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Within a local-density approximation, the different phases that appear in a trap with a spatially dependent coupling constant v (x) due to the decrease in the local Fermi wave vector kF (x) from the center of the trap to its edge simply follow from a parabolic (vertical at unitarity) line in Fig., h/ εF (x) = (2h / | Eb |) v2 (x), where | Eb | =1/ (m a2) for both positive and negative scattering lengths a. Its curvature 2h/ | Eb | is fixed by the global imbalance, while the initial point v (x=0) is determined by the local Fermi wave vector at the trap center x=0. Note that in a trap different phases are spatially separated for both continuous and first-order transition lines. By contrast, genuine phase separation is associated with a first-order transition and appears only in the homogeneous system.
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