The resonance amplitude associated with the Gamow states

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.

Light-Quark Baryon Spectroscopy from ANL-Osaka Dynamical Coupled-Channels Analysis

“…This was a rather unsatisfactory situation [33] because the Breit-Wigner parameters are nothing more than "approximation" of the resonance parameters defined by poles of scattering amplitudes in the complex energy plane [34], where the latter has a clear physical meaning: the resonance states defined by poles are associated with the exact (complex) energy eigenstates of the full Hamiltonian of the system under the purely outgoing boundary condition (see, e.g., Refs. [35,36]). …”

q-Gamow states for intermediate energies

“…Gamow-states [9,10], can be seen as (Sebastiao e Silva's) Ultradistributions [11,12,13]. Their treatment needs appealing to Rigged Hilbert Space [14,15,16,17]. It was demonstrated in [18] that associated resonances, called q-Gamow states, constitute a useful generalization of the GS concept adapted to Tsallis' q-statistics [4].…”

Replacing the Breit-Wigner Amplitude by the Complex Delta Function to Describe Resonances

“…This result provides another argument in favor of seeing the complex delta function as the exact resonant amplitude, and the Breit-Wigner amplitude as an approximate resonant amplitude that is valid whenever we can neglect the lower bound of the energy. 14) This result also shows that what Fermi's, Weisskopf-Wigner's and similar approximations do is basically to neglect the effect of the lower bound of the energy.…”

“…As shown in Ref. 14), describing resonances by means of the complex delta function is the same as describing resonances by means of the Gamow state. 15)- 49) We recall that the Gamow states are eigenfunctions of the Hamiltonian subject to purely boundary conditions.…”