Vol. 12, No. 4, 2019

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Quantum dynamical bounds for ergodic potentials with underlying dynamics of zero topological entropy

Rui Han and Svetlana Jitomirskaya

Vol. 12 (2019), No. 4, 867–902
DOI: 10.2140/apde.2019.12.867
Abstract

We show that positive Lyapunov exponents imply upper quantum dynamical bounds for Schrödinger operators Hf,θu(n) = u(n + 1) + u(n 1) + ϕ(fnθ)u(n), where ϕ : is a piecewise Hölder function on a compact Riemannian manifold , and f : 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.

Keywords
transport exponent, multifrequency quasiperiodic, skew-shift
Mathematical Subject Classification 2010
Primary: 47B36, 81Q10
Milestones
Received: 6 December 2016
Revised: 30 May 2018
Accepted: 5 July 2018
Published: 20 October 2018
Authors
Rui Han
Department of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States
Svetlana Jitomirskaya
Department of Mathematics
University of California
Irvine, CA
United States