Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Avila A., Damanik D.: Generic singular spectrum for ergodic Schrdinger operators. Duke Math. J. 130, 393–400 (2005)
Avila A., Damanik D.: Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling. Invent. Math. 172, 439–453 (2008)
Avron J., van Mouche P., Simon B.: On the measure of the spectrum of the almost Mathieu operator. Commun. Math. Phys. 132, 117–142 (1990)
Avron J., Simon B.: Almost periodic Schrödinger operators, I. Limit periodic potentials. Commun. Math. Phys. 82, 101–120 (1982)
Simon B.: Equilibrium measures and capacities in spectral theory. Inverse Probl. Imaging 1(4), 713–772 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by B. Simon
Rights and permissions
About this article
Cite this article
Avila, A. On the Spectrum and Lyapunov Exponent of Limit Periodic Schrödinger Operators. Commun. Math. Phys. 288, 907–918 (2009). https://doi.org/10.1007/s00220-008-0667-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0667-2