Bragg spectrum, K-theory, and gap labeling of aperiodic solids

Abstract

The diffraction spectrum of an aperiodic solid is related to the group of eigenvalues of the dynamical system associated with the solid. Those eigenvalues with continuous eigenfunctions constitute the topological Bragg spectrum. We relate the topological Bragg spectrum to topological invariants (Chern numbers) of the solid and to the gap-labeling group, which is the group of possible gap labels for the spectrum of a Schrödinger operator describing the electronic motion in the solid.