The bound-states in quantum field theory : review of some analytic problems raised by the variational perturbation method

In the framework of the weakly-coupled P(φ)2 models we summarize some of the problems raised by a new method for finding bound states, called the variational perturbation method. To show first its interest, we present a result of this method, from which the existence of a bound state follows simply by solving a Schroedinger equation, and which allows to find time-zero eigenvectors at first perturbation orders. The main part of this paper is devoted to the review of the problems encountered by the restriction to zero-time vectors (existence of zero-time vectors in the domain of the Hamiltonian, asymptotic series of zero-time vectors approaching any vector, and particularly those of the one-particle subspace). Lastly we present a new quantum and almost-relativistic model for the two-particle system at low energy, deduced from the P(φ)2 models by these considerations. (orig.)