Square-Root Profiles and Sharp Fourier Decay of Quasiperiodic Spectral Measures
Abstract
We study spectral measures of one-dimensional quasiperiodic Schrödinger operators. For subcritical Schrödinger operator with Diophantine frequency, we prove sharp measurable-set bounds, answering a question of Avila and Jitomirskaya \cite{AJ}; we prove optimal Fourier decay and the sharp decay order of the time-averaged quantum return probability, confirming a conjecture of Aloisio, Carvalho, and de Oliveira \cite{ACO} and a prediction of Zhong and Mosseri \cite{ZhongMosseri}. For the subcritical almost Mathieu operator, we establish summable square-root envelopes for the spectral density, with an optimal density-one formulation of the lower bound, thereby answering a question of Avila \cite{Aqi}. These results follow from a general theorem for reflectionless measures under a summable square-root Craig-type condition. We establish an exact residue expansion of the squared Borel transform, which connects local density blow-up with long-time averaged return probabilities: the endpoint residues determine both the square-root edge profiles and the leading coefficient of the time-averaged return probability.
Related
- A Classification of Fourier Summation Formulas and Crystalline Measures
- Null-Space-Free 6D Spectral Embedding with Local Rayleigh Quotient Recovery for 3D Quasiperiodic Maxwell's Eigenproblems
- Fourier Decay of SRB Measures for Skew Products
- Sharp L6 Fourier moments and (3,δ)-energy estimates for fractal measures on the truncated cone in R3
- Some properties of Fourier quasicrystals and measures on a strip
BibTeX
@misc{wang2026squareroot,
title = {{Square-Root Profiles and Sharp Fourier Decay of Quasiperiodic Spectral Measures}},
author = {Zhenfu Wang and Qi Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18259},
eprint = {2610.18259},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18259},
}
RIS
TY - UNPB
TI - Square-Root Profiles and Sharp Fourier Decay of Quasiperiodic Spectral Measures
AU - Zhenfu Wang
AU - Qi Zhou
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18259
UR - https://arxiv.org/abs/2610.18259
AB - We study spectral measures of one-dimensional quasiperiodic Schrödinger operators. For subcritical Schrödinger operator with Diophantine frequency, we prove sharp measurable-set bounds, answering a question of Avila and Jitomirskaya \cite{AJ}; we prove optimal Fourier decay and the sharp decay order of the time-averaged quantum return probability, confirming a conjecture of Aloisio, Carvalho, and de Oliveira \cite{ACO} and a prediction of Zhong and Mosseri \cite{ZhongMosseri}. For the subcritical almost Mathieu operator, we establish summable square-root envelopes for the spectral density, with an optimal density-one formulation of the lower bound, thereby answering a question of Avila \cite{Aqi}. These results follow from a general theorem for reflectionless measures under a summable square-root Craig-type condition. We establish an exact residue expansion of the squared Borel transform, which connects local density blow-up with long-time averaged return probabilities: the endpoint residues determine both the square-root edge profiles and the leading coefficient of the time-averaged return probability.
ER -