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 time. 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–Nordström 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.

中文翻译：

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极值 Reissner–Nordström 波动方程的非退化散射理论

众所周知，由于事件视界处的红移效应在时间向后方向上充当不稳定的蓝移机制，因此次极值黑洞背景在外部区域不承认（双射）非退化散射理论. 然而，在极端情况下，红移效应会退化，因此在向后观察时会产生更温和的蓝移效应。在本文中，我们为极端 Reissner-Nordström 背景上的波动方程构建了一个确定的（双射的）非退化散射理论。我们利用在事件视界和零无穷处都是非退化的物理空间能量规范。作为我们理论的应用，我们提出了一大类平滑、指数衰减模式的构造。