Tuomas Sahlsten: Quantum dispersion, Fourier decay and dynamics

Abstract:  How fast can a quantum particle spread through a crystal material? Dispersion estimates measure this spreading through the decay of quantum probability amplitudes over time. I will discuss joint work with Gaétan Leclerc and Mostafa Sabri constructing one-dimensional crystals with long-range hopping and arbitrarily fast polynomial dispersion. The link is through Fourier decay for images of Lebesgue measure under Weierstrass functions. Our main tool is an adaptation of Dolgopyat’s method from hyperbolic dynamics, using the self-similarity of these functions to obtain a version of van der Corput’s lemma for non-smooth phases. This also answers a question of Geman and Horowitz from 1980 on the existence of local times for classical Weierstrass functions with sufficiently large lacunarity.