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Spectral properties of one dimensional quasi-crystals. (English) Zbl 0825.58010
Summary: We prove that the one-dimensional Schrödinger operator on \(l^ 2(\mathbb{Z})\) with potential given by: \[ v(n)= \lambda_{\chi[1- \alpha,1[}(x+ n\alpha),\qquad \alpha\not\in\mathbb{Q} \] has a Cantor spectrum of zero Lebesgue measure for any irrational \(\alpha\) and any \(\lambda> 0\). We can thus extend the Kotani result on the absence of absolutely continuous spectrum for this model, to all \(x\in \mathbb{T}\).

37E99 Low-dimensional dynamical systems
37A30 Ergodic theorems, spectral theory, Markov operators
82D25 Statistical mechanical studies of crystals
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