Abstract
We discuss discrete one-dimensional Schr\odinger operators whose potentials are generated by an invertible ergodic transformation of a compact metric space and a continuous real-valued sampling function. We pay particular attention to the case where the transformation is a minimal interval exchange transformation. Results about the spectrum and the spectral type of these operators are established. In particular, we provide the first examples of transformations for which the associated Schr\odinger operators have purely singular spectrum for every non-constant continuous sampling function.
Highlights
Consider a probability space (Ω, μ) and an invertible ergodic transformation T : Ω → Ω
Given a bounded measurable sampling function f : Ω → R, one can consider discrete one-dimensional Schrodinger operators acting on ψ ∈ l2(Z) as
One can essentially force this setting by mapping Ω ∋ ω → Vω ∈ Ω, where Ωis an infinite product of compact intervals
Summary
Consider a probability space (Ω, μ) and an invertible ergodic transformation T : Ω → Ω. The typical interval exchange transformations form a prominent class of examples For these we are able to use weak mixing to prove the absence of absolutely continuous spectrum for Lipschitz functions: Theorem 1.1. For more general continuous functions, it is rather the discontinuity of an interval exchange transformation that we exploit and not the weak mixing property We regard it as an interesting open question whether the Lipschitz assumption in Theorem 1.1 can be removed and the conclusion holds for every non-constant continuous f. (iii) Even more to the point, to the best of our knowledge, Theorem 1.2 is the first result of this kind, that is, one that identifies an invertible transformation T for which the absolutely continuous spectrum is empty for all non-constant continuous sampling functions.. This is an extension of a result of Avron and Simon
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