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