Derivation and analysis of the multichannel Dyson equation

In a recent letter [Riva et al., Phys. Rev. Lett. 131, 216401 (2023)] we presented the multichannel Dyson equation (MCDE), in which two or more many-body Green's functions are coupled. In this work we will give further details of the MCDE approach. In particular we will discuss (1) the derivation of the MCDE and the definition of the space in which it is to be solved, (2) the rationale of the approximation to the multichannel self-energy, (3) a diagrammatic analysis of the MCDE, and (4) the recasting of the MCDE on an eigenvalue problem with an effective Hamiltonian that can be solved using standard numerical techniques. This work mainly focuses on the coupling between the one-body Green's function and the three-body Green's function to describe photoemission spectra, but the MCDE method can be generalized to the coupling of other many-body Green's functions and to other spectroscopies.