A scattering theory for the wave equation on kerr black hole exteriors

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.