  We consider a quantum model of two-channel scattering to describe the
mechanism of a Feshbach resonance. We perform a rigorous analysis in order to
count and localize the energy resonances in the perturbative regime, i.e., for
small inter-channel coupling, and in the non-perturbative one. We provide an
expansion of the effective scattering length near the resonances, via a
detailed study of an effective Lippmann-Schwinger equation with
energy-dependent potential.
