  Spectral function is a key tool for understanding the behavior of
Bose-Einstein condensates of cold atoms in random potentials generated by a
laser speckle. In this paper we introduce a new method for computing the
spectral functions in disordered potentials. Using a combination of the
Wigner-Weyl approach with the landscape theory, we build an approximation for
the Wigner distributions of the eigenstates in the phase space and show its
accuracy in all regimes, from the deep quantum regime to the intermediate and
semiclassical. Based on this approximation, we devise a method to compute the
spectral functions using only the landscape-based effective potential. The
paper demonstrates the efficiency of the proposed approach for disordered
potentials with various statistical properties without requiring any adjustable
parameters.
