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Many-body theory is largely based on self-consistent equations that areconstructed in terms of the physical quantity of interest itself, for examplethe density. Therefore, the calculation of important properties such as totalenergies or photoemission spectra requires the solution of non-linear equationsthat have unphysical and physical solutions. In this work we show in whichcircumstances one runs into an unphysical solution, and we indicate how one canovercome this problem. Moreover, we solve the puzzle of when and why theinteracting Green's function does not unambiguously determine the underlyingsystem, given in terms of its potential, or non-interacting Green's function.Our results are general since they originate from the fundamental structure ofthe equations. The absorption spectrum of lithium fluoride is shown as oneillustration, and observations in the literature for some widely used modelsare explained by our approach. Our findings apply to both the weak andstrong-correlation regimes. For the strong-correlation regime we show that onecannot use the expressions that are obtained from standard perturbation theory,and we suggest a different approach that is exact in the limit of stronginteraction.

Unphysical and Physical Solutions in Many-Body Theories: from Weak to Strong Correlation