  We theoretically investigate the physics of interacting Bose-Einstein
condensates at equilibrium in a weak (possibly random) potential. We develop a
perturbation approach to derive the condensate wavefunction for an amplitude of
the potential smaller than the chemical potential of the condensate and for an
arbitrary spatial variation scale of the potential. Applying this theory to
disordered potentials, we find in particular that, if the healing length is
smaller than the correlation length of the disorder, the condensate assumes a
delocalized Thomas-Fermi profile. In the opposite situation where the
correlation length is smaller than the healing length, we show that the random
potential can be significantly smoothed and, in the meanfield regime, the
condensate wavefunction can remain delocalized, even for very small correlation
lengths of the disorder.
