Communications in Mathematical Physics

, Volume 288, Issue 3, pp 907-918

First online:

On the Spectrum and Lyapunov Exponent of Limit Periodic Schrödinger Operators

  • Artur AvilaAffiliated withCNRS UMR 7599, Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie–Boîte Courrier 188IMPA Email author 

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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.