  We present a novel and comprehensive microscopic study of Luther-Emery-paired
phases in a strongly interacting atomic Fermi gas inside a parabolic trap and a
one-dimensional (1D) optical lattice. Our work is based on a lattice version of
density-functional theory, which uses as reference system the 1D homogeneous
Hubbard model. We test our approach for repulsive interactions against Quantum
Monte Carlo data and show that, for sufficiently strong attractions, an
atomic-density wave (ADW) in the central portion of the trap breaks the
discrete translational symmetry of the underlying lattice. We demonstrate that
the emergence of an ADW has a dramatic impact on experimental observables such
as the Fraunhofer diffraction pattern and the momentum distribution.
