  This paper analyses the GW method for finite electronic systems. In a first
step, we provide a mathematical framework for the usual one-body operators that
appear naturally in many-body perturbation theory. We then discuss the GW
equations which construct an approximation of the one-body Green's function,
and give a rigorous mathematical formulation of these equations. Finally, we
study the well-posedness of the GW0 equations, proving the existence of a
unique solution to these equations in a perturbative regime.
