Continuum Schrödinger Operators Associated With Aperiodic Subshifts

Abstract

We study Schrödinger operators on the real line whose potentials are generated by an underlying ergodic subshift over a finite alphabet and a rule that replaces symbols by compactly supported potential pieces. We first develop the standard theory that shows that the spectrum and the spectral type are almost surely constant, and that identifies the almost sure absolutely continuous spectrum with the essential closure of the set of energies with vanishing Lyapunov exponent. Using results of Damanik-Lenz and Klassert-Lenz-Stollmann, we also show that the spectrum is a Cantor set of zero Lebesgue measure if the subhift satisfies the Boshernitzan condition and the potentials are aperiodic and irreducible. We then study the case of the Fibonacci subshift in detail and prove results for the local Hausdorff dimension of the spectrum at a given energy in terms of the value of the associated Fricke-Vogt invariant. These results are elucidated for some simple choices of the local potential pieces, such as piecewise constant ones and local point interactions. In the latter special case, our results explain the occurrence of so-called pseudo bands, which have been pointed out in the physics literature.

19 pages

References in corpus (8)

• The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
• Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian
• Bers and Hénon, Painlevé and Schroedinger
• Hyperbolicity of the Trace Map for the Weakly Coupled Fibonacci Hamiltonian
• On the spectrum and Lyapunov exponent of limit periodic Schrodinger operators
• Spectral analysis of tridiagonal Fibonacci Hamiltonians
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Cited by in corpus (9)

• The Fibonacci Hamiltonian
• Absolutely Continuous Convolutions of Singular Measures and an Application to the Square Fibonacci Hamiltonian
• Multidimensional Almost-Periodic Schrödinger Operators with Cantor Spectrum
• Singularly continuous spectrum of a self-similar Laplacian on the half-line
• Zero Measure and Singular Continuous Spectra for Quantum Graphs
• Finding spectral gaps in quasicrystals
• Schrödinger Operators with Thin Spectra
• Spectral Properties of Continuum Fibonacci Schrödinger Operators
• Positive Lyapunov Exponents and a Large Deviation Theorem for Continuum Anderson Models, Briefly