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arXiv · 2608.27326

Arbitrarily Fast Quantum Dispersion in Long-Range Crystals

Gaétan Leclerc·Mostafa Sabri·Tuomas Sahlsten
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Abstract

We construct the first examples of long-range crystals exhibiting arbitrarily fast polynomial quantum dispersion. The Floquet functions of our Hamiltonians are highly oscillatory Weierstrass functions, whose rough autosimilar structure drives the fast dispersion. The proof develops a new Fourier decay theory for $C^\alpha$ images of Lebesgue measure, based on a Dolgopyat-inspired transfer operator method, and yields a van der Corput lemma for Weierstrass functions. As a consequence, the local time of classical Weierstrass functions of sufficiently large lacunarity exists and is $C^k$, answering a question raised by Geman and Horowitz in 1980.

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Gaétan Leclerc, Mostafa Sabri, Tuomas Sahlsten. 2026-08-27. Arbitrarily Fast Quantum Dispersion in Long-Range Crystals. https://arxiv.org/abs/2608.27326

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