Nonadiabatic contribution to the quasiparticle self-energy in systems with strong electron-phonon interaction

We investigate effects of a nonadiabatic electron-phonon(boson) interaction on the quasiparticle self-energy in the lowest order in the coupling constant. Existing approaches either overestimate or underestimate these effects because of different approximations for momentum and frequency dependences of the vertex corrections. The connection between the nonadiabaticity and a possible instability of the interacting Fermi system is discussed as well.