Some remarks on discrete aperiodic Schr�dinger operators

Abstract

We consider Schrodinger operators onl2(ℤν) with deterministic aperiodic potential and Schrodinger operators on the l2-space of the set of vertices of Penrose tilings and other aperiodic self-similar tilings. The operators onl2(ℤν) fit into the formalism of ergodic random Schrodinger operators. Hence, their Lyapunov exponent, integrated density of states, and spectrum are almost-surely constant. We show that they are actually constant: the Lyapunov exponent for one-dimensional Schrodinger operators with potential defined by a primitive substitution, the integrated density of states, and the spectrum in arbitrary dimension if the system is strictly ergodic. We give examples of strictly ergodic Schrodinger operators that include several kinds of “almost-periodic” operators that have been studied in the literature. For Schrodinger operators on Penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular Penrose tiling under consideration.