The Dry Ten Martini Problem for Sturmian Hamiltonians

TL;DR

This paper proves that all spectral gaps are open for Sturmian Hamiltonians with non-zero coupling, using a novel representation of the spectrum as the boundary of an infinite tree constructed through periodic approximations.

Contribution

It provides a complete proof of the Dry Ten Martini Problem for Sturmian Hamiltonians, establishing the openness of all spectral gaps.

Findings

All spectral gaps are open for Sturmian Hamiltonians.

Spectrum represented as the boundary of an infinite tree.

Spectral characteristics encoded via periodic approximations.

Abstract

The Dry Ten Martini Problem for Sturmian Hamiltonians is affirmatively solved. Concretely, we prove that all spectral gaps are open for Schr\"odinger operators with Sturmian potentials and non-vanishing coupling constant. A key approach towards the solution is a representation of the spectrum as the boundary of an infinite tree. This tree is constructed via particular periodic approximations and it encodes substantial spectral characteristics.