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A problem of collisions of atoms with two-channel zero-range
interaction under cylindrical harmonic confinement is solved by using of a renormalization procedure. A matching of the solution to a solution of the related one-dimensional problem leads to relation between the one-dimensional and three-dimensional scattering parameters.
At low collision energies the scattering amplitude for the confined system can be approximated by the scattering amplitude for the one-dimensional one. At higher energies the opening of transverse channels leads to resonances in the confined scattering amplitude. Its average behavior can be approximated by the amplitude of three-dimensional free collisions. The confined two-body system has two or one bound states below or above the resonance, respectively. Shallow bound states are similar to ones of the related one-dimensional system, while deep ones are similar to bound states of two free atoms. A Feshbach resonance also affects a one-dimensional three-body scattering. |