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

Abstract

Avila and Jitomirskaya prove that the spectral measure ${\mu }_{\lambda v,\alpha ,x}^f$ of quasi-periodic Schrödinger operator is $1/2$-Hölder 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 $1/2$-Hölder continuous with small $\lambda$, if $v$ is real analytic in a neighbor of $\{|\Im x|\le C\beta \}$, where $C$ is a large absolute constant. In particular, the spectral measure ${\mu }_{\lambda ,\alpha ,x}^f$ of almost Mathieu operator is $1/2$-Hölder continuous if $|\lambda |<e^{-C\beta }$ with $C$ a large absolute constant.

References in corpus (2)

• The absolutely continuous spectrum of the almost Mathieu operator
• Holder continuity of absolutely continuous spectral measures for one-frequency Schrodinger operators

Cited by in corpus (5)

• Quantitative inductive estimates for Green's functions of non-self-adjoint matrices
• Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation
• Hölder continuity of the integrated density of states for Extended Harper's Model with Liouville frequency
• Holder Continuity of Absolutely Continuous Spectral Measure for Multi-frequency Schrodinger Operators
• Absolutely Continuous Spectrum for the Quasi-periodic Schrödinger Operator in Exponential Regime