  Nonadiabatic effects in the electron-phonon coupling are important whenever
the ratio between the phononic and the electronic energy scales, the adiabatic
ratio, is non negligible. For superconducting systems, this gives rise to
additional diagrams in the superconducting self-energy, the vertex and cross
corrections. In this work we explore these corrections in a two-dimensional
single-band system through the crossover between the weak-coupling BCS and
strong-coupling Bose-Einstein regimes. By focusing on the pseudogap phase, we
identify the parameter range in which the pairing amplitude is amplified by
nonadiabatic effects and map them throughout the BCS-BEC crossover. These
effects become stronger as the system is driven deeply in the crossover regime,
for phonon frequencies of the order of the hopping energy and for large enough
electron-phonon coupling. Finally, we provide the phase space regions in which
the effects of nonadiabaticity are more relevant for unconventional
superconductors.
