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The GW approximation for the electronic self-energy is an important tool forthe quantitative prediction of excited states in solids, but its mathematicalexploration is hampered by the fact that it must, in general, be evaluatednumerically even for very simple systems. In this paper I describe a nontrivialmodel consisting of two electrons on the surface of a sphere, interacting withthe normal long-range Coulomb potential, and show that the GW self-energy, inthe absence of self-consistency, can in fact be derived completely analyticallyin this case. The resulting expression is subsequently used to analyze theconvergence of the energy gap between the highest occupied and the lowestunoccupied quasiparticle orbital with respect to the total number of statesincluded in the spectral summations. The asymptotic formula for the truncationerror obtained in this way, whose dominant contribution is proportional to thecutoff energy to the power -3/2, may be adapted to extrapolate energy gaps inother systems.

Analytic evaluation of the electronic self-energy in the GW approximation for two electrons on a sphere