  We show that the equations underlying the $GW$ approximation have a large
number of solutions. This raises the question: which is the physical solution?
We provide two theorems which explain why the methods currently in use do, in
fact, find the correct solution. These theorems are general enough to cover a
large class of similar algorithms. An efficient algorithm for including
self-consistent vertex corrections well beyond $GW$ is also described and
further used in numerical validation of the two theorems.
