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

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