  We examine bosons hopping on a one-dimensional lattice in the presence of a
random potential at zero temperature. Bogoliubov excitations of the
Bose-Einstein condensate formed under such conditions are localized, with the
localization length diverging at low frequency as $\ell(\omega)\sim
1/\omega^\alpha$. We show that the well known result $\alpha=2$ applies only
for sufficiently weak random potential. As the random potential is increased
beyond a certain strength, $\alpha$ starts decreasing. At a critical strength
of the potential, when the system of bosons is at the transition from a
superfluid to an insulator, $\alpha=1$. This result is relevant for
understanding the behavior of the atomic Bose-Einstein condensates in the
presence of random potential, and of the disordered Josephson junction arrays.
