Isospectrum of non-self-adjoint almost-periodic Schrodinger operators

TL;DR

This paper establishes a criterion for non-self-adjoint almost-periodic Schrödinger operators to share the same spectrum and Lyapunov exponents as the free Laplacian, revealing limitations of traditional gap-opening methods.

Contribution

It introduces a new criterion for spectral and Lyapunov exponent equivalence and demonstrates the inapplicability of the Moser-Pöschel gap-opening argument in the non-self-adjoint setting.

Findings

Non-self-adjoint operators can have identical spectra and Lyapunov exponents as the free Laplacian.

The Moser-Pöschel argument may fail for non-self-adjoint operators.

A specific criterion guarantees spectral and Lyapunov exponent equivalence.

Abstract

For non-self-adjoint almost-periodic Schr\"odinger operators, a criterion is given to guarantee that they have both the same spectrum and same Lyapunov exponents with the discrete free Laplacian. As a byproduct, we show that the Moser-P\"oschel argument for opening gaps may not be valid for non-self-adjoint operators.