$ [@Leggett-80; @Nozieres-SchmittRink-85; @Andrenacci-Perali-Pieri-Strinati-99]. The amplitudes $ \Delta _{l,m}^{(0)} $ are related to the expectation values of the pair annihilation operators by $ \Delta _{l,m}^{(0)}=V_{l}\left\langle \hat{b}_{l,m,0}\right\rangle  $. The equations determining the saddle point take the following form:$$\begin{aligned}
\Delta _{l,m}^{(0)} & = & -V_{l}\sum _{l',m'}\left\{ \int \frac{d^{3}\mathbf{k}}{\left( 2\pi \right) ^{3}}\frac{\phi _{l,m,\mathbf{k}}^{*}\phi _{l',m',\mathbf{k}}}{2E_{\mathbf{k}}}\right\} \Delta _{l',m'}^{(0)}\label{gap eq DL} \end{aligned}$$ where $ E_{\mathbf{k}}\equiv \sqrt{\varepsilon _{\mathbf{k}}^{2}+\left| \Delta _{\mathbf{k}}\right| ^{2}} $ with $ \Delta _{\mathbf{k}}\equiv \sum _{l,m}\Delta ^{(0)}_{l,m}R_{l}\left( \left| \mathbf{k}\right| \right) Y_{l,m}\left( \hat{\mathbf{k}}\right)  $ the usual BCS “gap function”. Evidently this non-linear system of equations may have many different solutions, each corresponding to a different superposition of angular momentum states. To determine their relative stability one has to evaluate the appropriate potential. If, as usual, we fix the density $ n $, treating the chemical potential $ \mu  $ as a parameter to be determined self-consistenly [@Nozieres-SchmittRink-85], the energy $ \mathcal{U}_{0}=\left\langle \hat{H}\right\rangle  $ has to be calculated. Quite generally, it is given by$$\label{gs energy eigen V}
\frac{1}{L^{3}}\mathcal{U}_{0}=\int \frac{d^{3}\mathbf{k}}{(2\pi )^{3}}\frac{\hbar ^{2}\left| \mathbf{k}\right| ^{2}}{2m^{*}}\left( 1-\frac{\varepsilon _{\mathbf{k}}}{E_{\mathbf{k}}}\right) +\sum _{l,m}\frac{\left| \Delta ^{(0)}_{l,m}\right| ^{2}}{V_{l}}$$ which results from taking the $ T\rightarrow 0 $ limit of $ L^{-3}\left\langle \hat{H}\right\rangle =-L^{-3}\beta ^{-1}\ln Z+\mu n $.

The above equations are entirely equivalent to the 