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 Gevrey type perturbation. We establish the large deviation theorem (LDT) for multi-dimensional quasi-periodic operator under sub-exponentially decaying long-range perturbation, and then prove its pure point spectrum property. Based on the LDT and Aubry duality, we show the absence of point spectrum for 1D (exponentially decaying) long-range quasi-periodic operator with multi-frequency and small Gevrey potential (without transversality restriction). We also prove the positivity of Lebesgue measure of spectrum.

Absence of eigenvalues of analytic quasi-periodic Schrodinger operators on $\mathbb{R}^d$

“…We refer to [YZ14] for the phase transition parameter involving the coupling and energy: log E log λ . (5) Our results can be extended to the case H = (−∆) l + λV ( θ + Kx ω), where l ∈ Z, l ≥ 1 and V (•) is Gevrey regular (see [Shi19b]).…”

Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

“…φ has pure point spectrum, while the localized eigenfunctions almost have the same decay rate. As a comparison, we mention that Shi [53] proved that for multi-frequency quasi-periodic operator with Gevrey type perturbations, that is,…”

Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

“…Recently, Ge-You-Zhou [18] showed the same holds for d ≥ 2, but is a perturbative result, in the sense that the largeness of λ depends on α. Theorem 1.3 extends their result to Gevrey class. As a comparison, we mention that Shi [33] proved that for multi-frequency quasi-periodic operator with Gevrey type perturbations, that is,…”