Convergence of resonant state expansions for transmission in double delta resonators

Abstract

We investigate the transmission features of symmetric double Dirac delta resonators using a non-Hermitian quantum approach, based on resonant state expansions involving complex energy eigenvalues. We focus on the convergence properties of these expansions, which involve a sum of resonance and anti-resonance terms We demonstrate that systems with transmission profiles featuring sharp and isolated resonances converge more rapidly, whereas those with overlapping resonances exhibit slower convergence. We also show that for the former systems, by taking into account only the resonance terms, we can accurately describe the transmission coefficient as a sum of Breit-Wigner resonances, each distinctly characterized by its energy and width. We demonstrate that there is a one-to-one correlation between transmission peaks and the resonance energies of the system. We also compare the convergence properties of transmission using Mittag-Leffler expansions, noting their slower rates compared to resonant state expansions. These findings emphasize the advantage of using non-Hermitian resonant expansions for analyzing quantum mechanical systems, providing a clearer understanding of systems characterized by resonant features.