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The choice of the “contact” potential (2.2) does not allow the fermionic attraction to extend over a finite range. If a finite-range fermionic attraction would instead have been adopted, the effective boson-boson potential in the strong-coupling limit would acquire an attractive part which would dominate over the usual repulsive part due to Pauli principle, in the sense that the bosonic scattering length associated with the attractive part would not vanish in that limit. The attractive part would thus lead to an instability of the bosonic system when the finite-range fermionic attraction gets sufficiently strong. To avoid this instability, a condition of the type (Formula presented) has to be imposed on the BE limit, where (Formula presented) is the wave vector specifying the finite range of the potential and (Formula presented) is the bound-state energy of the two-body problem [cf. footnote 43 of Ref. 15]. In this context, it is worth mentioning the recent work by G. Röpke et. al [Phys. Rev. Lett. 80, 3177 (1998)] where it has been shown that, in the strong-coupling limit, there exists a competition between pair and quartet condensation in a Fermi liquid with finite-range attraction. It is then clear that adopting a finite-range fermionic potential merely makes the BCS-BE crossover more involved, and it is thus not relevant to the purposes of the present paper.
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