Duke Mathematical Journal
- Duke Math. J.
- Volume 146, Number 2 (2009), 253-280.
Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts
We consider continuous -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 -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 -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
Duke Math. J. Volume 146, Number 2 (2009), 253-280.
First available in Project Euclid: 5 January 2009
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Mathematical Reviews number (MathSciNet)
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