Perturbation theory for bound states and resonances where potentials and propagators have arbitrary energy dependence

Abstract

Standard derivations of ``time-independent perturbation theory'' of quantum mechanics cannot be applied to the general case where potentials are energy dependent or where the inverse free Green function is a nonlinear function of energy. Such derivations cannot be used, for example, in the context of relativistic quantum field theory. Here we solve this problem by providing a new, general formulation of perturbation theory for calculating the changes in the energy spectrum and wave function of bound states and resonances induced by perturbations to the Hamiltonian. Although our derivation is valid for energy-dependent potentials and is not restricted to inverse free Green functions that are linear in the energy, the expressions obtained for the energy and wave function corrections are compact, practical, and maximally similar to the ones of quantum mechanics. For the case of relativistic quantum field theory, our approach provides a direct covariant way of obtaining corrections to wave functions that are not in the center of mass frame.