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Bound and scattering states in harmonic waveguides in the vicinity of free space Feshbach resonances

The two-body bound and scattering properties in an one-dimensional harmonic waveguide close to free space magnetic Feshbach resonances are investigated based on the local frame transformation approach within a single partial wave approximation. An energy and magnetic field dependent free space phase shift is adopted in the current theoretical framework. For both $s$- and $p$-wave interaction, the least bound state in the waveguide dissociates into the continuum at the resonant magnetic field where the effective one-dimensional scattering length $a_{\rm 1D}$ diverges. Consequently, the association of atoms into molecules in the waveguide occurs when the magnetic field is swept adiabatically across the pole of $a_{\rm 1D}$. In the vicinity of broad $s$-wave resonances, the resonant magnetic field is nearly independent on the transverse confining frequency $ω_{\perp}$ of the waveguide. Close to $p$-wave and narrow $s$-wave resonances, the resonant magnetic field changes as $ω_{\perp}$ varies.