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Generic singular continuous spectrum for ergodic Schr\"odinger operators

Dynamical Systems 2015-02-24 v1 Mathematical Physics math.MP

Abstract

We consider Schr\"odinger operators with ergodic potential Vω(n)=f(Tn(ω))V_\omega(n)=f(T^n(\omega)), nZn \in \Z, ωΩ\omega \in \Omega, where T:ΩΩT:\Omega \to \Omega is a non-periodic homeomorphism. We show that for generic fC(Ω)f \in C(\Omega), the spectrum has no absolutely continuous component. The proof is based on approximation by discontinuous potentials which can be treated via Kotani Theory.

Keywords

Cite

@article{arxiv.math/0409061,
  title  = {Generic singular continuous spectrum for ergodic Schr\"odinger operators},
  author = {Artur Avila and David Damanik},
  journal= {arXiv preprint arXiv:math/0409061},
  year   = {2015}
}

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6 pages