Real eigenvalues in the non-Hermitian Anderson model

TL;DR

This paper demonstrates that in the non-Hermitian Anderson model, eigenvalues are real and close to Hermitian eigenvalues when the Lyapunov exponent exceeds the non-Hermiticity parameter, expanding understanding of spectral properties.

Contribution

It provides new insights into the spectral behavior of non-Hermitian Anderson matrices, specifically showing conditions under which eigenvalues are real and close to Hermitian counterparts.

Findings

Eigenvalues are real when Lyapunov exponent exceeds non-Hermiticity parameter.

Eigenvalues are exponentially close to Hermitian eigenvalues in this regime.

Complements previous results on eigenvalue distribution in non-Hermitian models.

Abstract

The eigenvalues of the Hatano--Nelson non-Hermitian Anderson matrices, in the spectral regions in which the Lyapunov exponent exceeds the non-Hermiticity parameter, are shown to be real and exponentially close to the Hermitian eigenvalues. This complements previous results, according to which the eigenvalues in the spectral regions in which the non-Hermiticity parameter exceeds the Lyapunov exponent are aligned on curves in the complex plane.