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Annales Henri Poincaré

, Volume 16, Issue 4, pp 961–1031 | Cite as

The Dynamics of a Class of Quasi-Periodic Schrödinger Cocycles

  • Kristian BjerklövEmail author
Article

Abstract

Let \({f:\mathbb{T} \to\mathbb{R}}\) be a Morse function of class C 2 with exactly two critical points, let \({\omega \in \mathbb{T}}\) be Diophantine, and let λ > 0 be sufficiently large (depending on f and ω). For any value of the parameter \({E\in \mathbb{R}}\), we make a careful analysis of the dynamics of the skew-product map
$$\Phi_E(\theta,r)=\left(\theta+\omega, {\rm \lambda} f(\theta)-E-1/r\right),$$
acting on the “torus” \({\mathbb{T} \times \widehat{\mathbb{R}}}\). Here, \({\widehat{\mathbb{R}}}\) denotes the projective space \({\mathbb{R} \cup\{\infty\}}\). The map Φ E is intimately related to the quasi-periodic Schrödinger cocycle \({(\omega, A_E): \mathbb{T}\times \mathbb{R}^2 \to \mathbb{T}\times \mathbb{R}^2,\, (\theta,x)\mapsto (\theta+\omega, A_E(\theta)\cdot x)}\), where \({A_E:\mathbb{T}\to {\rm SL}(2,\mathbb{R})}\) is given by
$$A_{E}(\theta)=\left( \begin{array}{ll}0 \quad \quad \quad 1\\ -1 \quad {\rm \lambda} f(\theta)-E \\\end{array}\right),\quad E \in \mathbb{R}.$$
More precisely, (ω, A E ) naturally acts on the space \({\mathbb{T} \times \widehat{\mathbb{R}}}\), and Φ E is the map thus obtained. The cocycle (ω, A E ) arises when investigating the eigenvalue equation H θ u = Eu, where H θ is the quasi-periodic Schrödinger operator
$$(H_\theta u)_n=-(u_{n+1}+u_{n-1}) + {\rm \lambda} f(\theta+(n-1)\omega)u_n,$$
acting on the space \({l^2(\mathbb{Z})}\). It is well known that the spectrum of \({H_\theta,\, \sigma(H)}\), is independent of the phase \({\theta \in \mathbb{T}}\). Under our assumptions on f, ω and λ, Sinai (in J Stat Phys 46(5–6):861–909, 1987) has shown that σ(H) is a Cantor set, and the operator H θ has a pure-point spectrum, with exponentially decaying eigenfunctions, for a.e. \({\theta \in \mathbb{T}}\) The analysis of Φ E allows us to derive three main results:
  1. (1)

    The (maximal) Lyapunov exponent of the Schrödinger cocycle (ω, A E ) is \({\gtrsim {\rm log} {\rm \lambda}}\), uniformly in \({E\in \mathbb{R}}\). This implies that the map Φ E has exactly two ergodic probability measures for all \({E \in \mathbb{R}}\);

     
  2. (2)

    If E is on the edge of an open gap in the spectrum σ(H), then there exist a phase \({\theta \in\mathbb{T}}\) and a vector \({u \in l^2(\mathbb{Z})}\), exponentially decaying at \({\pm\infty}\), such that H θ u = Eu;

     
  3. (3)

    The map Φ E is minimal iff \({E\in \sigma(H){\setminus} \{\text{edges of open gaps} \}}\). In particular, Φ E is minimal for all \({E\in \mathbb{R}}\) for which the fibered rotation number α(E) associated with (ω, A E ) is irrational with respect to ω.

     

Keywords

Lyapunov Exponent Inductive Step Rotation Number Positive Lyapunov Exponent Single Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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