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arXiv · 2610.18259

Square-Root Profiles and Sharp Fourier Decay of Quasiperiodic Spectral Measures

Zhenfu Wang·Qi Zhou
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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.

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Zhenfu Wang, Qi Zhou. 2026-10-13. Square-Root Profiles and Sharp Fourier Decay of Quasiperiodic Spectral Measures. https://arxiv.org/abs/2610.18259

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