On non-perturbative Anderson localization for C^αpotentials generated by shifts and skew-shifts

2006/07/13 by Chan, Jackson, Goldstein, Michael, Schlag, Wilhelm
#39A70 #47B39 #47N55 #81Q10 #82B44 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.math/0607302

Abstract

In this paper we address the question of proving Anderson localization (AL) for the operator [H(x,ω)ψ](n) := - \vp(n+1) - \vp(n-1) + V(Tⁿ_ωx)ψ(n), n∈\mathbb Z where T:\tor²→\tor² is either the shift or the skew-shift and V is only C^α(\tor²) for some α>0. We show that under the assumption of positive Lyapunov exponents, (AL) takes place for a.e. frequency, phase, and energy.

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