Communications in Mathematical Physics

, Volume 288, Issue 3, pp 907–918

On the Spectrum and Lyapunov Exponent of Limit Periodic Schrödinger Operators


    • CNRS UMR 7599, Laboratoire de Probabilités et Modèles AléatoiresUniversité Pierre et Marie Curie–Boîte Courrier 188
    • IMPA

DOI: 10.1007/s00220-008-0667-2

Cite this article as:
Avila, A. Commun. Math. Phys. (2009) 288: 907. doi:10.1007/s00220-008-0667-2


We exhibit a dense set of limit periodic potentials for which the corresponding one-dimensional Schrödinger operator has a positive Lyapunov exponent for all energies and a spectrum of zero Lebesgue measure. No example with those properties was previously known, even in the larger class of ergodic potentials. We also conclude that the generic limit periodic potential has a spectrum of zero Lebesgue measure.

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© Springer-Verlag 2008