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

In a previous paper [Phys. Rev. A 105, 042205 (2022)], the distribution of resonance poles in the complex plane of the wave number 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 wave function over the disorder. Then, that the band at large k can be described by the Bethe-Salpeter equation for the square modulus of the wave function. This study is supported by careful comparisons with numerical simulations.