  It is shown that the non-equilibrium self-energy of an interacting
lattice-fermion model has a unique Lehmann representation. Based on the
construction of a suitable non-interacting effective medium, we provide an
explicit and numerically practicable scheme to construct the Lehmann
representation for the self-energy, given the Lehmann representation of the
single-particle non-equilibrium Green's function. This is of particular
importance for an efficient numerical solution of Dyson's equation in the
context of approximations where the self-energy is obtained from a reference
system with a small Hilbert space. As compared to conventional techniques to
solve Dyson's equation on the Keldysh contour, the effective-medium approach
allows to reach a maximum propagation time which can be several orders of
magnitude longer. This is demonstrated explicitly by choosing the
non-equilibrium cluster-perturbation theory as a simple approach to study the
long-time dynamics of an inhomogeneous initial state after a quantum quench in
the Hubbard model on a 10 x 10 square lattice. We demonstrate that the
violation of conservation laws is moderate for weak Hubbard interaction and
that the cluster approach is able to describe prethermalization physics.
