Spectral gaps of almost Mathieu operators in the exponential regime

Abstract

For almost Mathieu operator

the dry version of the 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 $\lambda$.