, Volume 98, Issue 5-6, pp 1135-1148

Dynamical Localization for the Random Dimer Schrödinger Operator

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Abstract

We study the one-dimensional random dimer model, with Hamiltonian H ω =Δ+V ω , where for all x \(\mathbb{Z}\) , 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 Vand with probability one in ω, the spectrum of His pure point. If V≤1 and V≠1/ \(\sqrt 2\) , the Lyapunov exponent vanishes only at the two critical energies given by EV. For the particular value V=1/ \(\sqrt 2\) , respectively, V= \(\sqrt 2\) , we show the existence of new additional critical energies at E=±3/ \(\sqrt 2\) , respectively, E=0. On any compact interval Inot 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 \(\ell\) 2( \(\mathbb{Z}\) ) with sufficiently rapid decrease $${\mathop {\sup }\limits_t} r_{\psi ,I}^{\left( q \right)} {\kern 1pt} \left( t \right): = {\mathop {\sup }\limits_t} \left\langle {P_I \left( {H\omega } \right)\psi _t ,\left| X \right|^q P_I \left( {H\omega } \right)\psi _t } \right\rangle < \infty $$ Here \(\psi _t = e^{- iH_{\omega ^t}} \psi\) , and P I(H ω) is the spectral projector of H ωonto the interval I. In particular, if V>1 and V \(\sqrt 2\) , these results hold on the entire spectrum [so that one can take I=σ(H ω)].