  A controversial issue on whether the electron-phonon interaction (EPI) is
crucial for high-temperature superconductivity or it is weak and inessential
has remained one of the most challenging problems of contemporary condensed
matter physics. We employ a continuum RPA approximation for the dielectric
response function allowing for a selfconsistent semi-analytical evaluation of
the EPI strength, electron-electron attractions, and the carrier mass
renormalisation in layered high-temperature superconductors. We show that the
Fr\"{o}hlich EPI with high-frequency optical phonons in doped ionic lattices is
the key pairing interaction, which is beyond the BCS-Migdal-Eliashberg
approximation in underdoped superconductors, and it remains a significant
player in overdoped compounds.

Superconductivity;Quantum Physics