# Numerical calculation of the decay widths, the decay constants, and the decay energy spectra of the resonances of the delta-shell potential

## TL;DR

This paper provides a numerical analysis of the decay properties and energy spectra of delta-shell potential resonances, revealing deviations from traditional Lorentzian lineshapes and limitations of the Golden Rule.

## Contribution

It introduces a method to compute resonance energies, decay widths, and constants using the Lambert W function, and analyzes the interference effects in decay spectra.

## Key findings

- Decay widths and constants are expressed via Lambert W function.
- Decay energy spectrum can significantly differ from Lorentzian lineshape.
- Standard Golden Rule fails to describe interference between resonances.

## Abstract

We express the resonant energies of the delta-shell potential in terms of the Lambert $W$ function, and we calculate their decay widths and decay constants. The ensuing numerical results strengthen the interpretation of such decay widths and constants as a way to quantify the coupling between a resonance and the continuum. We calculate explicitly the decay energy spectrum of the resonances of the delta-shell potential, and we show numerically that the lineshape of such spectrum is not the same as, and can be very different from, the Breit-Wigner (Lorentzian) distribution. We argue that the standard Golden Rule cannot describe the interference of two resonances, and we show how to describe such interference by way of the decay energy spectrum of two resonant states.