Effective-medium approach to the resonance distribution of wave scattering in a random point field

TL;DR

This paper develops a theoretical framework based on wave transport theory to explain and predict the distribution of resonance poles in wave scattering within a random point field, confirming findings with numerical simulations.

Contribution

It introduces an effective-medium approach to analytically describe the resonance distribution structures observed in wave scattering in random media.

Findings

Peaks at small $k$ explained by effective wave equation

Large $k$ band described by Bethe-Salpeter equation

Theoretical predictions match numerical simulations

Abstract

In a previous paper [Phys. Rev. A 105, 042205 (2022)], the distribution of resonance poles in the complex plane of the wavenumber $k$ associated to the multiple scattering of a quantum particle in a random point field was numerically discovered. This distribution presented two distinctive structures: a set of peaks at small $k$ when the wavelength is larger than the interscatterer distance, and a band almost parallel to the real axis at larger $k$. In this paper, a theoretical study based on wave transport theory is proposed to explain the origin of these structures and to predict their distribution in the complex $k$ plane. First, it is shown that the peaks at small $k$ can be understood using the effective wave equation for the average wavefunction over the disorder. Then, that the band at large $k$ can be described by the Bethe-Salpeter equation for the square modulus of the wavefunction. This study is supported by careful comparisons with numerical simulations.