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{sub perpendicular}. 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{sub ad} and b{sub ad}. In the tightly bound 'dimer limit' a{sub perpendicular}/a{yields}{infinity}, we find b{sub ad}=0 and a{sub ad} is linked to the 3D atom-dimer scattering length. In the weakly bound 'BCS limit' a{sub perpendicular}/a{yields}-{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{sub ad} and b{sub ad} depend both on a{sub perpendicular}/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.