  We study 1D fermions with photoassociation or with a narrow Fano-Feshbach
resonance described by the Boson-Fermion resonance model. Using thebosonization
technique, we derive a low-energy Hamiltonian of the system. We show that at
low energy, the order parameters for the Bose Condensation and fermion
superfluidity become identical, while a spin gap and a gap against the
formation of phase slips are formed. As a result of these gaps, charge density
wave correlations decay exponentially in contrast with the phases where only
bosons or only fermions are present. We find a Luther-Emery point where the
phase slips and the spin excitations can be described in terms of
pseudofermions. This allows us to provide closed form expressions of the
density-density correlations and the spectral functions. The spectral functions
of the fermions are gapped, whereas the spectral functions of the bosons remain
gapless. The application of a magnetic field results in a loss of coherence
between the bosons and the fermion and the disappearance of the gap. Changing
the detuning has no effect on the gap until either the fermion or the boson
density is reduced to zero. Finally, we discuss the formation of a Mott
insulating state in a periodic potential. The relevance of our results for
experiments with ultracold atomic gases subject to one-dimensional confinement
is also discussed.
