Dynamical Localization for the Random Dimer Schrödinger Operator

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 E=±V. 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)].