Almost Sure Frequency Independence of the Dimension of the Spectrum of Sturmian Hamiltonians

TL;DR

This paper proves that for large coupling, the Hausdorff and upper box counting dimensions of the spectrum of Sturmian Hamiltonians are almost surely equal across almost all frequencies, revealing a form of spectral dimension stability.

Contribution

It establishes the almost sure equality of Hausdorff and box counting dimensions of the spectrum for Sturmian Hamiltonians at high coupling, a novel result in spectral theory.

Findings

Hausdorff and box counting dimensions coincide for large coupling

Dimension equality holds for Lebesgue almost every frequency

Spectral dimension stability at high coupling

Abstract

We consider the spectrum of discrete Schr\"odinger operators with Sturmian potentials and show that for sufficiently large coupling, its Hausdorff dimension and its upper box counting dimension are the same for Lebesgue almost every value of the frequency.