  We present a nontrivial model system of interacting electrons that can be
solved analytically in the GW approximation. We obtain the particle number from
the GW Green's function strictly analytically, and prove that there is a
genuine violation of particle number conservation if the self-energy is
calculated non-self-consistently from a zeroth order Green's function, as done
in virtually all practical implementations. We also show that a simple shift of
the self-energy that partially restores self-consistency reduces the numerical
deviation significantly.
