Arithmetic version of Anderson localization via reducibility

Reducibility of ultra-differentiable quasi-periodic cocycles under an adapted arithmetic condition

We prove a reducibility result for sl(2,R) quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which extends the Brjuno-R{ü}ssmann condition in the analytic case. The proof is based on an elementary property of the fibered rotation number and deals with ultra-differentiable functions with a weighted Fourier norm. We also show that a weaker arithmetic condition is necessary for reducibility, and that it can be compared to a sufficient arithmetic condition.

精确的谱间隙、算术定位与通过共振分析的可约性

Sharp Spectral Gaps, Arithmetic Localization, and Reducibility via Resonance Analys

本文通过共振分析，研究了准周期Schrödinger算子的谱间隙、算术定位及可约性问题，解决了多个开放性猜想并揭示了亚临界算子的分层结构。

Reducibility of cocycles under a Brjuno-Rüssmann arithmetical condition

The arithmetics of the frequency and of the rotation number play a fundamental role in the study of reducibility of analytic quasi-periodic cocycles which are sufficiently close to a constant. In this paper we show how to generalize previous works by L.H.Eliasson which deal with the diophantine case so as to implement a Brjuno-Russmann arithmetical condition both on the frequency and on the rotation number. Our approach adapts the Poschel-Russmann KAM method, which was previously used in the problem of linearization of vector fields, to the problem of reducing cocycles.

Arithmetic version of anderson localization for quasiperiodic Schrödinger operators with even cosine type potentials

We propose a new method to prove Anderson localization for quasiperiodic Schrödinger operators and apply it to the quasiperiodic model considered by Sinai and Fröhlich-Spencer-Wittwer. More concretely, we prove Anderson localization for even $C^2$ cosine type quasiperiodic Schrödinger operators with large coupling constants, Diophantine frequencies and Diophantine phases.

Embedding of Analytic Quasi-Periodic Cocycles into Analytic Quasi-Periodic Linear Systems and its Applications

In this paper, we prove that any analytic quasi-periodic cocycle close to constant is the Poincaré map of an analytic quasi-periodic linear system close to constant. With this local embedding theorem, we get fruitful new results. We show that the almost reducibility of an analytic quasi-periodic linear system is equivalent to the almost reducibility of its corresponding Poincaré cocycle. By the local embedding theorem and the equivalence, we transfer the recent local almost reducibility results of quasi-periodic linear systems \cite{HoY} to quasi-periodic cocycles, and the global reducibility …

算术$k$多项式动力学局域化：$\n\ell^2(\mathbb{Z}^d)$上$k$幂律拟周期长程算子

Arithmetic $k$-Polynomial Dynamical Localization for $k$ Power-Law Quasi-Periodic Long-Range Operators on $\ell^2(\mathbb{Z}^d)$

本文基于定量$C^{k}$可约性，建立了幂律长程拟周期算子的算术$k$多项式谱与动力学局域化判据，并证明了几乎马蒂厄算子的幂律长程扰动在强耦合与丢番图频率下具有局域化性质。