PreprintCC BY 4.0arXiv
Arbitrarily Fast Quantum Dispersion in Long-Range Crystals
Abstract
We construct the first examples of one-dimensional long-range crystals exhibiting arbitrarily fast polynomial dispersive bounds. 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 Cα images of the 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 Ck, answering a question raised by Geman and Horowitz in 1980.
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BibTeX
@misc{leclerc2026arbitrarily,
title = {{Arbitrarily Fast Quantum Dispersion in Long-Range Crystals}},
author = {Gaétan Leclerc and Mostafa Sabri and Tuomas Sahlsten},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.27326},
eprint = {2608.27326},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27326},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Arbitrarily Fast Quantum Dispersion in Long-Range Crystals
AU - Gaétan Leclerc
AU - Mostafa Sabri
AU - Tuomas Sahlsten
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2608.27326
UR - https://arxiv.org/abs/2608.27326
AB - We construct the first examples of one-dimensional long-range crystals exhibiting arbitrarily fast polynomial dispersive bounds. 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^α images of the 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.
ER -