  We prove that as a one-dimensional Fermi gas is brought across the resonance
adiabatically from large repulsion to large attraction, the singlet ground
state will give way to the maximum spin state, which is the lowest energy state
among the states accessible to the system in this process. In the presence of
tiny symmetry breaking fields that destroy spin conservation, the singlet
ground state can evolve to the ferromagnetic state or a spin segregated state.
We have demonstrated these effects by exact calculations on fermion cluster
relevant to current experiments, and have worked out the quantum mechanical
wavefunction that exhibits phase separation.
