Dynamical Localization for the Random Dimer Model

We study the one-dimensional random dimer model, with Hamiltonian $H_\omega=\Delta + V_\omega$, where for all $x\in\Z, V_\omega(2x)=V_\omega(2x+1)$ 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 $V\leq1$ and $V\neq 1/\sqrt{2}$, 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 $\psi\in\ell^2(\Z)$ with sufficiently rapid decrease: $$\sup_t r^{(q)}_{\psi,I}(t) \equiv \sup_t < P_I(H_\omega)\psi_t, |X|^q P_I(H_\omega)\psi_t > <\infty.$$ Here $\psi_t=e^{-iH_\omega t} \psi$, and $P_I(H_\omega)$ is the spectral projector of $H_\omega$ onto the interval $I$. In particular if $V>1$ and $V\neq \sqrt{2}$, these results hold on the entire spectrum (so that one can take $I=\sigma(H_\omega)$).