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


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.



Lyapunov Exponent Inductive Step Rotation Number Positive Lyapunov Exponent Single Interval 
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© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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