Hölder Continuity of the Spectral Measures for One-Dimensional Schrödinger Operator in Exponential Regime

Abstract

Avila and Jitomirskaya prove that the spectral measure $μ_{λv, α,x}^f$ of quasi-periodic Schrödinger operator is $1/2$-Hölder continuous with appropriate initial vector $f$, if $α$ satisfies Diophantine condition and $λ$ is small. In the present paper, the conclusion is extended to that for all $α$ with $β(α)<\infty$, the spectral measure $μ_{λv, α,x}^f$ is $1/2$-Hölder continuous with small $λ$, if $v$ is real analytic in a neighbor of $\{|\Im x|\leq Cβ\}$, where $C$ is a large absolute constant. In particular, the spectral measure $μ_{λ, α,x}^f$ of almost Mathieu operator is $1/2$-Hölder continuous if $|λ|<e^{-Cβ}$ with $C$ a large absolute constant.