Regular Spacings of Complex Eigenvalues in the One-dimensional non-Hermitian Anderson Model

TL;DR

This paper proves that in the one-dimensional non-Hermitian Anderson model with a broad class of self-averaging potentials, the non-real eigenvalues are regularly spaced, simplifying previous approaches and enabling new insights.

Contribution

It introduces a wide class of self-averaging potentials and provides a simpler method to analyze eigenvalue spacings and properties of the density of states.

Findings

Non-real eigenvalues are regularly spaced in 1D NHA model.

The approach applies to stationary and quasi-periodic potentials.

Simplified proofs of previous results on non-real eigenvalues.

Abstract

We prove that in dimension one the non-real eigenvalues of the non-Hermitian Anderson (NHA) model with a selfaveraging potential are regularly spaced. The class of selfaveraging potentials which we introduce in this paper is very wide and in particular includes stationary potentials (with probability one) as well as all quasi-periodic potentials. It should be emphasized that our approach here is much simpler than the one we used before. It allows us a) to investigate the above mentioned spacings, b) to establish certain properties of the integrated density of states of the Hermitian Anderson models with selfaveraging potentials, and c) to obtain (as a by-product) much simpler proofs of our previous results concerned with non-real eigenvalues of the NHA model.