The Dry Ten Martini Problem for Sturmian Hamiltonians

Abstract

The dry ten Martini problem for Sturmian Hamiltonians is solved. Concretely, we prove that all the predicted spectral gaps "are there" for all the Schrödinger 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 using periodic approximations and encodes substantial spectral characteristics.

This revised version originates from the manuscript first posted as arXiv:2402.16703v1, which has since been split into two parts. The present version contains the second part of that manuscript. The first part now appears as a separate preprint, posted as arXiv:2607.06361. Both parts were carefully edited to form coherent and complementary manuscripts