This approximation becomes exact in the limit when $ r_{c}\rightarrow 0 $ and $ V\left( r_{0}\right) \rightarrow -\infty  $, keeping $ V\left( r_{0}\right) r_{c}\equiv -g $ constant. Then we obtain the DSM [@Quintanilla-Gyorffy-00], featuring the central “delta shell” potential [@Gottfried-66]$$\label{delta-shell potential}
V\left( \mathbf{r}-\mathbf{r}'\right) =-g\delta \left( \left| \mathbf{r}-\mathbf{r}'\right| -r_{0}\right) ,$$ for which [@Quintanilla-Gyorffy-Annett-Wallington-02] $$\label{V separable DSM}
K_{l}\left( \left| \mathbf{k}\right| ,\left| \mathbf{k}'\right| \right) =-g4\pi r_{0}^{2}\left( -1\right) ^{l}j_{l}\left( \left| \mathbf{k}\right| r_{0}\right) j_{l}\left( \left| \mathbf{k}'\right| r_{0}\right) .$$ More generally, it must be regarded as a convenient approximation, valid when the potential well in Fig. \[FIG-two\_plots\] (b) is sufficiently deep and narrow.

Having set up our Hamiltonian, we can now use the standard method reviewed by Randeria [@Randeria-95], which has been applied to specific models by several authors [@Zwerger-92; @SaDeMelo-Randeria-Engelbrecht-93; @Zwerger-97; @Engelbrecht-Randeria-SaDeMelo-97], to discuss the ground state at $ T=0 $ and the superconducting instability at $ T_{c} $. In short we introduce bosonic fields $ \Delta _{l,m,\mathbf{q}}\left( \tau \right) ,\Delta ^{*}_{l,m,\mathbf{q}}\left( \tau \right)  $ coupling to the pair creation and annihilation operators $ \hat{b}^{+}_{l,m,\mathbf{q}},\hat{b}_{l,m,\mathbf{q}} $, respectively. In terms of these fields one can define an effective action $ S_{eff}\left[ \Delta ^{*},\Delta \right]  $ that determines the partition function of the system: $$\label{Z with Seff}
