  The 40-year-old Calogero-Sutherland (CS) model remains a source of
inspirations for understanding 1d interacting fermions. At $\beta=1,
\text{or}0$, the CS model describes a free non-relativistic fermion, or boson
theory, while for generic $\beta$, the system can be interpreted either as
interacting fermions or bosons, or free anyons depending on the context.
However, we shall show in this letter that the fermionic picture is
advantageous in diagonalizing the CS Hamiltonian. Comparing to the previously
known multi-integral representation or the Dunkl operator formalism for the CS
wave functions, our method depends on the (upper or lower) triangular nature of
the fermion interaction, which is resolved in perturbation theory of the second
quantized form. The eigenstate is constructed from a multiplet of unperturbed
states and the perturbation is of finite order. The full construction is a
similarity transformation from the free fermion theory, in the same spirit as
the Landau Fermi liquid theory and the 1d Luttinger liquid theory. That means
quasi-particles or anyons can be represented in terms of free fermion modes (or
bosonic modes via bosonization). The method is applicable to other (higher than
one space dimension) systems for which the adiabatic theorem applies.
