# Spectral Properties of Continuum Fibonacci Schr\"odinger Operators

## TL;DR

This paper investigates the spectral characteristics of continuum Fibonacci Schr"odinger operators, revealing that their spectrum's Hausdorff dimension approaches one in specific regimes, independent of potential shape.

## Contribution

It demonstrates that the Hausdorff dimension of the spectrum approaches one in small-coupling and high-energy limits for continuum Fibonacci Schr"odinger operators.

## Key findings

- Hausdorff dimension tends to one in small-coupling regime
- Hausdorff dimension tends to one in high-energy regime
- Results are independent of potential shape

## Abstract

We study continuum Schr\"odinger operators on the real line whose potentials are comprised of two compactly supported square-integrable functions concatenated according to an element of the Fibonacci substitution subshift over two letters. We show that the Hausdorff dimension of the spectrum tends to one in the small-coupling and high-energy regimes, regardless of the shape of the potential pieces.