  Dressed states forming when quantum emitters or atoms couple to a photonic
bath underpin a number of phenomena and applications, in particular dispersive
effective interactions occurring within photonic bandgaps.
  Here, we present a compact formulation of the resolvent-based theory for
calculating atom-photon dressed states built on the idea that the atom behaves
as an effective impurity. This establishes an explicit connection with the
standard impurity problem in condensed matter. Moreover, it allows us to
formulate and settle in a model-independent context a number of properties
previously known only for specific models or not entirely formalized. The
framework is next extended to the case of more than one emitter, which is used
to derive a general expression of dissipationless effective Hamiltonians
explicitly featuring the overlap of single-emitter dressed bound states.
