A review of a work by Raymond: Sturmian Hamiltonians with a large coupling constant -- periodic approximations and gap labels

Abstract

We present a review of the work L. Raymond from 1995. The review aims at making this work more accessible and offers adaptations of some statements and proofs. In addition, this review forms an applicable framework for the complete solution of the Dry Ten Martini Problem for Sturmian Hamiltonians as appears in the work arXiv:2402.16703 by R. Band, S. Beckus and R. Loewy. A Sturmian Hamiltonian is a one-dimensional Schr\"odinger operator whose potential is a Sturmian sequence multiplied by a coupling constant, $V\in\mathbb{R}$. The spectrum of such an operator is commonly approximated by the spectra of designated periodic operators. If $V>4$, then the spectral bands of the periodic operators exhibit a particular combinatorial structure. This structure provides a formula for the integrated density of states. Employing this, it is shown that if $V>4$, then all the gaps, as predicted by the gap labelling theorem, are there.