Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coexistence of ac and pp spectrum for kicked quasi-periodic potentials

Kristian Bjerklöv, Raphaël Krikorian

Abstract

We introduce a class of real analytic "peaky" potentials for which the corresponding quasi-periodic 1D Schrödinger operators exhibit, for quasiperiodic frequencies in a set of positive Lebesgue measure, both absolutely continuous and pure point spectrum.

Coexistence of ac and pp spectrum for kicked quasi-periodic potentials

Abstract

We introduce a class of real analytic "peaky" potentials for which the corresponding quasi-periodic 1D Schrödinger operators exhibit, for quasiperiodic frequencies in a set of positive Lebesgue measure, both absolutely continuous and pure point spectrum.

Paper Structure

This paper contains 41 sections, 45 theorems, 117 equations, 2 figures.

Table of Contents

  1. Introduction and Results
  2. Quasiperiodic Schrödinger operators
  3. Main results
  4. Dynamics of cocycles and spectral theory
  5. On the genesis of the paper
  6. Criteria for ac and pp spectrum
  7. Background
  8. Norms
  9. Cocycles
  10. Lyapunov exponent
  11. Rotation number
  12. Schrödinger cocycles
  13. Diophantine conditions
  14. Criterium for ac spectrum
  15. Criterium for pp spectrum
  16. ...and 26 more sections

Key Result

Theorem A

There exists $s_{0}\in{\mathbb N}^*$ such that the following holds. Let $V\in\mathcal{P}^\infty$ and $q\in {\mathbb N}^*$ be such that $K(V)>10$ and $L(V)<1/q$. Then, there exists $\varepsilon>0$ such that for any $\widetilde{V}\in\mathcal{P}^\omega(V;s_{0},\varepsilon)$ there exists a set of freque

Figures (2)

  • Figure 1: Numerical computations of (A) the Lyapunov exponent $LE(\alpha,S_{E-V_{K,\lambda}})$ and (B) the rotation number $\rho_\alpha(E)$ for $\alpha=(\sqrt{5}-1)/2, K=10$ and $\lambda=10000$.
  • Figure 2: Numerical computations of $E\mapsto LE(\alpha,S_{E-V_{K,\lambda}})$ on $[-2,2]$ for $\alpha=(\sqrt{5}-1)/2, K=10$ and $\lambda=10000$.

Theorems & Definitions (77)

  • Theorem A
  • Remark 1
  • Remark 2
  • Theorem B
  • Theorem 2.1: Eliasson, El-cmp
  • Remark 3
  • Theorem 2.2
  • Theorem 2.3: Extension of Eliasson's Theorem
  • proof
  • Theorem 2.4: Bourgain-Goldstein Theorem
  • ...and 67 more