Multiplicity of solutions to GW -type approximations

We show that the equations underlying the GW approximation have a large number of solutions. This raises the question, how can we find the physical solution? We provide two theorems which explain why the methods currently in use usually find the correct solution. These theorems are general enough to cover a large class of similar methods. An efficient algorithm for including self-consistent vertex corrections well beyond GW is described and used in a numerical validation of the two theorems. The effect of a simple mixing scheme on solutions obtained iteratively is also investigated.