  Using the asymptotic Bethe Ansatz, we obtain an exact solution of the
many-body problem for 1D spin-polarized fermions with resonant p-wave
interactions, taking into account the effects of both scattering volume and
effective range. Under typical experimental conditions, accounting for the
effective range, the properties of the system are significantly modified due to
the existence of "shape" resonances. The excitation spectrum of the considered
model has unexpected features, such as the inverted position of the particle-
and hole-like branches at small momenta, and roton-like minima. We find that
the frequency of the "breathing" mode in the harmonic trap provides an
unambiguous signature of the effective range.
