  As a means of realizing oscillatory pairing between fermions, we study
superfluid pairing between two fermion "spin" species that are confined to
adjustable spin-dependent trapping potentials. Focusing on the one-dimensional
limit, we find that with increasing separation between the spin-dependent traps
the fermions exhibit distinct phases, including a fully paired phase, a
spin-imbalanced phase with oscillatory pairing, and an unpaired fully
spin-polarized phase. We obtain the phase diagram of fermions in such a
spin-split trap and discuss signatures of these phases in cold-atom
experiments.
