Evolution from BCS superconductivity to Bose-Einstein condensation: Mapping of the fermionic onto a bosonic system in the strong-coupling limit

We consider a system of fermions with an effective attractive interaction, for which formation of (real-space) bound pairs is expected to occur in the strong-coupling limit. We provide a direct mapping of this fermionic system onto an effective system of bosons with residual boson-boson interaction by means of functional integrals. We determine in this way the strength and the range of the residual boson-boson interaction.