On the image of the scattering map for horizon-regular solutions of the linear scalar wave equation on the Schwarzschild black hole exterior

We construct Hilbert-space isomorphisms identifying the spaces of radiation fields on the event horizon and null infinity that are induced by the forward scattering map on the Schwarzschild exterior for fixed spherical harmonic mode solutions to the linear scalar wave equation arising from Cauchy data sets at $t=0$ which are regular at the event horizon, in the sense that the local energy at the event horizon is finite.

We show that fixed spherical harmonic mode solutions with no radiation on the event horizon must decay exponentially in time, at a rate that is determined by the surface gravity.

On the other hand, we construct examples of fixed spherical harmonic mode solutions with no radiation on null infinity and which have polynomial decay along the event horizon.

Finally, we construct examples of polynomially decaying solutions to the linear scalar wave equation which are regular at the event horizon, have unbounded support in spherical harmonic modes, and induce no radiation on the event horizon.