DSNT Seminar: Quantum dispersion and dynamical Cantor sets

Abstract: A way to understand quantum properties of physical materials is to study Schrodinger operators associated to various electric potentials. In crystals, the potential is periodic, and the spectrum of the associated Schrodinger operators in a union of interval. Spectral measures are then absolutely continuous, and transport or quantum dispersion is easy to understand. But for quasicrystals (such as the aluminium–palladium–manganese alloy), the potential is quasiperiodic, and the spectrum is typically a Cantor set. Spectral measures, that drive the quantum dynamics, are then fractal measures. How can one use our knowledge of geometric measure theory to prove results related to the spectral theory of such Schrodinger operators? We will study in particular the case of the "Fibonacci Hamiltonian", for which the spectrum is in fact a dynamical cantor set, related to some hyperbolic diffeomorphism on a surface: the Fibonacci Trace map. In this setting, pointwise power Fourier decay for the density of state measure can be established.

This is a seminar in our seminar series on Dynamical Systems and Number Theory (DSNT).