Spectral transitions for the square Fibonacci Hamiltonian

Abstract

We study the spectrum and the density of states measure of the square Fibonacci Hamiltonian. We describe where the transitions from positive-measure to zero-measure spectrum and from absolutely continuous to singular density of states measure occur. This shows in particular that for almost every parameter from some open set, a positive-measure spectrum and a singular density of statesmeasure coexist. This provides the first physically relevant example exhibiting this phenomenon.