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It is known that sub-extremal black hole backgrounds do not admit a(bijective) non-degenerate scattering theory in the exterior region due to thefact that the redshift effect at the event horizon acts as an unstableblueshift mechanism in the backwards direction. In the extremal case, however,the redshift effect degenerates and hence yields a much milder blueshift effectwhen viewed in the backwards direction. In this paper, we construct adefinitive (bijective) non-degenerate scattering theory for the wave equationon extremal Reissner-Nordstrom backgrounds. We make use of physical-spaceenergy norms which are non-degenerate both at the event horizon and at nullinfinity. As an application of our theory we present a construction of a largeclass of smooth exponentially decaying modes. We also derive scattering resultsin the black hole interior region.

A non-degenerate scattering theory for the wave equation on extremal Reissner-Nordstrom