可容许傅里叶长度、KAM可约化性及谱应用
Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications
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中文总结 AI 辅助
本文提出基于可容许傅里叶长度的KAM可约化理论，统一处理解析与Gevrey情形，并应用于准周期薛定谔算子，获得绝对连续谱、态密度Hölder连续性及Aubry对偶纯点谱等结果。
中文摘要 AI 辅助
我们基于可容许傅里叶长度 $\ell$ 发展了一套针对单频 $\mathrm{SL}(2,\mathbb{R})$ 余环的扰动KAM可约化理论。与迭代相关的正则性由正的适应傅里叶宽度度量，而非欧几里得长度 $|n|$ 下的普通光滑性。同一长度同时控制傅里叶衰减、截断与共振尺度，以及控制小除数的算术条件。该框架包含经典解析和Gevrey情形，而非单调的 $\ell$ 选择允许经典的无处可微的Weierstrass型扰动以及所有正Hölder类之外的连续扰动。作为谱应用，我们为相关的准周期薛定谔算子获得了每个相位下的纯绝对连续谱以及态密度积分密度的 $1/2$-Hölder连续性。Aubry对偶在勒贝格几乎每个对偶相位下具有纯点谱，其特征函数在由 $\ell$ 诱导的度量下指数局部化。我们还构造了具有纯绝对连续Cantor谱的无处可微准周期势。
英文摘要
We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive Hölder class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-Hölder continuity of the integrated density of states for the associated quasiperiodic Schrödinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.
发表机构
- Texas A&M University（德克萨斯农工大学）
- Nankai University（南开大学）
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