Dynamical Localization for the Random Dimer Model release_gli33pran5en5eutx4qv6zxl34

by S. De Bièvre, F. Germinet

Released as a article .

1999  

Abstract

We study the one-dimensional random dimer model, with Hamiltonian H_ω=Δ + V_ω, where for all x∈, V_ω(2x)=V_ω(2x+1) and where the V_ω(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≠ 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^2() with sufficiently rapid decrease: _t r^(q)_ψ,I(t) ≡_t < P_I(H_ω)ψ_t, |X|^q P_I(H_ω)ψ_t > <∞. Here ψ_t=e^-iH_ω tψ, and P_I(H_ω) is the spectral projector of H_ω onto the interval I. In particular if V>1 and V≠√(2), these results hold on the entire spectrum (so that one can take I=σ(H_ω)).
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Date   1999-07-07
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Language   en ?
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