Level spacing distribution of a Lorentzian matrix at the spectrum edge

TL;DR

This paper derives the level spacing distribution for eigenvalues of Lorentzian matrices at the spectrum edge, revealing a divergence in mean spacing and a unique decay law, relevant for understanding chaotic scattering.

Contribution

It provides the first explicit distribution of eigenvalue spacings for Lorentzian matrices at the spectrum edge, highlighting their unique statistical properties.

Findings

Distribution has no finite mean level spacing

Eigenvalue spacings follow a geometrical decay law

Relevance to chaotic scattering processes

Abstract

Effective Hamiltonians can explain in a much simpler way the physics behind a scattering process. Chaotic scattering is directly related to Lorentzian Hamiltonians which, because of their properties, can be reduced to a $2\times 2$ matrix problem in the case of two modes scattering. In this framework, we provide the distribution of level spacing of its eigenvalues and show that this special kind of distribution has no mean level spacing (divergent) and is characterized by a geometrical decay law. We discuss the relation of this distribution to the averaged level spacing at the edge of the spectrum of $N \times N$ Lorentzian matrices.