Atom-trimer scattering lengths and four-body bound states for the one-dimensional ``HHHL'' system

The energy spectrum and atom-trimer scattering length for a four-body system under one-dimensional confinement are presented. We consider an ``HHHL'' system of three heavy atoms and one light atom within the Born-Oppenheimer approximation with zero-range interactions. We present a ``phase-diagram'' on the $(m_H/m_L) - (a_{HH}/a_{HL})$ plane indicating the number of tetramer states and the sign of the atom-trimer scattering length.