On the Spectrum of Sturmian Hamiltonians of Bounded Type in a Small Coupling Regime

TL;DR

This paper proves that the Hausdorff dimension of the spectrum of Sturmian Hamiltonians approaches one as the coupling constant decreases to zero, using trace map formalism.

Contribution

It establishes the limiting behavior of the spectrum's Hausdorff dimension for bounded type Sturmian Hamiltonians in low coupling regimes.

Findings

Hausdorff dimension tends to one as coupling approaches zero

Uses trace map formalism for the proof

Focuses on Sturmian potentials of bounded type

Abstract

We prove that the Hausdorff dimension of the spectrum of a discrete Schr\"odinger operator with Sturmian potential of bounded type tends to one as coupling tends to zero. The proof is based on the trace map formalism.