he effective phonon-induced electron-electron interaction $$\begin{aligned}   \label{00}   V_{{\bf k},{\bf q}} &=&    - \frac{     0 \left| g_{\bf q} \right|^{0}     \Theta\left[       \omega_{\bf q} -        \left| \varepsilon_{\bf k} - \varepsilon_{({\bf k}+{\bf q})} \right|     \right]   }{     \left|        \varepsilon_{\bf k} - \varepsilon_{({\bf k}+{\bf q})}      \right| + \omega_{\bf q}   }