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This chapter represents an attempt to summarize some of the direct andindirect connections that exist between ray theory, wave theory and potentialscattering theory. Such connections have been noted in the past, and have beenexploited to some degree, but in the opinion of this author, there is much moreyet to be pursued in this regard. This article provides the framework for moredetailed analysis in the future. In order to gain a better appreciation for atopic, it is frequently of value to examine it from as many complementarylevels of description as possible, and that is the objective here. Drawing inpart on the work of Nussenzveig, Lock, Debye and others, the mathematicalnature of the rainbow is discussed from several perspectives. The primary bowis the lowest-order bow that can occur by scattering from a spherical drop withconstant refractive index n, but zero-order (or direct transmission) bows canexist when the sphere is radially inhomogeneous. The refractive index profileautomatically defines a scattering potential, but with a significant differencecompared 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. Thecorrespondences between the resonant modes in scattering by a potential of thewell-barrier type and the behavior of electromagnetic rays in a transparent (ordielectric) sphere are discussed. The poles and saddle points of the associatedscattering matrix have quite profound connections to electromagnetic tunneling,resonances and bows arising within and from the sphere. The links between thevarious mathematical and physical viewpoints are most easily appreciated in thecase of constant n, thus providing insight into possible extensions to thesedescriptions for bows of arbitrary order in radially inhomogeneous spheres (andcylinders).

Rainbows in homogeneous and radially inhomogeneous spheres: connections with ray, wave and potential scattering theory