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Several widely used methods for the calculation of band structures and photoemission spectra, such as the GW approximation, rely on Many-Body PerturbationTheory. They can be obtained by iterating a set of functional differentialequations relating the one-particle Green's function to its functionalderivative with respect to an external perturbing potential. In the presentwork we apply a linear response expansion in order to obtain insights invarious approximations for Green's functions calculations. The expansion leadsto an effective screening, while keeping the effects of the interaction to allorders. In order to study various aspects of the resulting equations wediscretize them, and retain only one point in space, spin, and time for allvariables. Within this one-point model we obtain an explicit solution for theGreen's function, which allows us to explore the structure of the generalfamily of solutions, and to determine the specific solution that corresponds tothe physical one. Moreover we analyze the performances of establishedapproaches like $GW$ over the whole range of interaction strength, and weexplore alternative approximations. Finally we link certain approximations forthe exact solution to the corresponding manipulations for the differentialequation which produce them. This link is crucial in view of a generalizationof our findings to the real (multidimensional functional) case where only thedifferential equation is known.

Approximations for many-body Green's functions: insights from the fundamental equations