Dynamical Localization for the Random Dimer Schrödinger Operator

We study the one-dimensional random dimer model, with Hamiltonian Hω = ∆ + Vω, where for all x ∈ Z, Vω(2x) = Vω(2x + 1) and where the Vω(2x) are i.i.d. Bernoulli random variables taking the values ±V, V > 0. We show that, for all values of V and with probability one in ω, the spectrum of H is pure point. If V ≤ 1 and V 6= 1/ √ 2, the Lyapounov exponent vanishes only at the two critical energies given by E = ±V . For the particular value V = 1/ √ 2, respectively V = √ 2, we show the existence of additional critical energies at E = ±3/ √ 2, resp. E = 0. On any compact interval I not containing the critical energies, the eigenfunctions are then shown to be semi-uniformly exponentially localized, and this implies dynamical localization: for all q > 0 and for all ψ ∈ l(Z) with sufficiently rapid decrease: sup t r (q) ψ,I(t) ≡ sup t 〈PI(Hω)ψt, |X |PI(Hω)ψt〉 < ∞. Here ψt = e ωψ, and PI(Hω) is the spectral projector of Hω onto the interval I. In particular if V > 1 and V 6= √ 2, these results hold on the entire spectrum (so that one can take I = σ(Hω)).