Solvable three-boson model with attractive $\delta$-function interactions

A one-parameter solvable model for three bosons subject to $\delta$-function attractive interactions in one dimension with periodic boundary conditions is studied. The energy levels and wave functions are classified and given explicitly in terms of three momenta. In particular, eigenstates and eigenvalues are described as functions of the model parameter $c$. Some of the states are given in terms of complex momenta and represent dimer or trimer configurations for large negative $c$. The asymptotic behavior for small and large values of the parameter, and at thresholds between real and complex momenta, is provided. The properties of the potential energy are also discussed.