  It is known that sub-extremal black hole backgrounds do not admit a
(bijective) non-degenerate scattering theory in the exterior region due to the
fact that the redshift effect at the event horizon acts as an unstable
blueshift mechanism in the backwards direction. In the extremal case, however,
the redshift effect degenerates and hence yields a much milder blueshift effect
when viewed in the backwards direction. In this paper, we construct a
definitive (bijective) non-degenerate scattering theory for the wave equation
on extremal Reissner-Nordstrom backgrounds. We make use of physical-space
energy norms which are non-degenerate both at the event horizon and at null
infinity. As an application of our theory we present a construction of a large
class of smooth exponentially decaying modes. We also derive scattering results
in the black hole interior region.
