Cross section versus time delay and trapping probability

TL;DR

This paper investigates the relationship between cross section, time delay, and trapping probability in quantum scattering, revealing that their maxima occur at different wave numbers and clarifying the nature of resonances.

Contribution

It provides a detailed numerical analysis showing that maxima of cross section, trapping probability, and time delay do not coincide, offering new insights into the interpretation of resonances in quantum scattering.

Findings

Resonance features occur at different wave numbers for each quantity.

Time delay remains positive at its local maxima.

Resonance maxima do not necessarily align with peaks in cross section or trapping probability.

Abstract

We study the behavior of the $s$-wave partial cross section $\sigma(k)$, the Wigner-Smith time delay $\tau(k)$, and the trapping probability $P(k)$ as function of the wave number $k$. The $s$-wave central square well is used for concreteness, simplicity, and to elucidate the controversy whether it shows true resonances. It is shown that, except for very sharp structures, the resonance part of the cross section, the trapping probability, and the time delay, reach their local maxima at different values of $k$. We show numerically that $\tau(k)>0$ at its local maxima, occuring just before the resonant part of the cross section reaches its local maxima. These results are discussed in the light of the standard definition of resonance.