The one body density matrix is given by the Fourier transform of
the momentum distribution (see (1.45)). There are three
contributions to the one-body density matrix coming from
condensate, particle-particle interaction effects and
particle-impurity interaction effects:
(2.23)
The result for has been given in (1.47).
Here we calculate the contribution due to disorder
(2.24)
where
is the healing length.
Notice that at the value of the integral equals to the
density of particles which are scattered out of the condensate due
to the presence of the external field.