$ \hbar \mathbf{k} $ is the momentum of one of the two components of the pair with respect to its centre of mass. To put the generic Hamiltonian (\[our H in k-space\]) in the form (\[our H in k-space separated\]) it suffices to define the “kernel factors” $ R_{\kappa ,l}\left( \left| \mathbf{k}\right| \right)  $ and “coupling constants” $ V_{\kappa ,l} $ so that they yield the following parametrization:$$\label{central V amplitude in terms of its eigenfunctions}
K_{l}\left( \left| \mathbf{k}\right| ,\left| \mathbf{k}'\right| \right) =\frac{1}{4\pi }\sum _{\kappa }V_{\kappa ,l}R_{\kappa ,l}\left( \left| \mathbf{k}\right| \right) R^{*}_{\kappa ,l}\left( \left| \mathbf{k}'\right| \right) .$$ We may construct the $ \phi _{\kappa ,l,m,\mathbf{k}} $ and $ V_{\kappa ,l} $ as a complete set of solutions to the eigenvalue problem$$\label{V eigenvalue eq}
\sum _{\mathbf{k}'}V\left( \mathbf{k}-\mathbf{k}'\right) \phi _{\kappa ,l,m,\mathbf{k}'}=V_{\kappa ,l}\phi _{\kappa ,l,m,\mathbf{k}},$$ which can be regarded as the result of neglecting, for strong attraction, the kinetic energy contribution to the Schrödinger equation for a two-body bound state.

Eq. (\[our H in k-space separated\]) generalises the usual BCS Hamiltonian by including interaction terms corresponding to pairs with $ \hbar \mathbf{q}\neq 0 $ and different values of the the internal angular momentum, given by $ l,m $. This is required for the correct 