  This chapter represents an attempt to summarize some of the direct and
indirect connections that exist between ray theory, wave theory and potential
scattering theory. Such connections have been noted in the past, and have been
exploited to some degree, but in the opinion of this author, there is much more
yet to be pursued in this regard. This article provides the framework for more
detailed analysis in the future. In order to gain a better appreciation for a
topic, it is frequently of value to examine it from as many complementary
levels of description as possible, and that is the objective here. Drawing in
part on the work of Nussenzveig, Lock, Debye and others, the mathematical
nature of the rainbow is discussed from several perspectives. The primary bow
is the lowest-order bow that can occur by scattering from a spherical drop with
constant refractive index n, but zero-order (or direct transmission) bows can
exist when the sphere is radially inhomogeneous. The refractive index profile
automatically defines a scattering potential, but with a significant difference
compared to the standard quantum mechanical form: the potential is k-dependent.
A consequence of this is that there are no bound states for this system. The
correspondences between the resonant modes in scattering by a potential of the
well-barrier type and the behavior of electromagnetic rays in a transparent (or
dielectric) sphere are discussed. The poles and saddle points of the associated
scattering matrix have quite profound connections to electromagnetic tunneling,
resonances and bows arising within and from the sphere. The links between the
various mathematical and physical viewpoints are most easily appreciated in the
case of constant n, thus providing insight into possible extensions to these
descriptions for bows of arbitrary order in radially inhomogeneous spheres (and
cylinders).
