directeur de recherches FNRS; E-mail: [email protected] 1 In recent years, the Sutherland one-dimensional N -particle model [1] and its rational limit, the Calogero model [2], have received considerable attention in the literature because they are relevant to several important physical problems (for a list of references, see e.g. [3]). The Sutherland problem can be analyzed in terms of a set of N commuting first-order differential-difference operators [4], related to the root system of the AN−1 algebra [5] and known in the mathematical literature as Dunkl operators [6]. Use of the latter leads to a Hamiltonian with exchange terms, connected with an extension of the model for particles with internal degrees of freedom, referred to as the spin Sutherland problem [7, 8]. A similar type of approach can be employed [9] for other integrable models related to root systems of Lie algebras [5]. In the present letter, we shall deal with a generalized Sutherland three-particle problem including an extra three-body trigonometric potential. Such a problem is related to the exceptional Lie algebra G2, whose Weyl group, of order 12, is the dihedral group D6 [5]. In contrast with the approaches used elsewhere [4, 6, 7, 8, 9], our starting point will be an analysis of the problem in supersymmetric quantum mechanics (SUSYQM), thereby emphasizing the link between Dunkl operators and SUSYQM. Let us consider a system of three particles on a circle of length π/a interacting via long-range twoand three-body potentials. Its Hamiltonian is defined by H = − 3 ∑