  Anderson localization of matter waves was recently observed with cold atoms
in a weak 1D disorder realized with laser speckle potential [J. Billy et al.,
Nature 453, 891 (2008)]. The latter is special in that it does not have spatial
frequency components above certain cutoff $q_{c}$. As a result, the Lyapunov
exponent (LE), or inverse localization length, vanishes in Born approximation
for particle wavevector $k>{1/2}q_{c}$, and higher orders become essential.
These terms, up to the order four, are calculated analytically and compared
with numerical simulations. For very weak disorder, LE exhibits a sharp drop at
$k$ $={1/2}q_{c}$. For moderate disorder (a) the drop is less dramatic than
expected from the fourth order approximation and (b) LE becomes very sensitive
to the sign of the disorder skewness (which can be controlled in cold atom
experiments). Both observations are related to the strongly non-Gaussian
character of the speckle intensity.

Disordered Systems and Neural Networks;Statistical Mechanics