Feshbach resonance in atomic binary collisions in the Wigner threshold law regime

We analyze, within the generalized multichannel quantum defect theory framework, atomic binary collision cross sections over the 1 muK to 10 mK energy range just above the entrance channel threshold, a domain where the Wigner threshold law should apply. By adjusting the strength of a constant external magnetic field a Feshbach resonance is tuned at will over this energy range. In the threshold regime, the quasibound state interacts with an opening continuum whose wave function presents a strong energy dependence which reflects the breakdown of the WKB approximation. The effective discrete-continuum interactions become very sensitive to the energy. The consequences of these variations have been investigated by including threshold effects in the analysis of the Fano configuration-interaction theory [ Phys. Rev. 124, 1866 (1961) ] in terms of quantum defect theory quantities proposed by [ Lecomte J. Phys. B 20, 3645 (1987) ]. This analysis shows that the energy variations are in general so important that it becomes meaningless to associate a width with a Feshbach resonance. However, it is still possible to define the resonance energy, as long as the energy variation of the shift of the resonance, induced by the effective discrete-continuum interaction, remains linear over an energy range corresponding to the magnitude of the shift.