Complete hierarchical structure of the spectral bands in the Kohmoto model

Abstract

We study the Kohmoto model, a family of discrete Schrödinger operators with Sturmian potentials depending on a frequency and a coupling constant. We prove that, for all non-vanishing coupling constants, all spectral bands admit a hierarchical structure. This structure offers a variety of applications, including a detailed description of the Kohmoto butterfly and a central step towards the resolution of the dry ten Martini problem for Sturmian Hamiltonians, which we carry out in a subsequent work.

This preprint originates from the manuscript first posted as arXiv:2402.16703v1, which has since been split into two parts. The present preprint contains the first part and therefore overlaps with arXiv:2402.16703v1. The revised version arXiv:2402.16703v2 contains the second part. Both parts were carefully edited to form coherent and complementary manuscripts