Reducibility or nonuniform hyperbolicity for quasiperiodic Schrodinger cocycles

By Artur Avila, R. Krikorian. 2003

225 citations recorded by the source corpus.

We show that for almost every frequency ?? ?? R\Q, for every C?O potential v : R/Z ?? R, and for almost every energy E the corresponding quasiperiodic Schr?Nodinger cocycle is either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schr?Nodinger operator, and allows us to complete the proof of the Aubry-Andr?Le conjecture on the measure of the spectrum of the Almost Mathieu Operator.

Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/00718f39a4cb290da40039017f577956fe9508ce

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. David Damanik
2. Jiangong You
3. Qi Zhou
4. Barry Simon
5. Joaquim Puig
6. Artur Avila

In Mathematical Physics

1. David Damanik
2. Jiangong You
3. Qi Zhou
4. Barry Simon
5. Artur Avila
6. Joaquim Puig
7. Doron S. Lubinsky
8. A. Zorich

In Statistical and Nonlinear Physics

1. David Damanik
2. Jiangong You
3. Qi Zhou
4. Artur Avila
5. Barry Simon
6. Joaquim Puig
7. Marcelo Viana
8. A. Zorich

Citations in the corpus

• Cantor Spectrum for the Almost Mathieu Operator (2004, background): “This last result has been obtained recently for the remaining Diophantine frequencies by Avila & Krikorian [2].”
• Sharp phase transitions for the almost Mathieu operator (2015, background): “In the case λ = 1, since Lebesgue measure of Σ1,α is zero for any α ∈ R\Q [11, 34], then H1,θ,α also doesn’t support any absolutely continuous spectrum, thus to prove Theorem 3.”
• Schrödinger operators with dynamically defined potentials (2014, methodology): “Therefore, the zero-measure results of Last [162] and Avila-Krikorian [18] described in the previous subsection imply Cantor spectrum in this case.”
• EQUILIBRIUM MEASURES AND CAPACITIES IN SPECTRAL THEORY (2007, background): “At the special point λ = 2, it is known that for all irrational α, the spectrum has measure zero [74, 10], and therefore for all irrational α and a.”