  The mapping, exact or approximate, of a many-body problem onto an effective
single-body problem is one of the most widely used conceptual and computational
tools of physics. Here, we propose and investigate the inverse map of effective
approximate single-particle equations onto the corresponding many-particle
system. This approach allows us to understand which interacting system a given
single-particle approximation is actually describing, and how far this is from
the original physical many-body system. We illustrate the resulting reverse
engineering process by means of the Kohn-Sham equations of density-functional
theory. In this application, our procedure sheds light on the non-locality of
the density-potential mapping of DFT, and on the self-interaction error
inherent in approximate density functionals.
