Quantum Calogero-Moser models: integrability for all root systems

Abstract

The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. 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 total sum of the Lax matrix L, i.e. ∑µ,ν∊(Ln)µν, in which is a set of r vectors invariant under the action of the Coxeter group. They form a single Coxeter orbit. (ii) Proof of Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. Generalized Jack polynomials are defined for all root systems as unique eigenfunctions of the Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (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.