  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 non-linear 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 bound and resonance state masses, as well as to wave functions
that are not in the centre of mass frame.
