  Multiple scattering formulations have been employed for more than 30 years as
a method of studying the quantum vacuum or Casimir interactions between
distinct bodies. Here we review the method in the simple context of
$\delta$-function potentials, so-called semitransparent bodies. (In the limit
of strong coupling, a semitransparent boundary becomes a Dirichlet one.) After
applying the method to rederive the Casimir force between two semitransparent
plates and the Casimir self-stress on a semitransparent sphere, we obtain
expressions for the Casimir energies between disjoint parallel semitransparent
cylinders and between disjoint semitransparent spheres. Simplifications occur
for weak and strong coupling. In particular, after performing a power series
expansion in the ratio of the radii of the objects to the separation between
them, we are able to sum the weak-coupling expansions exactly to obtain
explicit closed forms for the Casimir interaction energy. The same can be done
for the interaction of a weak-coupling sphere or cylinder with a Dirichlet
plane. We show that the proximity force approximation (PFA), which becomes the
proximity force theorem when the objects are almost touching, is very poor for
finite separations.
