Abstract
We consider Schrödinger operators onl 2(ℤ ν) with deterministic aperiodic potential and Schrödinger operators on the l2-space of the set of vertices of Penrose tilings and other aperiodic self-similar tilings. The operators onl 2(ℤ ν) fit into the formalism of ergodic random Schrödinger 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 Schrödinger 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 Schrödinger operators that include several kinds of “almost-periodic” operators that have been studied in the literature. For Schrödinger 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.
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Hof, A. Some remarks on discrete aperiodic Schrödinger operators. J Stat Phys 72, 1353–1374 (1993). https://doi.org/10.1007/BF01048190
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DOI: https://doi.org/10.1007/BF01048190