I have a Hamiltonian $H=H_0+H_1.$ $H_0$ is completely solvable; I know its ground state wave-funtion, Green function,etc. If $H_0$ was furthermore a free hamiltonian, standar diagrammatic perturbation theory would be a suitable approach to attack the problem of solving the full Hamiltonian $H.$ However, if $H_0$ contains many-body terms (interactions), Wick's theorem doesn't apply, and it seems that there's no perturbation theory available. Of course, there's perturbation theory for computing the energies, but I'm after the self-energy, the Green functions and the Density of States. Is there any way around this? Any generalization of Wick's theorem/Diagrammatic perturbation theory that can be applied for cases in which the exactly solvable hamiltonian is non-free?
Also, in case it is a relevant simplification, the operator $H_1$ in the specific problem I'm considering is a one-electron operator (this might simplify matters considerably)