Dynamical Localization for the Random Dimer Model

Abstract

We study the one-dimensional random dimer model, with Hamiltonian $H_{\omega }=\Delta +V_{\omega }$, where for all $x\in \mathbb{Z},V_{\omega }(2x)=V_{\omega }(2x+1)$ and where the $V_{\omega }(2x)$ are i.i.d. Bernoulli random variables taking the values $\pm V,V>0$. We show that, for all values of $V$ and with probability one in $\omega$, the spectrum of $H$ is pure point. If $V\le 1$ and $V\ne 1/\sqrt{2}$, the Lyapounov exponent vanishes only at the two critical energies given by $E=\pm V$. For the particular value $V=1/\sqrt{2}$, respectively $V=\sqrt{2}$, we show the existence of additional critical energies at $E=\pm 3/\sqrt{2}$, 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 $\psi \in {\ell }^2(\mathbb{Z})$ with sufficiently rapid decrease: $$\underset{t}{\sup }{r}_{\psi ,I}^{(q)}(t)\equiv \underset{t}{\sup }<P_I(H_{\omega }){\psi }_t,|X{|}^q{P}_I(H_{\omega }){\psi }_t><\infty .$$ Here ${\psi }_t=e^{-i{H}_{\omega }t}\psi$, and $P_I(H_{\omega })$ is the spectral projector of $H_{\omega }$ onto the interval $I$. In particular if $V>1$ and $V\ne \sqrt{2}$, these results hold on the entire spectrum (so that one can take $I=\sigma (H_{\omega })$).

14 pages

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