  The algebraic structure and the relationships between the eigenspaces of the
Calogero-Sutherland model (CSM) and the Sutherland model (SM) on a circle are
investigated through the Cherednik operators. We find an exact connection
between the simultaneous non-symmetric eigenfunctions of the $A_{N-1}$
Cherednik operators, from which the eigenfunctions of the CSM and SM are
constructed, and the monomials. This construction, not only, allows one to
write down a harmonic oscillator algebra involving the Cherednik operators,
which yields the raising and lowering operators for both of these models, but
also shows the connection of the CSM with free oscillators and the SM with free
particles on a circle. We also point out the subtle differences between the
excitations of the CSM and the SM.
