Distribution of Eigenvalues in Non-Hermitian Anderson Model

TL;DR

This paper develops a theoretical framework to describe the distribution of eigenvalues of certain one-dimensional non-Hermitian random operators, revealing their concentration along a specific complex curve and relating this to underlying Hermitian spectral properties.

Contribution

It introduces a new theory linking eigenvalue distributions of non-Hermitian Anderson models to spectral features of Hermitian systems, providing explicit equations and density descriptions.

Findings

Eigenvalues are distributed along a specific curve in the complex plane.

The density of eigenvalues is expressed in terms of Hermitian spectral characteristics.

The spectrum exhibits coexistence of real and complex eigenvalues.

Abstract

We develop a theory which describes the behaviour of eigenvalues of a class of one-dimensional random non-Hermitian operators introduced recently by Hatano and Nelson. Under general assumptions on random parameters we prove that the eigenvalues are distributed along a curve in the complex plane. An equation for the curve is derived and the density of complex eigenvalues is found in terms of spectral characteristics of a ``reference'' hermitian disordered system. Coexistence of the real and complex parts in the spectrum and other generic properties of the eigenvalue distribution for the non-Hermitian problem are discussed.