V\left( r\right)  $ has to be most attractive at distances at which the weighting factors for a finite value of the angular momentum quantum number are largest. That is achieved by non-monotonic potentials that lead to maximum attraction at some finite distance $ r_{0} $, such as the one in Fig. \[FIG-two\_plots\] (b). For example, provided that the width of the potential well centred on $ r_{0} $, namely $ r_{c} $ (see figure), is sufficiently small, choosing $ r_{0} $ so that $ k_{F}r_{0}\sim 3 $ yields $ l_{max}=2 $.

Basic theory of the BCS to Bose crossover\[SEC-basic theory\]
=============================================================

The above discussion implies that, in the BCS limit, exotic pairing can be described by an effectively anisotropic interaction, Eq. (\[angle decomp of V with wc approx\]). On the other hand we would like to describe the ground state and the superconducting instability also in the BE limit, where such approximation may not (and, as we shall see, does not) apply. For this we need to develop our theory in a more general framework, and in particular it is important to work with a complete description of the interaction potential. For concreteness let us write down explicitely the grand-canonical 