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
1T 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 nonmagnetic 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 near
k=0 in the tunneling matrix elements.