Speaker
Gaetan Lecrec, University of Helsinki
Abstract
A way to understand quantum properties of physical materials is to study Schrödinger operators associated to various electric potentials. In crystals, the potential is periodic, and the spectrum of associated Schrödinger operators is a finite union of intervals. 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 Schrödinger 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. In this setting, dispersion can be reduced to establishing Fourier decay of equilibrium states of 2D hyperbolic diffeomorphisms.