Spectra of Discrete Schrödinger Operators with Primitive Invertible Substitution Potentials

Abstract

We study the spectral properties of discrete Schrödinger operators with potentials given by primitive invertible substitution sequences (or by Sturmian sequences whose rotation angle has an eventually periodic continued fraction expansion, a strictly larger class than primitive invertible substitution sequences). It is known that operators from this family have spectra which are Cantor sets of zero Lebesgue measure. We show that the Hausdorff dimension of this set tends to $1$ as coupling constant $\lambda$ tends to $0$. Moreover, we also show that at small coupling constant, all gaps allowed by the gap labeling theorem are open and furthermore open linearly with respect to $\lambda$. Additionally, we show that, in the small coupling regime, the density of states measure for an operator in this family is exact dimensional. The dimension of the density of states measure is strictly smaller than the Hausdorff dimension of the spectrum and tends to $1$ as $\lambda$ tends to $0$.

References in corpus (3)

• Dynamical Green Functions and Discrete Schrödinger Operators with Potentials Generated by Primitive Invertible Substitution
• The Spectrum of the Almost Mathieu Operator
• Tridiagonal substitution Hamiltonians

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• On the Hausdorff dimension of the spectrum of Thue-Morse Hamiltonian
• The spectral properties of the strongly coupled Sturm Hamiltonian of eventually constant type
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• The spectral characteristics of the Sturm Hamiltonian with eventually periodic type
• Exact-dimensional property of density of states measure of Sturm Hamiltonian