  We investigate the effect of strong electron-electron repulsion on the
electron-phonon interaction from a Fermi-liquid point of view: the strong
interaction is responsible for vertex corrections, which are strongly dependent
on the $v_Fq/\omega$ ratio. These corrections generically lead to a strong
suppression of the effective coupling between quasiparticles mediated by a
single phonon exchange in the $v_Fq/\omega \gg 1$ limit. However, such effect
is not present when $v_Fq/\omega \ll 1$. Analyzing the Landau stability
criterion, we show that a sizable electron-phonon interaction can push the
system towards a phase-separation instability. A detailed analysis is then
carried out using a slave-boson approach for the infinite-U three-band Hubbard
model. In the presence of a coupling between the local hole density and a
dispersionless optical phonon, we explicitly confirm the strong dependence of
the hole-phonon coupling on the transferred momentum versus frequency ratio. We
also find that the exchange of phonons leads to an unstable phase with negative
compressibility already at small values of the bare hole-phonon coupling. Close
to the unstable region, we detect Cooper instabilities both in s- and d-wave
channels supporting a possible connection between phase separation and
superconductivity in strongly correlated systems.

Condensed Matter