Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation

Abstract

In this paper, we study the multi-frequency quasi-periodic operator with a Gevrey type perturbation. We first establish the large deviation theorem (LDT) for the multi-dimensional operator with a sub-exponential (or Gevrey) long-range hopping and then prove the pure point spectrum property. Based on the LDT and the Aubry duality, we show the absence of point spectrum for the $1D$ exponential long-range operator with a multi-frequency and a Gevrey potential. We also prove the spectrum has the positive Lebesgue measure.

arXiv commentA revised version. To appear in Journal d'Analyse Mathematique

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Cited by in corpus (3)

• Absence of eigenvalues of analytic quasi-periodic Schrodinger operators on ${\mathbb{R}}^d$
• Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications
• A Multi-scale Analysis Proof of the Power-law Localization for Random Operators on ${\mathbb{Z}}^d$