  Motivated by current interest in strongly correlated quasi-one-dimensional
(1D) Luttinger liquids subject to axial confinement, we present a novel
density-functional study of few-electron systems confined by power-low external
potentials inside a short portion of a thin quantum wire. The theory employs
the 1D homogeneous Coulomb liquid as the reference system for a Kohn-Sham
treatment and transfers the Luttinger ground-state correlations to the
inhomogeneous electron system by means of a suitable local-density
approximation (LDA) to the exchange-correlation energy functional. We show that
such 1D-adapted LDA is appropriate for fluid-like states at weak coupling, but
fails to account for the transition to a ``Wigner molecules'' regime of
electron localization as observed in thin quantum wires at very strong
coupling. A detailed analyzes is given for the two-electron problem under axial
harmonic confinement.
