  We have calculated wave functions and matrix elements of the dipole operator
in the two- and three-dimensional Anderson model of localization and have
studied their statistical properties in the limit of weak disorder. In
particular, we have considered two cases. First, we have studied the
fluctuations as an external Aharonov-Bohm flux is varied. Second, we have
considered the influence of incipient localization. In both cases, the
statistical properties of the eigenfunctions are non-trivial, in that the joint
probability distribution function of eigenvalues and eigenvectors does no
longer factorize. We report on detailed comparisons with analytical results,
obtained within the non-linear sigma model and/or the semiclassical approach.
