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The extended RPA theories are analyzed from the point of view of the problemof stability of their solutions. Three kinds of such theories are considered:the second RPA and two versions of the quasiparticle-phonon coupling modelwithin the time-blocking approximation: the model including 1p1h*phononconfigurations and the two-phonon model. It is shown that stability is ensuredby making use of the subtraction method proposed previously to solve doublecounting problem in these theories. This enables one to generalize the famousThouless theorem proved in the case of the RPA. These results are illustratedby an example of schematic model.

Subtraction method and stability condition in the extended RPA theories