External theorems actually used

For the massless scalar on the Schwarzschild exterior, the past and future horizon and infinity channels give complete stationary scattering. In the s-wave reduction, the incoming generalized eigenfunctions define a unitary spectral transform

intertwining the radial operator with multiplication by ${\omega }^2$. The flux-normalized stationary scattering matrix is unitary.

We use the complete wave-scattering result of Dimock Dimock, together with the ordinary one-dimensional spectral resolution for a real short-range potential DeiftTrubowitz. Baskin–Wang provides a complementary geometric account of the two radiation ends and their energy normalization BaskinWang. The exact mode and test normalizations used below are displayed explicitly.

A formally self-adjoint normally hyperbolic operator on a globally hyperbolic spacetime has the advanced and retarded Green operators and Cauchy evolution used below. Adding a smooth compactly supported symmetric zeroth-order matrix coupling gives a well-posed coupled theory and a symplectic scattering map. It induces the corresponding automorphism of the Weyl algebra.

The PDE input is the standard Green-operator and Cauchy theory BGP. The local system–probe construction and its scattering interpretation are those of Fewster–Verch FewsterVerch. Its Weyl-algebra version is described by Fewster–Jubb–Ruep FewsterJubbRuep. We give the particular test transformation and estimates required here.

The affine-horizon and Killing-infinity product prescription defines a quasifree Hadamard Unruh state on the Schwarzschild exterior.

This is the Dappiaggi–Moretti–Pinamonti construction DMP, already declared in Em03-A. Its global microlocal proof is not reproduced.

None of these external statements supplies the compact bulk-to-packet approximation proved in Section sec:packetbridge.

boundary. The manuscript supplies analytical constructions. Compilation, formal verification, numerical evaluation, calibration, and experimental realization are not asserted.