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}
}
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… 
View PDF on arXiv
5 Citations

5 Citations

Arithmetic version of anderson localization for quasiperiodic Schr\"odinger operators with even cosine type potentials

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…

Stability of the non-critical spectral properties I: arithmetic absolute continuity of the integrated density of states

. 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…

The Arithmetic Version of the Frequency Transition Conjecture: New Proof and Generalization

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…

Ballistic transport for limit-periodic Schr\"odinger operators in one dimension

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…

Ballistic Transport for Periodic Jacobi Operators on $\mathbb{Z}^d$

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

Ballistic Transport for Limit-Periodic Jacobi Matrices with Applications to Quantum Many-Body Problems

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…

On the relation between strong ballistic transport and exponential dynamical localization

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,…

Global theory of one-frequency Schrödinger operators

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.…

Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation

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…

Absence of ballistic motion

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}…

Quantum Dynamics of Periodic and Limit-Periodic Jacobi and Block Jacobi Matrices with Applications to Some Quantum Many Body Problems

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…

Reducibility or nonuniform hyperbolicity for quasiperiodic Schrodinger cocycles

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…

Almost localization and almost reducibility

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…

Continuity of the spectrum of quasi-periodic Schrödinger operators with finitely differentiable potentials

  • 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…

Operators with singular continuous spectrum: III. Almost periodic Schrödinger operators

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…
...