Positive lyapounov exponents for most energies

Abstract

Consider the ID lattice Schrödinger operator

$$H = \lambda \cos (2\pi n^\rho \omega + \theta )\delta _{nn'} + \Delta$$

((1))

with 1 < ρ < 2 and define

$$\gamma (E,\lambda )\mathop {\underline {\lim } }\limits_{N \to \infty } \frac{1}{N}\int_\mathbb{T} {\log } \left\| {\prod\limits_N^0 {\left( {\begin{array}{*{20}c}{E - \lambda cos(2\pi n^\rho \omega + \theta ) - 1} \\{10} \\\end{array} } \right)} } \right\|d\theta$$

. If λ > 2, M. Herman’s [H] argument implies that γ(E,λ)≥log λ/2 >0, for all E. We are interested here in small λ and show that for all E∈E _λ⊂[−2,2]

$$mes([ - 2,2]\backslash \mathcal{E}_\lambda )\xrightarrow{{\lambda \to 0}}0$$

((2))

we have that γ(E, λ) > 0. See Proposition 4.

Considering the skew shift on $\mathbb{T}^2$

$$T(x,y) = (x + y,y + \omega )$$

((3))

and the Hamiltonian

$$H = \lambda \cos (\pi _1 T^m (x,y))\delta _{nn'} + \Delta$$

((4))

where $\pi _1 T^m (x,y) = x + ny + \frac{{n(n - 1)}}{2}\omega$we show that the Lyapounov exponent

$$\gamma (E,\lambda )^{\underline{\underline {a.e.}} } \mathop {\underline {\lim } }\limits_{N \to \infty } \frac{1}{N}\log \left\| {\prod\limits_N^0 {\left( {\begin{array}{*{20}c}{E - \lambda \cos \pi _1 T^n (x,y) - 1} \\{10} \\\end{array} } \right)} } \right\|$$

is strictly positive for E∈E _λ⊂[−2,2] satisfying (2), provided we assume in (3) that

$$\left| \omega \right| < \varepsilon (\lambda )$$

. See Proposition 5.

The method is based on a local approximation of (1), (4) by the almost Mathieu model

$$H_{\alpha ,\lambda ,\theta } = \lambda \cos (2\pi \alpha + \theta )\delta _{nn'} + \Delta$$

((5))

and uses the fact (see Corollary 3) that for λ small and E∈E _λ⊂[−2,2] satisfying (2),

$$\int_\mathbb{T} {\gamma (\alpha ,\lambda ,E)d\alpha > 0}$$

((6))

where γ(α, λ, E) refers to the Lyapounov exponents of (5). The proof of (6) does rely on the Aubry duality, [A-A], [La]).

Added in Proof. Concerning lattice Schrödinger operators of the form (1), related references were pointed out to the author by Y. Last. First, it is shown in the paper [L-S] that H=λcos(m^ρ)+Δ on ℤ has no absolutely continuous spectrum for λ > 2, ρ > 1. In fact, Theorem 1.4 from [L-S] provides an alternative proof of Proposition 4 in this paper. Other numerical and heuristic studies appear in [G-F],[B-F]. The particular case 1 < ρ < 2 was studied in [Th]. See [L-S] for further details.