Absolutely Continuous Spectrum for the Quasi-periodic Schrödinger Operator in Exponential Regime

Abstract

Avila and Jitomirskaya prove that the quasi-periodic Schrödinger operator $H_{\lambda v,\alpha ,\theta }$ has purely absolutely continuous spectrum for $\alpha$ in sub-exponential regime (i.e., $\beta (\alpha )=0$) with small $\lambda$, if $v$ is real analytic in a strip of real axis. In the present paper, we show that for all $\alpha$ with $0<\beta (\alpha )<\infty$, $H_{\lambda v,\alpha ,\theta }$ has purely absolutely continuous spectrum with small $\lambda$, if $v$ is real analytic in strip $|\Im x|<C\beta$, where $C$ is a large absolute constant.

arXiv admin note: substantial text overlap with arXiv:1311.0669

References in corpus (2)

• The absolutely continuous spectrum of the almost Mathieu operator
• Hölder Continuity of the Spectral Measures for One-Dimensional Schrödinger Operator in Exponential Regime