External field inputs

The incoming horizon and infinity modes give complete stationary scattering for the massless Schwarzschild scalar. In the s-wave sector there is a unitary spectral transform

intertwining $\mathsf{L}$ with multiplication by ${\omega }^2$. The flux-normalized two-channel scattering matrix is unitary. The free Cauchy problem and advanced/retarded Green operators have their usual causal support.

The classical scattering input is the Schwarzschild wave-scattering construction Dimock; the Green-operator input is standard normally hyperbolic theory BGP. The specific weighted estimates needed here are proved below.

The initial field is the centered, regular, gauge-invariant quasifree Unruh prescription. The incoming horizon sector has occupation $n_{\mathrm{H}}(\omega )=(e^{\beta \omega }-1{)}^{-1}$; the incoming infinity sector is vacuum; the sectors are independent. This prescription has the bulk Hadamard identification of Dappiaggi–Moretti–Pinamonti DMP.

The global Hadamard proof is not reproduced. Hadamard regularity alone would not select this state.