  We present the multi-channel Dyson equation that combines two or more
many-body Green's functions to describe the electronic structure of materials.
In this work we use it to model photoemission spectra by coupling the one-body
Green's function with the three-body Green's function. We demonstrate that,
unlike methods using only the one-body Green's function, our approach puts the
description of quasiparticles and satellites on an equal footing. We propose a
multi-channel self-energy that is static and only contains the bare Coulomb
interaction, making frequency convolutions and self-consistency unnecessary.
Despite its simplicity, we demonstrate with a diagrammatic analysis that the
physics it describes is extremely rich. Finally, we present a framework based
on an effective Hamiltonian that can be solved for any many-body system using
standard numerical tools. We illustrate our approach by applying it to the
Hubbard dimer and show that it is exact both at 1/4 and 1/2 filling.
