Global theory of one-frequency Schrödinger operators

By Artur Avila. 2015

287 citations recorded by the source corpus.

We study Schrödinger operators with a one-frequency analytic potential, focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials. From the dynamical point of view, the transition signals the emergence of non-uniform hyperbolicity, so the dependence of the Lyapunov exponent with respect to parameters plays a central role in the analysis. Though often ill-behaved by conventional measures, we show that the Lyapunov exponent is in fact remarkably regular in a “stratified sense” which we define: the irregularity comes from the matching of nice (analytic or smooth) functions along sets with complicated geometry. This result allows us to establish that the “critical set” for the transition lies within countably many codimension one subvarieties of the (infinite-dimensional) parameter space. A more refined renormalization-based analysis shows that the critical set is rather thin within those subvarieties, and allows us to conclude that a typical potential has no critical energies. Such acritical potentials also form an open set and have several interesting properties: only finitely many “phase transitions” may happen, but never at any specific point in the spectrum, and the Lyapunov exponent is minorated in the region of the spectrum where it is positive. On the other hand, we do show that the number of phase transitions can be arbitrarily large.Key to our approach are two results about the dependence of the Lyapunov exponent of one-frequency SL$${(2,\mathbb{C})}$$(2,C) cocycles with respect to perturbations in the imaginary direction: on one hand there is a severe “quantization” restriction, and on the other hand “regularity” of the dependence characterizes uniform hyperbolicity when the Lyapunov exponent is positive. Our method is independent of arithmetic conditions on the frequency.

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

View Paper: https://www.semanticscholar.org/paper/dcecee05b027e755fab504caff15e8f5431cc93d

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. Artur Avila

In Mathematical Physics

1. David Damanik
2. Jiangong You
3. Qi Zhou
4. Artur Avila
5. A. Zorich
6. Marcelo Viana
7. Jiagang Yang
8. G. Forni

In Statistical and Nonlinear Physics

1. David Damanik
2. Jiangong You
3. Qi Zhou
4. Artur Avila
5. Marcelo Viana
6. A. Zorich
7. Jiansheng Geng
8. Massimiliano Berti

Citations in the corpus

• Schrödinger operators with dynamically defined potentials (2014, background): “Fixing the frequency α, Avila showed in [3] that, for a typical analytic f ‡, there are no critical energies, and hence the spectrum splits into a localized regime and a subcritical regime.”
• Sharp phase transitions for the almost Mathieu operator (2015, methodology): “The proof can be found in Appendix A of [4].”