  We study the tail of the spectrum for non-interacting bosons in a
blue-detuned random speckle potential. Using an instanton approach we derive
the asymptotic behavior of the density of states in d dimensions. The leading
corrections resulting from fluctuations around the saddle point solution are
obtained by means of the Gel'fand-Yaglom method generalized to functional
determinants with zero modes. We find a good agreement with the results of
numerical simulations in one dimension. The effect of weak repulsive
interactions in the Lifshitz tail is also discussed.
