Effect of Strong Correlation on the Electron-Phonon Interaction
  High-$T_c$ superconductors are usually described as strongly correlated electronic systems. This feature deeply affects the one-particle and two-particle properties of the system. In particular, a large incoherent background developes on the top of a narrow quasi-particle peak in the one-electron spectral function. We schematize this structure with a simple phenomenological form. The corresponding Green's function is employed to calculate the charge response of the system taking into account in a proper way strong correlation effects. The effective charge interaction acquires a structure in the exchanged momentum space with a predominance of forward scattering, in agreement with previous numerical calculations. The consequences of the momentum dependence of the interaction are discussed in the framework of the nonadiabatic theory of superconductivity proposed for the high-$T_c$ materials. 
L. Boeri, E. Cappelluti, C. Grimaldi and L. Pietronero