  Using the simple (symmetric) Hubbard dimer, we analyze some important
features of the $GW$ approximation. We show that the problem of the existence
of multiple quasiparticle solutions in the (perturbative) one-shot $GW$ method
and its partially self-consistent version is solved by full self-consistency.
We also analyze the neutral excitation spectrum using the Bethe-Salpeter
equation (BSE) formalism within the standard $GW$ approximation and find, in
particular, that i) some neutral excitation energies become complex when the
electron-electron interaction $U$ increases, which can be traced back to the
approximate nature of the $GW$ quasiparticle energies; ii) the BSE formalism
yields accurate correlation energies over a wide range of $U$ when the trace
(or plasmon) formula is employed; iii) the trace formula is sensitive to the
occurrence of complex excitation energies (especially singlet), while the
expression obtained from the adiabatic-connection fluctuation-dissipation
theorem (ACFDT) is more stable (yet less accurate); iv) the trace formula has
the correct behavior for weak (\ie, small $U$) interaction, unlike the ACFDT
expression.
