Continuum Fibonacci Schr\"odinger Operators in the Strongly Coupled Regime

TL;DR

This paper investigates the spectral properties of Fibonacci Schrödinger operators with zero potential segments in the strong coupling limit, revealing nuanced behaviors of the spectrum's dimension that challenge previous assumptions.

Contribution

It generalizes existing theorems on Fibonacci Schrödinger operators and provides a counterexample showing that spectrum dimension does not uniformly vanish at high coupling.

Findings

Spectrum dimension behavior varies with coupling strength.

Counterexample disproves naive generalizations of earlier results.

Local Hausdorff dimension may not tend to zero uniformly.

Abstract

We study Schr\"odinger operators on the real line whose potentials are generated by the Fibonacci substitution sequence and a rule that replaces symbols by compactly supported potential pieces. We consider the case in which one of those pieces is identically zero, and study the dimension of the spectrum in the large-coupling regime. Our results include a generalization of theorems regarding explicit examples that were studied previously and a counterexample that shows that the na\"ive generalization of previously established statements is false. In particular, in the aperiodic case, the local Hausdorff dimension of the spectrum does not necessarily converge to zero uniformly on compact subsets as the coupling constant is sent to infinity.