Supersymmetric Calogero-Moser-Sutherland models: superintegrability structure and eigenfunctions

We first review the construction of the supersymmetric extension of the quantum CalogeroMoser-Sutherland (CMS) models. We stress the remarkable fact that this extension is completely captured by the insertion of a fermionic exchange operator in the Hamiltonian: sCMS models (s for supersymmetric) are nothing but special exchange-type CMS models. Under the appropriate projection, the conserved charges can thus be formulated in terms of the standard Dunkl operators. This is illustrated in the rational case, where the explicit form of the 4N (N being the number of bosonic variables) conserved charges is presented, together with their full algebra. The existence of 2N commuting bosonic charges settles the question of the integrability of the srCMS model. We then prove its superintegrability by displaying 2N − 2 extra independent charges commuting with the Hamiltonian. In the second part, we consider the supersymmetric version of the trigonometric case (stCMS model) and review the construction of its eigenfunctions, the Jack superpolynomials. This leads to closed-form expressions, as determinants of determinants involving supermonomial symmetric functions. Here we focus on the main ideas and the generic aspects of the construction: those applicable to all models whether supersymmetric or not. Finally, the possible Lie superalgebraic structure underlying the stCMS model and its eigenfunctions is briefly considered. [email protected] [email protected] [email protected] 1To appear in the proceedings of the Workshop on superintegrability in classical and quantum systems, September 16-22 2002, Centre de recherches mathématiques, Université de Montréal, ed. by P. Winternitz.