Almost Mathieu operators with completely resonant phases

Abstract

Let $\alpha \in \mathbb{R}\backslash \mathbb{Q}$ and $\beta (\alpha )={\mathrm{limsup}}_{n\to \infty }(\ln q_{n+1})/q_n<\infty$, where $p_n/q_n$ is the continued fraction approximations to $\alpha$. Let $(H_{\lambda ,\alpha ,\theta }u)(n)=u(n+1)+u(n-1)+2\lambda \cos 2\pi (\theta +n\alpha )u(n)$ be the almost Mathieu operator on ${\ell }^2(\mathbb{Z})$, where $\lambda ,\theta \in \mathbb{R}$. Avila and Jitomirskaya \cite{avila2009ten} conjectured that for $2\theta \in \alpha \mathbb{Z}+\mathbb{Z}$, $H_{\lambda ,\alpha ,\theta }$ satisfies Anderson localization if $|\lambda |>e^{2\beta (\alpha )}$. In this paper, we developed a method to treat simultaneous frequency and phase resonances and obtain that for $2\theta \in \alpha \mathbb{Z}+\mathbb{Z}$, $H_{\lambda ,\alpha ,\theta }$ satisfies Anderson localization if $|\lambda |>e^{3\beta (\alpha )}$.

Ergodic Theory Dynam. Systems to appear

References in corpus (10)

• Sharp Phase transitions for the almost Mathieu operator
• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Arithmetic Spectral Transitions for the Maryland Model
• Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase
• Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators
• Full measure reducibility and localization for Jacobi operators: a topological criterion
• Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies
• Central spectral gaps of the almost Mathieu operator
• Zero Hausdorff dimension spectrum for the almost Mathieu operator
• Dry Ten Martini Problem in the non-critical case

Cited by in corpus (1)

• Quantitative inductive estimates for Green's functions of non-self-adjoint matrices