$ [@Balian-64].

Let us now try to understand how the preference for a particular value of $ l $ comes about. To do this one has to examine the relationship between the coupling constant $ K_{l} $ and the interaction potential $ V\left( r\right)  $. It can be found by expanding in spherical waves the two exponentials $ e^{i\mathbf{k}.\mathbf{r}} $ and $ e^{-i\mathbf{k}'.\mathbf{r}} $ in Eq. (\[FT of V inverted opposite spins\]); comparison to (\[angle decomp of V\]) yields $$\label{kernel in general}
K_{l}\left( \left| \mathbf{k}\right| ,\left| \mathbf{k}'\right| \right) =\left( -1\right) ^{l}\int _{0}^{\infty }dr\, 4\pi r^{2}\, j_{l}\left( \left| \mathbf{k}\right| r\right) \, V\left( r\right) \, j_{l}\left( \left| \mathbf{k}'\right| r\right) ,$$ where $ j_{l}\left( x\right)  $ is a spherical Bessel function. Substituing (\[kernel in general\]) in (\[BCS limit coupling const\]) we obtain$$\label{Kl}
K_{l}=\left( -1\right) ^{l}\int _{0}^{\infty }dr\, 4\pi r^{2}\, j_{l}\left( k_{F}r\right) ^{2}\, V\left( r\right) .$$ Thus $ K_{l} $ is a weighted integral of the interaction potential $ V\left( r\right)  $. The weighting factors $ j_{l}\left( k_{F}r\right) ^{2} $ are shown in Fig. \[FIG-weighting factor\] for $ l=0,2,4,\ldots  $ Evidently for a purely attractive, monotonic potential, such as the one in Fig. \[FIG-two\_plots\] (a), $ l_{max}=0 $. To have $ l_{max}>0 $, $ 