Analysis & PDE

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

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

Rui Han and Svetlana Jitomirskaya

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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 : 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.

Article information

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

Dates
Received: 6 December 2016
Revised: 30 May 2018
Accepted: 5 July 2018
First available in Project Euclid: 30 October 2018

Permanent link to this document
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

Subjects
Primary: 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis

Keywords
transport exponent multifrequency quasiperiodic skew-shift

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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