  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.
