Quantum Calogero-Moser models for any root system

Calogero-Moser models and Toda models are 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 (Weierstra beta 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) Lionville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. M Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based on the A series, i.e. su(N) type, root systems. Solution methods of quantum Calogero-Moser models are expected to give some clues for understanding dynamical symmetries, non-perturbative methods for field theories, etc.