# Ballistic Transport for One‐Dimensional Quasiperiodic Schrödinger Operators

@article{Ge2020BallisticTF,
title={Ballistic Transport for One‐Dimensional Quasiperiodic Schr{\"o}dinger Operators},
author={Lingrui Ge and Ilya Kachkovskiy},
journal={Communications on Pure and Applied Mathematics},
year={2020}
}
• Published 7 September 2020
• Mathematics
• Communications on Pure and Applied Mathematics
In this paper, we show that one-dimensional discrete multi-frequency quasiperiodic Schrodinger operators with smooth potentials demonstrate ballistic motion on the set of energies on which the corresponding Schrodinger cocycles are smoothly reducible to constant rotations. The proof is performed by establishing a local version of strong ballistic transport on an exhausting sequence of subsets on which reducibility can be achieved by a conjugation uniformly bounded in the $\mathrm{C}^{\ell… 5 Citations • Mathematics • 2021 We propose a new method to prove Anderson localization for quasiperiodic Schrodinger operators and apply it to the quasiperiodic model considered by Sinai and Frohlich-Spencer-Wittwer. More • Mathematics • 2022 . We prove absolute continuity of the integrated density of states for frequency-independent analytic perturbations of the non-critical almost Mathieu operator under arithmetic conditions on • Mathematics Peking Mathematical Journal • 2021 The arithmetic version of the frequency transition conjecture for the almost Mathieu operators was recently proved by Jitomirskaya and Liu [34]. We give a new proof via reducibility theory and Abstract. In this paper, we consider the transport properties of the class of limit-periodic continuum Schrödinger operators whose potentials are approximated exponentially quickly by a sequence of In this expository work, we collect some background results and give a short proof of the following theorem: periodic Jacobi matrices on$\mathbb{Z}^d$exhibit strong ballistic motion. ## References SHOWING 1-10 OF 59 REFERENCES We study Jacobi matrices that are uniformly approximated by periodic operators. We show that if the rate of approximation is sufficiently rapid, then the associated quantum dynamics are ballistic in We establish strong ballistic transport for a family of discrete quasiperiodic Schr\"odinger operators as a consequence of exponential dynamical localization for the dual family. The latter has been, 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. We show that the 1-dimensional Schrödinger equation with a quasiperiodic potential which is analytic on its hull admits a Floquet representation for almost every energyE in the upper part of the AbstractFor large classes of Schrödinger operators and Jacobi matrices we prove that ifh has only one point spectrum then for φ0 of compact support$$\mathop {\lim }\limits_{t \to \infty } t^{ - 2} • Mathematics • 2014 We investigate quantum dynamics with the underlying Hamiltonian being a Jacobi or a block Jacobi matrix with the diagonal and the off-diagonal terms modulated by a periodic or a limit-periodic • Mathematics • 2003 We show that for almost every frequency ?? ?? R\Q, for every C?O potential v : R/Z ?? R, and for almost every energy E the corresponding quasiperiodic Schr?Nodinger cocycle is either reducible or • Mathematics • 2008 We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schr\"odinger cocycles associated to a non-perturbatively small analytic potential • Xin Zhao • Mathematics Ergodic Theory and Dynamical Systems • 2020 In this paper, we consider the spectrum of discrete quasi-periodic Schrödinger operators on$\ell ^{2}(\mathbb{Z})$with the potentials$v\in C^{k}(\mathbb{T})$. For sufficiently large$k\$ , we show
• Mathematics
• 1994
We prove that one-dimensional Schrödinger operators with even almost periodic potential have no point spectrum for a denseGδ in the hull. This implies purely singular continuous spectrum for the