Spectrum of Hatano-Nelson model with strictly ergodic potentials

TL;DR

This paper derives a precise spectral formula for the Hatano-Nelson model with strictly ergodic potentials, revealing real-complex spectrum transitions and spectral approximation properties, contrasting with random potential cases.

Contribution

It provides an exact spectral characterization for the model with ergodic potentials and explores spectrum transitions and approximation, differing from random potential scenarios.

Findings

Spectrum formula in terms of Lyapunov exponent

Observation of real-complex spectrum transition

Spectral approximation via finite-interval truncation

Abstract

We provide a precise formula for the spectrum of the Hatano-Nelson model with strictly ergodic potentials in terms of its Lyapunov exponent. As applications, one clearly observes the real-complex spectrum transition. Moreover, if the Lyapunov exponent is continuous, the spectrum of the Hatano-Nelson model in $\ell^{2}(\mathbb{Z})$ can be approximated by the spectrum of its finite-interval truncation with periodic boundary conditions. Both of these results are strikingly different from the Hatano-Nelson model with random potentials \cite{Dav01A, Dav01, Dav02}.