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

Abstract

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.