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On the Kotani-Last and Schrödinger conjectures


Author: Artur Avila
Journal: J. Amer. Math. Soc. 28 (2015), 579-616
MSC (2010): Primary 37H15; Secondary 47B39
DOI: https://doi.org/10.1090/S0894-0347-2014-00814-6
Published electronically: June 11, 2014
MathSciNet review: 3300702
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Abstract: In the theory of ergodic one-dimensional Schrödinger operators, the ac spectrum has been traditionally expected to be very rigid. Two key conjectures in this direction state, on the one hand, that the ac spectrum demands almost periodicity of the potential, and, on the other hand, that the eigenfunctions are almost surely bounded in the essential support of the ac spectrum. We show how the repeated slow deformation of periodic potentials can be used to break rigidity, and disprove both conjectures.


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Artur Avila
Affiliation: CNRS, IMJ-PRG, UMR 7586, Univ Paris Diderot, Sorbonne Paris Cité, Sorbonnes Universités, UPMC Univ Paris 06, F-75013, Paris, France; IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil
Email: artur@math.univ-paris-diderot.fr

DOI: https://doi.org/10.1090/S0894-0347-2014-00814-6
Received by editor(s): October 11, 2012
Received by editor(s) in revised form: April 11, 2014
Published electronically: June 11, 2014
Article copyright: © Copyright 2014 American Mathematical Society