Subtraction method and stability condition in the extended RPA theories

The extended RPA theories are analyzed from the point of view of the problem of stability of their solutions. Three kinds of such theories are considered: the second RPA and two versions of the quasiparticle-phonon coupling model within the time-blocking approximation: the model including 1p1h*phonon configurations and the two-phonon model. It is shown that stability is ensured by making use of the subtraction method proposed previously to solve double counting problem in these theories. This enables one to generalize the famous Thouless theorem proved in the case of the RPA. These results are illustrated by an example of schematic model.