Scattering theory for bulk and surface waves in an impedance half-space, I: radiation conditions and a Rellich-type theorem

@misc{watanabe2026scattering,
  title = {{Scattering theory for bulk and surface waves in an impedance half-space, I: radiation conditions and a Rellich-type theorem}},
  author = {Michiyuki Watanabe},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27885},
  eprint = {2610.27885},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27885},
}

TY  - UNPB
TI  - Scattering theory for bulk and surface waves in an impedance half-space, I: radiation conditions and a Rellich-type theorem
AU  - Michiyuki Watanabe
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27885
UR  - https://arxiv.org/abs/2610.27885
AB  - We study the negative Laplacian on the half-space ℝ^n_+, n ≥ 3, subject to a constant impedance (Robin) boundary condition with α> 0. The boundary then carries a bound transverse mode, and at each energy E > 0 the system propagates two channels with the distinct wavenumbers √E and √(E+α^2): bulk waves decaying like |x|^−(n−1)/2 and surface waves decaying like |x'|^−(n−2)/2. The two rates are not comparable pointwise. Each is critical for the Agmon-Hörmander space ℬ^*, however. This Part I treats the flat boundary, and its purpose is to develop, at that endpoint, the scattering theory of the two channels in full. We prove a limiting absorption principle ℬ → ℬ^*, construct the generalized Fourier transform, derive an expansion of the resolvent at infinity in which the two channels appear in a single relation, characterize the solutions of ℬ^* growth, and compute the scattering matrix. The radiation condition is imposed on the solution itself, with no decomposition into channels, and is shown to be equivalent to the vanishing of the incoming data. The main result is a Rellich-type theorem whose hypotheses are imposed only near infinity and which forces both channels to vanish simultaneously. The surface channel turns out to be rigid: its block of the scattering matrix is the identity and carries no information about α, so that on the band −α^2 < E < 0 the scattering matrix is trivial. The bulk block, by contrast, determines α.
ER  -