## Abstract

We develop a definitive physical-space scattering theory for the scalar wave equation □_{g} φ = 0 on Kerr exterior backgrounds in the general subextremal case |a| < M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting "scattering matrix" mapping radiation fields on the past horizon H^{-} and past null infinity I^{-} to radiation fields on H^{+} and I^{+} is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity I^{-} is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalized reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity IC of suitable wave-packets as above. The results make essential use of a refinement of our re-cent proof [30] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards.

Original language | English (US) |
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Pages (from-to) | 371-486 |

Number of pages | 116 |

Journal | Annales Scientifiques de l'Ecole Normale Superieure |

Volume | 51 |

Issue number | 2 |

DOIs | |

State | Published - 2018 |

## All Science Journal Classification (ASJC) codes

- General Mathematics