Faddeev description of two-hole one-particle motion and the single-particle spectral function

The Faddeev technique is employed to address the problem of describing the influence of both particle-particle and particle-hole phonons on the single-particle self-energy. The scope of the few-body Faddeev equations is extended to describe the motion of two-hole one-particle (two-particle one-hole) excitations. This formalism allows to sum both particle-particle and particle-hole phonons, obtained separately in the Random Phase Approximation. The appearance of spurious solutions for the present application of the Faddeev method is related to the inclusion of a consistent set of diagrams. The formalism presented here appears practical for finite nuclei and achieves a simultaneous inclusion of particle-particle and particle-hole phonons to all orders while the spurious solutions are properly eliminated.