  We consider the three-boson problem with $\delta$-function interactions in
one spatial dimension. Three different approaches are used to calculate the
phase shifts, which we interpret in the context of the effective range
expansion, for the scattering of one free particle a off of a bound pair. We
first follow a procedure outlined by McGuire in order to obtain an analytic
expression for the desired S-matrix element. This result is then compared to a
variational calculation in the adiabatic hyperspherical representation, and to
a numerical solution to the momentum space Faddeev equations. We find excellent
agreement with the exact phase shifts, and comment on some of the important
features in the scattering and bound-state sectors. In particular, we find that
the 1+2 scattering length is divergent, marking the presence of a zero-energy
resonance which appears as a feature when the pair-wise interactions are
short-range. Finally, we consider the introduction of a three-body interaction,
and comment on the cutoff dependence of the coupling.
