Continuum Schr\"odinger Operators Associated With Aperiodic Subshifts

TL;DR

This paper investigates the spectral properties of Schr"odinger operators generated by aperiodic subshifts, establishing conditions for spectrum Cantor sets of zero measure, and analyzing the local spectral dimension for Fibonacci subshifts.

Contribution

It develops a comprehensive theory linking subshift dynamics with spectral properties, including the spectrum's Cantor structure and local Hausdorff dimension analysis for Fibonacci models.

Findings

Spectrum is almost surely constant and can be a zero-measure Cantor set.

The spectrum's local Hausdorff dimension relates to the Fricke-Vogt invariant.

Pseudo bands are explained through local potential piece analysis.

Abstract

We study Schr\"odinger 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.