Sufficient conditions on the continuous spectrum for ergodic Schrödinger operators

Abstract

We investigate the spectral types of families of discrete one-dimensional Schrödinger operators {Hω}ω∈Ω , where each H ω has a potential function V ω (n) = f(T n ω) for n∈Z , T is an ergodic homeomorphism on a compact space Ω and f:Ω→R is a continuous function. The main analytical tool of the paper is Gordon’s lemma, of which we provide a complete proof. We demonstrate that a generic operator Hω∈{Hω}ω∈Ω has purely continuous spectrum if {Tnα}n≥0 is dense in Ω for a certain α ∈ Ω. Furthermore, we extend this result to the case where {Ω, T} satisfies topological repetition property, a concept introduced by Boshernitzan and Damanik (2008, []). Theorems presented in this paper weaken the hypotheses of previous research and enable us to reach the same conclusion as those authors. Our paper presents a comprehensive study of the topological and metric repetition properties, highlighting their binding role between the theory of dynamical systems and spectral theory. We also discuss the implications of these properties on the qualitative description of specific physical systems.