  We investigate one-dimensional three-body systems composed of two identical
bosons and one imbalanced atom (impurity) with two-body and three-body
zero-range interactions. For the case in the absence of three-body interaction,
we give a complete phase diagram of the number of three-body bound states in
the whole region of mass ratio via the direct calculation of the
Skornyakov-Ter-Martirosyan equations. We demonstrate that other low-lying
three-body bound states emerge when the mass of the impurity particle is not
equal to another two identical particles. We can obtain not only the binding
energies but also the corresponding wave functions. When the mass of impurity
atom is vary large, there are at most three three-body bound states. We then
study the effect of three-body zero-range interaction and unveil that it can
induces one more three-body bound state at a certain region of coupling
strength ratio under a fixed mass ratio.
