INTEGRABLE BCN ANALYTIC DIFFERENCE OPERATORS: HIDDEN PARAMETER SYMMETRIES AND EIGENFUNCTIONS

We consider integrable N-particle quantum systems of Calogero-Moser type, focusing on the ‘relativistic’ BC n setting, where commuting analytic difference operators arise. We show that the defining operators at the hyperbolic/elliptic levels, which depend on four/eight coupling constants, can be transformed to a manifestly D 4/D 8 symmetric form, resp. We survey various results on special eigenfunctions (including ‘ground states’) with regard to the latter symmetries and other ones. We also sketch a symmetry scenario for the arbitrary-N eigenfunctions, motivated by the hyperbolic BC 1 case, where our ‘relativistic’ hypergeometric function has all of the expected properties.