  We investigate, in the framework of the linearized quantum gravity and the
leading-order perturbation theory, the quantum correction to the classical
Newtonian interaction between a pair of gravitationally polarizable objects in
the presence of both Neumann and Dirichlet boundaries. We obtain general
results for the interaction potential and find that the presence of a boundary
always strengthens in the leading-order the interaction as compared with the
case in absence of boundaries. But different boundaries yield a different
degree of strengthening. In the limit when one partner of the pair is placed
very close to the Neumann boundary, the interaction potential is larger when
the pair is parallel with the boundary than when it is perpendicular to, which
is just opposite to the case when the boundary is Dirichlet where the latter is
larger than the former. In addition, we find that the pair-boundary separation
dependence of the higher-order correction term is determined by the orientation
of the pair with respect to boundary, with the parallel case giving a quadratic
behavior and the perpendicular case a linear one.
