  With the aim of identifying universal trends, we compare fully
self-consistent electronic spectra and total energies obtained from the GW
approximation with those from an extended GWGamma scheme that includes a
nontrivial vertex function and the fundamentally distinct Bethe-Goldstone
approach based on the T-matrix. The self-consistent Green's function G, as
derived from Dyson's equation, is used not only in the self-energy but also to
construct the screened interaction W for a model system. For all approximations
we observe a similar deterioration of the spectrum, which is not removed by
vertex corrections. In particular, satellite peaks are systematically broadened
and move closer to the chemical potential. The corresponding total energies are
universally raised, independent of the system parameters. Our results therefore
suggest that any improvement in total energy due to self-consistency, such as
for the electron gas in the GW approximation, may be fortuitous.
