H\"{o}lder Continuity of the Spectral Measures for One-Dimensional Schr\"{o}dinger Operator in Exponential Regime

Avila and Jitomirskaya prove that the spectral measure $\mu_{\lambda v, \alpha,x}^f$ of quasi-periodic Schr\"{o}dinger operator is $1/2$-H\"{o}lder continuous with appropriate initial vector $f$, if $\alpha$ satisfies Diophantine condition and $\lambda$ is small. In the present paper, the conclusion is extended to that for all $\alpha$ with $\beta(\alpha)<\infty$, the spectral measure $\mu_{\lambda v, \alpha,x}^f$ is $1/2$-H\"{o}lder continuous with small $\lambda$, if $v$ is real analytic in a neighbor of $\{|\Im x|\leq C\beta\}$, where $C$ is a large absolute constant. In particular, the spectral measure $\mu_{\lambda, \alpha,x}^f$ of almost Mathieu operator is $1/2$-H\"{o}lder continuous if $|\lambda|<e^{-C\beta}$ with $C$ a large absolute constant

Spectral Gaps of Almost Mathieu Operator in Exponential Regime

For almost Mathieu operator $(H_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+2\lambda \cos2\pi(\theta+n\alpha)u_n$, 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 $\lambda\neq 0$ 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 $\lambda$