Free harmonic oscillators, Jack polynomials, and Calogero-Sutherland systems

The algebraic structure and the relationships between the eigenspaces of th e Calogero-Sutherland model (CSM) and the Sutherland model (SM) on a circle are investigated through the Cherednik operators. We find an exact connect ion between the simultaneous nonsymmetric eigenfunctions of the A(N-1) Cher ednik operators, from which the eigenfunctions of the CSM and SM are constr ucted, and the monomials. This construction allows us to simultaneously dia gonalize both CSM and SM (after gauging away the Hamiltonians by suitable m easures) and also enables us to write down a harmonic oscillator algebra in volving the Cherednik operators, which yields the raising and lowering oper ators for both of these models. The connections of the CSM with free oscill ators and the SM with free particles on a circle are established in a novel way. We also point out the subtle differences between the excitations of t he CSM and the SM.