De Bi\`evre S., Germinet F.
Dynamical Localization for the Random Dimer Model
(36K, Latex)

ABSTRACT.  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:
$$
\sup_t r^{(q)}_{\psi,I}(t) \equiv \sup_t \langle P_I(H_\omega)\psi_t, \ |X|^q
P_I(H_\omega)\psi_t \rangle\ <\infty.
$$
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 $I=\sigma(H_\omega)$).