The Dry Ten Martini Problem for Sturmian Hamiltonians

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.