  We theoretically show that a two-band system with very different masses
harbors a resonant pair scattering that leads to novel pairing properties, as
highlighted by the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein
condensation (BEC) crossover. Most importantly, the interband pair-exchange
coupling induces an effective intraband attraction in each band, enhancing the
superfluidity/superconductivity. The effect, a kind of Suhl-Kondo mechanism, is
specifically enhanced when the second band has a heavy mass and is incipient
(lying close to, but just above, the chemical potential, $\mu$), which we call
a resonant pair scattering. By elucidating the dependence of the effective
interactions and gap functions on $\mu$, we can draw an analogy between the
resonant pair scattering and the Feshbach resonance.
