  The quantum vacuum fluctuations of a neutral scalar field induced by
background zero-range potentials concentrated on a flat hyperplane of
co-dimension $1$ in $(d+1)$-dimensional Minkowski spacetime are investigated.
Perfectly reflecting and semitransparent surfaces are both taken into account,
making reference to the most general local, homogeneous and isotropic boundary
conditions compatible with the unitarity of the quantum field theory. The
renormalized vacuum polarization is computed for both zero and non-zero mass of
the field, implementing a local version of the zeta regularization technique.
The asymptotic behaviours of the vacuum polarization for small and large
distances from the hyperplane are determined to leading order. It is shown that
boundary divergences are soften in the specific case of a pure Dirac delta
potential.
