Quantum dynamical bounds for ergodic potentials with underlying dynamics of zero topological entropy
We show that positive Lyapunov exponents imply upper quantum dynamical bounds for Schrödinger operators Hf,θ u(n) = u(n+1) + u(n-1) + ϕ (fnθ)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.
Publication Source (Journal or Book title)
Analysis and PDE
Han, R., & Jitomirskaya, S. (2018). Quantum dynamical bounds for ergodic potentials with underlying dynamics of zero topological entropy. Analysis and PDE, 12 (4), 867-892. https://doi.org/10.2140/apde.2019.12.867