INTEGRABILITY OF DIFFERENCE CALOGERO-MOSER SYSTEMS

A general class of n-particle difference Calogero-Moser systems with e lliptic potentials is introduced. Besides the step size and two period s, the Hamiltonian depends on nine coupling constants. We prove the qu antum integrability of the model for n = 2 and present partial results for n greater-than-or-equal-to 3. degenerate cases (rational, hyperbo lic, or trigonometric limit), the integrability follows for arbitrary particle number from previous work connected with the multivariable q- polynomials of Koornwinder and Macdonald. Liouville integrability of t he corresponding classical systems follows as a corollary. Limit trans itions lead to various well-known models such as the nonrelativistic C alogero-Moser systems associated with classical root systems and the r elativistic Calogero-Moser system.