The double correlation function

The same type of approximation may be performed for the correlation function. Namely in the Wyld equation (4.7) one may replace $G({\bf k},\omega)$ by $\tilde G({\bf k},\omega)$ and to neglect the $\omega$ dependence of $\Phi ({\bf k},\omega )$ by putting $\Phi({\bf k},\omega)\rightarrow\tilde\Phi(k)=\Phi({\bf k},\omega_*)$, or

$$\tilde n({\bf k},\omega)= \vert\tilde G({\bf k},\omega)\vert^2\left[D(k)+\tilde\Phi(k)\right]\,,$$ (77)

We will call this one-pole approximation for the correlation function.