above questions may guide us as to whether exotic pairing is also a possibility.

Recently, we and our co-authors have studied the above questions in a simple model that features exotic pairing via a central attraction [@Quintanilla-Gyorffy-Annett-Wallington-02]. In this contribution we revisit them with a more general point of view, using some of the results obtained for the “delta shell model” (DSM) of Ref. [@Quintanilla-Gyorffy-Annett-Wallington-02] as an illustration. We begin section \[SEC-central potentials leading to exotic pairing\] by reminding the reader how, as a consequence of the weak-coupling theory of superconductivity, a central potential can, *in principle*, lead to exotic pairing [@Balian-64; @Anderson-Brinkman-74]. Then we take this old argument one little step further by asking: what is the essential feature that makes a particular interaction potential, *in practice*, lead to this effect? Having established the existence of such potentials, and knowing what they look like, we move on to outline, in section \[SEC-basic theory\], the main features of a simple, but fairly general theory of the BCS to Bose crossover. Our formalism follows a recipe that has, by now, become standard [@Randeria-95] (though not the only one [@Gyorffy-Staunton-Stocks-91; @Chen-Kosztin-Levin-00]): to introduce bosonic pairing fields via a Hubbard-Stratonovich transformation (HST) [@Bender-Orszag-78; 