Calculating exciton Green's function on a tight binding chain

[physicsxanime]

Homework Statement

On a 1-D tight binding chain, when you put an electron from a site in the valence band to another site in the conduction band, you will can create an exciton if there is Hubbard attraction between the electron and hole. I would like to calculate the green's function of such two particle excited state.
The Hamiltonian of interest:
$$H = t\sum c^\dagger_{n+1}c_{n} - t\sum v^\dagger_{n+1}v_{n} + h.c. + U\sum c^\dagger_{n}c_{n}v^\dagger_{n}v_{n}$$

Relevant Equations

Lehmann representation of Green's function $$G(x,x';E) = \sum_n \frac{<\Psi^N_0|\psi(x)|\Psi^{N+1}_n><\Psi^{N+1}_n|\psi^\dagger(x')|\Psi^N_0>}{E + E^N_0-E^{N+1}_n +i\eta}$$

Honestly, I have no real idea. I know for sure the equation connects the initial state $|\Psi^N_0>$to a final state $|\Psi^{N+1}_0>$, $E$ is the energy and $E^N_0$ etc are the energy of the initial state and final state. I also know that these energy are related to conduction band and valence band energy like $2t\cos(ka)$. But I don't really know what $\psi(x)$ represent here, how to get the wave functions $|\Psi^{N+1}_0>$, and what it is summing over.

As you can see, I have a severe lack of knowledge here, so any help will be greatly appreciated and if possible, point to me some introductory resources on topics of condensed matter and many particle green's function.

 

[physicsxanime]

I can solve it using recurrence relation:
https://app.conceptboard.com/board/n11a-y61a-o584-90z4-rofy

Not sure how to proceed with Lehmann representation, maybe it is not possible I don't know.