Replacing the Breit-Wigner amplitude by the complex delta function to describe resonances

Whenever the Breit-Wigner amplitude appears in a calculation,there are many instances (e.g., Fermi's two-level system and the Weisskopf-Wigner approximation) where energy integrations are extended from the scattering spectrum of the Hamiltonian to the whole real line. Such extensions are performed in order to obtain a desirable, causal result. In this paper, we recall several of those instances and show that substituting the Breit-Wigner amplitude by the complex delta function allows us to recover such desirable results without having to extend energy integrations outside of the scattering spectrum.