=V\left( \left| \mathbf{r}-\mathbf{r}'\right| \right)  $. For simplicity we will assume the interaction to take place between electrons with opposite spins and limit our discussion to the case of singlet pairing, with angular momentum quantum number $ l=0,2,4,\ldots  $ As is well-known the Fourier transform $$\label{FT of V inverted opposite spins}
V\left( \mathbf{k}-\mathbf{k}'\right) =\int d^{3}\mathbf{r}e^{i\left( \mathbf{k}-\mathbf{k}'\right) .\mathbf{r}}V\left( \mathbf{r}\right)$$ admits the partial-wave decomposition

$$\label{angle decomp of V}
V\left( \mathbf{k}-\mathbf{k}'\right) =\sum _{l=0}^{\infty }K_{l}\left( \left| \mathbf{k}\right| ,\left| \mathbf{k}'\right| \right) \left( 2l+1\right) P_{l}\left( \hat{\mathbf{k}}.\hat{\mathbf{k}}'\right)$$

in terms of the Legendre polynomials $ P_{l}\left( \hat{\mathbf{k}}.\hat{\mathbf{k}}'\right) =\left( 2l+1\right) ^{-1}4\pi \sum _{m}Y_{l,m}\left( \hat{\mathbf{k}}\right) Y_{l,m}^{*}\left( \hat{\mathbf{k}}'\right)  $. It was soon pointed out [@Balian-64] that in the weak-coupling limit one can use the approximation$$\label{angle decomp of V with wc approx}
V\left( \mathbf{k}-\mathbf{k}'\right) \approx K_{l_{max}}\left( 2l_{max}+1\right) P_{l_{max}}\left( \hat{\mathbf{k}}.\hat{\mathbf{k}}'\right) ,$$ where $ l_{max} $ is the value of $ l $ for which the coupling constant on the Fermi surface, $$\label{BCS limit coupling const}
K_{l}\equiv K_{l}\left( k_{F},k_{F}\right) ,$$ is largest ($ k_{F} $ is the Fermi vector). The approximate form (\[angle decomp of V with wc approx\]) of the potential $ V\left( \mathbf{k}-\mathbf{k}'\right)  $ is, for $ l_{max}>0 $, anisotropic, and it leads to pairing with finite angular momentum quantum number $ l_{max} 