19-48 Svetlana Jitomirskaya, Sasa Kocic

Spectral theory of Schr\"odinger operators over circle diffeomorphisms (599K, pdf) Aug 28, 19

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Abstract. We initiate the study of Schr\"odinger operators with ergodic potentials defined over circle map dynamics, in particular over circle diffeomorphisms. For analytic circle diffeomorphisms and a set of rotation numbers satisfying Yoccoz's $\HH$ arithmetic condition, we discuss an extension of Avila's global theory. We also prove a sharp Gordon-type theorem which implies that for every $C^{1+BV}$ circle diffeomorphism, with a Liouville rotation number and an invariant measure $\mu$, for $\mu$-almost all $x\in\Tt^1$, the corresponding Schr\"odinger operator has purely continuous spectrum for every H\"older continuous potential $V.$

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