  On the basis of the self-consistent calculation scheme for the electron
self-energy with the use of the three-point vertex function always satisfying
the Ward identity, we find that the obtained quasiparticle dispersion in the
normal state in gapped systems such as semiconductors, insulators, and
molecules is well reproduced by that in the one-shot GW (or G0W0)
approximation. In calculating the superconducting transition temperature Tc, we
also find a similar situation; the result for Tc in the gauge-invariant
self-consistent (GISC) framework including the effect of the vertex corrections
satisfying the Ward identity is different from that in the conventional
Eliashberg theory (which amounts to the GW approximation for superconductivity)
but is close to that in the G0W0 approximation. Those facts indicate that the
G0W0 approximation actually takes proper account of both vertex and high-order
self-energy corrections in a mutually cancelling manner and thus we can
understand that the G0W0 approximation is better than the fully self-consistent
GW one in obtaining some of physical quantities.
