## On the Kotani-Last and Schrödinger conjectures

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**28**(2015), 579-616 Request permission

## 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.## References

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## Additional Information

**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
- Received by editor(s): October 11, 2012
- Received by editor(s) in revised form: April 11, 2014
- Published electronically: June 11, 2014
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3300702