Arbitrarily Fast Quantum Dispersion in Long-Range Crystals
Gaétan Leclerc, Mostafa Sabri, Tuomas Sahlsten
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 Cα 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 Ck, answering a question raised by Geman and Horowitz in 1980.
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