Quantum Calogero-Moser models: Complete integrability for all the root systems

Complete integrability of Calogero-Moser models is discussed from various angles. Calogero-Moser models and Toda models axe best known examples of solvable many-particle dynamics on a line which are based on root systems. At the classical level, the former (C-M models) is integrable for elliptic potentials (Weierstrass p function) and their various degenerate limits. The latter (Toda) has exponential potentials, which is obtained from the former as a special limit of the elliptic potential. Here we discuss quantum Calogero-Moser models based on any root system. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a Lax pair. (ii) Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalizations of the results known for the models based on the A series, i.e. su(N) type, root systems, which are treated rather explicitly in the Introduction. The independence of the conserved quantities (necessary for the Liouville integrability) obtained from the universal Lax pair is discussed in some detail.