Spectral Gaps of Almost Mathieu Operator in Exponential Regime

Abstract

For almost Mathieu operator $(H_{\lambda ,\alpha ,\theta }u{)}_n=u_{n+1}+u_{n-1}+2\lambda \cos 2\pi (\theta +n\alpha )u_n$, the dry version of Ten Martini problem predicts that the spectrum ${\Sigma }_{\lambda ,\alpha }$ of $H_{\lambda ,\alpha ,\theta }$ has all gaps open for all $\lambda \ne 0$ and $\alpha \in \mathbb{R}\backslash \mathbb{Q}$. Avila and Jitomirskaya prove that ${\Sigma }_{\lambda ,\alpha }$ has all gaps open for Diophantine $\alpha$ and $0<|\lambda |<1$. In the present paper, we show that ${\Sigma }_{\lambda ,\alpha }$ has all gaps open for all $\alpha \in \mathbb{R}\backslash \mathbb{Q}$ with small $\lambda$.

References in corpus (2)

• The absolutely continuous spectrum of the almost Mathieu operator
• Global theory of one-frequency Schrodinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity

Cited by in corpus (7)

• Central spectral gaps of the almost Mathieu operator
• Anderson localization for the completely resonant phases
• Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation
• Almost Mathieu operators with completely resonant phases
• Polynomial decay of the gap length for C^k quasi-periodic Schrodinger operators and spectral application
• Exponential Decay of the lengths of Spectral Gaps for Extended Harper's Model with Liouvillean Frequency
• Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications