  Based on the integrable Gaudin model and local density approximation, we
discuss the phase structure of one-dimensional trapped Fermi gases with
imbalanced spin populations for arbitrary attractive interactions. A phase
separation state, with a polarized superfluid core immersed in an unpolarized
superfluid shell, emerges below a critical spin polarization. Above it, a
coexistence of polarized superfluid matter and a fully polarized normal gas is
favored. These two exotic states could be realized experimentally in highly
elongated atomic traps, and diagnosed by measuring the lowest density
compressional mode. We identify the polarized superfluid as having an FFLO
structure, and predict the resulting mode frequency as a function of the spin
polarization.
