Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications
Abstract
We develop a perturbative KAM reducibility theory for one-frequency SL(2,ℝ) cocycles based on an admissible Fourier length ℓ. 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 ℓ 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 ℓ. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.
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BibTeX
@misc{wang2026admissible,
title = {{Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications}},
author = {Xueyin Wang and Jiangong You},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11025},
eprint = {2609.11025},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11025},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications
AU - Xueyin Wang
AU - Jiangong You
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11025
UR - https://arxiv.org/abs/2609.11025
AB - We develop a perturbative KAM reducibility theory for one-frequency SL(2,ℝ) cocycles based on an admissible Fourier length ℓ. 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 ℓ 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 ℓ. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.
ER -