  Two-component Fermi gases with tunable repulsive or attractive interactions
inside quasi-one-dimensional (Q1D) harmonic wells may soon become the cleanest
laboratory realizations of strongly correlated Luttiger and Luther-Emery
liquids under confinement. We present a microscopic Kohn-Sham
density-functional theory of these systems, with specific attention to a gas on
the approach to a confinement-induced Feshbach resonance. The theory employs
the one-dimensional Gaudin-Yang model as the reference system and transfers the
appropriate Q1D ground-state correlations to the confined inhomogeneous gas
{\it via} a suitable local-density approximation to the exchange and
correlation energy functional. Quantitative understanding of the role of the
interactions in the bulk shell structure of the axial density profile is
thereby achieved. While repulsive intercomponent interactions depress the
amplitude of the shell structure of the noninteracting gas, attractive
interactions stabilize atomic-density waves through spin pairing. These should
be clearly observable in atomic clouds containing of the order of up to a
hundred atoms.
