  We use the Hartree-Fock method to study an interacting one-dimensional
electron system on a finite wire, partially depleted at the center by a smooth
potential barrier. A uniform one-Tesla Zeeman field is applied throughout the
system. We find that with the increase in the potential barrier, the low
density electrons under it go from a non-magnetic state to an antiferromagnetic
state, and then to a state with a well-localized spin-aligned region isolated
by two antiferromagnetic regions from the high density leads. At this final
stage, in response to a continuously increasing barrier potential, the system
undergoes a series of abrupt density changes, corresponding to the successive
expulsion of a single electron from the spin-aligned region under the barrier.
Motivated by the recent momentum-resolved tunneling experiments in a parallel
wire geometry, we also compute the momentum resolved tunneling matrix elements.
Our calculations suggest that the eigenstates being expelled are spatially
localized, consistent with the experimental observations. However, additional
mechanisms are needed to account for the experimentally observed large spectral
weight at near $k=0$ in the tunneling matrix elements.
