  We introduce approximate, functional renormalization group based schemes to
obtain correlation functions in pure excited eigenstates of large fermionic
many-body systems at arbitrary energies. The algorithms are thouroughly
benchmarked and their strengths and shortcomings are documented using a
one-dimensional interacting tight-binding chain as a prototypical testbed. We
study two `toy applications' from the world of Luttinger liquid physics: the
survival of power laws in lowly-excited states as well as the spectral function
of high-energy `block' excitations which feature several single-particle Fermi
edges.
