Born-Oppenheimer study of two-component few-particle systems under one-dimensional confinement

The energy spectrum, atom-dimer scattering length, and atom-trimer scattering length for systems of three and four ultracold atoms with $\delta$-function interactions in one dimension are presented as a function of the relative mass ratio of the interacting atoms. The Born-Oppenheimer approach is used to treat three-body (``HHL'') systems of one light and two heavy atoms, as well as four-body (``HHHL'') systems of one light and three heavy atoms. Zero-range interactions of arbitrary strength are assumed between different atoms, but the heavy atoms are assumed to be noninteracting among themselves. Both fermionic and bosonic heavy atoms are considered.