  We present a study of a gas of bosons confined in one dimension with Feshbach
resonant interactions, at zero temperature. Unlike the gas of one dimensional
bosons with non-resonant interactions, which is known to be equivalent to a
system of interacting spinless fermions and can be described using the
Luttinger liquid formalism, the resonant gas possesses novel features.
Depending on its parameters, the gas can be in one of three possible regimes.
In the simplest of those, it can still be described by the Luttinger liquid
theory, but its Fermi momentum cannot be larger than a certain cutoff momentum
dependent on the details of the interactions. In the other two regimes, it is
equivalent to a Luttinger liquid at low density only. At higher densities its
excitation spectrum develops a minimum, similar to the roton minimum in helium,
at momenta where the excitations are in resonance with the Fermi sea. As the
density of the gas is increased further, the minimum dips below the Fermi
energy, thus making the ground state unstable. At this point the standard
ground state gets replaced by a more complicated one, where not only the states
with momentum below the Fermi points, but also the ones with momentum close to
that minimum, get filled, and the excitation spectrum develops several
branches. We are unable so far to study this new regime in detail due to the
lack of the appropriate formalism.
