Three-body problem for ultracold atoms in quasi-one-dimensional traps

We study the three-body problem for both fermionic and bosonic cold-atom gases in a parabolic transverse trap of length scale a(perpendicular to). For this quasi-one-dimensional (quasi-1D) problem, there is a two-body bound state (dimer) for any sign of the 3D scattering length a and a confinement-induced scattering resonance. The fermionic three-body problem is universal and characterized by two atom-dimer scattering lengths a(ad) and b(ad). In the tightly bound dimer limit a(perpendicular to)/a ->infinity, we find b(ad)=0 and a(ad) is linked to the 3D atom-dimer scattering length. In the weakly bound BCS limit a(perpendicular to)/a ->-infinity, a connection to the Bethe ansatz is established, which allows for exact results. The full crossover is obtained numerically. The bosonic three-body problem, however, is nonuniversal: a(ad) and b(ad) depend both on a(perpendicular to)/a and on a parameter R-* related to the sharpness of the resonance. Scattering solutions are qualitatively similar to fermionic ones. We predict the existence of a single confinement-induced three-body bound state (trimer) for bosons.