Speaker: Gaetan Leclerc (Helsinki)

Title: Fourier Decay and the Fibonacci Hamiltonian

Abstract: The aim of this talk is to explain recent results about dispersive bounds in some quasicrystals https://arxiv.org/abs/2507.23731 .
The main question of study is the following. Consider a quasiperiodic potential and its associated Schrodinger operator in l^2(Z). What can be said about quantum transport in this setting ? The spectrum of these Schrodinger operators is typically a Cantor set, which makes the question difficult to answer, as spectral measures are not absolutely continuous. Studying the Hausdorff dimension of the spectrum may reveal natural decay rate for time-averaged (and phase-averaged) dispersion, but going past the time-averaged setting is difficult.  In the specific case of the Fibonacci Hamiltonian, we prove a pointwise dispersion bound without the need to take an average in time. More precisely, we show that the phase-averaged residual probability decay to zero at a polynomial rate. We will explain how this is related to the study of the Fourier transform of the density of state measure for the Fibonacci Hamiltonian, which is itself related to the measure of maximal entropy for a well chosen hyperbolic dynamical system: the Fibonacci trace map.