Duke Mathematical Journal

Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts

Artur Avila, Jairo Bochi, and David Damanik

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We consider continuous SL(2,R)-cocycles over a strictly ergodic homeomorphism that fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle that is not uniformly hyperbolic can be approximated by one that is conjugate to an SO(2,R)-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be C0-perturbed to become uniformly hyperbolic. For cocycles arising from Schrödinger operators, the obstruction vanishes, and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schrödinger operator is a Cantor set

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Duke Math. J., Volume 146, Number 2 (2009), 253-280.

First available in Project Euclid: 5 January 2009

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Zentralblatt MATH identifier

Primary: 37D
Secondary: 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations 47B80: Random operators [See also 47H40, 60H25] 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis


Avila, Artur; Bochi, Jairo; Damanik, David. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J. 146 (2009), no. 2, 253--280. doi:10.1215/00127094-2008-065. https://projecteuclid.org/euclid.dmj/1231170940

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