Perturbation theory for projected states II: Convergence criteria and a soluble model

It is shown that a perturbation expansion for projected states proposed earlier satisfies the Sohrödinger equation. Modified forms of the series are suggested, and criteria for their convergence are discussed. The method is applied to the ‘Lipkin model’ for a two-level manybody problem, which is exactly soluble. The second-order term in the perturbation series is shown to give a good approximation, and is compared with alternative methods. The convergence factor of the series is shown to be better than that of ordinary perturbation theory, which in most cases of this model is divergent.