The extended random-phase approximation (RPA) theories are analyzed from the point of view of the problem of the 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 $1p1h\otimes$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 the 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 the schematic model.

• Received 8 August 2013

DOI:https://doi.org/10.1103/PhysRevC.88.054301