Quantum completely integrable systems connected with semi-simple Lie algebras

The quantum version of the dynamical systems whose integrability is related to the root systems of semi-simple Lie algebras are considered. It is proved that the operators Jsub(k) introduced by Calogero et al. are integrals of motion and that they commute. The explicit form of another class of integrals of motion is given for systems with few degrees of freedom. (Auth.)