Analysis & PDE

• Anal. PDE
• Volume 12, Number 4 (2019), 867-902.

Quantum dynamical bounds for ergodic potentials with underlying dynamics of zero topological entropy

Abstract

We show that positive Lyapunov exponents imply upper quantum dynamical bounds for Schrödinger operators $H f , θ u ( n ) = u ( n + 1 ) + u ( n − 1 ) + ϕ ( f n θ ) u ( n )$, where $ϕ : ℳ → ℝ$ is a piecewise Hölder function on a compact Riemannian manifold $ℳ$, and $f : ℳ → ℳ$ is a uniquely ergodic volume-preserving map with zero topological entropy. As corollaries we also obtain localization-type statements for shifts and skew-shifts on higher-dimensional tori with arithmetic conditions on the parameters. These are the first localization-type results with precise arithmetic conditions for multifrequency quasiperiodic and skew-shift potentials.

Article information

Source
Anal. PDE, Volume 12, Number 4 (2019), 867-902.

Dates
Revised: 30 May 2018
Accepted: 5 July 2018
First available in Project Euclid: 30 October 2018

https://projecteuclid.org/euclid.apde/1540864855

Digital Object Identifier
doi:10.2140/apde.2019.12.867

Mathematical Reviews number (MathSciNet)
MR3869380

Zentralblatt MATH identifier
06991221

Citation

Han, Rui; Jitomirskaya, Svetlana. Quantum dynamical bounds for ergodic potentials with underlying dynamics of zero topological entropy. Anal. PDE 12 (2019), no. 4, 867--902. doi:10.2140/apde.2019.12.867. https://projecteuclid.org/euclid.apde/1540864855

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