The resonance amplitude associated with the Gamow states

TL;DR

This paper establishes a theoretical connection between Gamow states, complex delta functions, and Breit-Wigner amplitudes, explaining the origin of Lorentzian peaks in cross sections as arising from unstable particles.

Contribution

It demonstrates that the resonance amplitude for Gamow states is given by the complex delta function and reduces to Breit-Wigner form under near-resonance approximation, linking several concepts.

Findings

Resonance amplitude is given by the complex delta function.

Under near-resonance approximation, it simplifies to Breit-Wigner amplitude.

Provides a theoretical basis for Lorentzian peaks from unstable particles.

Abstract

The Gamow states describe the quasinormal modes of quantum systems. It is shown that the resonance amplitude associated with the Gamow states is given by the complex delta function. It is also shown that under the near-resonance approximation of neglecting the lower bound of the energy, such resonance amplitude becomes the Breit-Wigner amplitude. This result establishes the precise connection between the Gamow states, Nakanishi's complex delta function and the Breit-Wigner amplitude. In addition, this result provides another theoretical basis for the phenomenological fact that the almost-Lorentzian peaks in cross sections are produced by intermediate, unstable particles.