Anderson Localization for the Almost Mathieu Operator in Exponential Regime

Abstract

For the almost Mathieu operator $(H_{\lambda ,\alpha ,\theta }u{)}_n=u_{n+1}+u_{n-1}+2\lambda \cos 2\pi (\theta +n\alpha )u_n$, Avila and Jitomirskaya guess that for a.e. $\theta$, $H_{\lambda ,\alpha ,\theta }$ satisfies Anderson localization if $|\lambda |>e^{\beta }$, and they establish this for $|\lambda |>e^{\frac{16}{9}\beta }$. In the present paper, we extend their result to regime $|\lambda |>e^{\frac{3}{2}\beta }$.

Cited by in corpus (4)

• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Quantitative inductive estimates for Green's functions of non-self-adjoint matrices
• Pure point spectrum for the Maryland model: a constructive proof
• Almost Mathieu operators with completely resonant phases