Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces. (English) Zbl 0837.35107

Melrose, Richard B.

Ikawa, Mitsuru (ed.), Spectral and scattering theory. Proceedings of the Taniguchi international workshop, held at Sanda, Hyogo, Japan. Basel: Marcel Dekker. Lect. Notes Pure Appl. Math. 161, 85-130 (1994).

The author uses a microlocal view to investigate Euclidean scattering theory. Both the spectral and scattering theory are covered for the Laplacian of a “scattering metric” on any compact manifold with boundary. The theory is therefore universal. The author then goes into a detailed analysis from stereographic projection and structure algebra through the scattering Laplacian and calculus, to wavefront set and scattering matrix.
The author’s main contribution in this work is providing a bridge between the Euclidean scattering theory, with the explicit inversion of the Euclidean Laplacian, and the geometric problems typical of a metric singular at the boundary of a compact manifold with boundary. Proofs are provided for a wider context than the Euclidean setting. This work is useful since the methods used could be generalized to give a unified treatment of the \(N\)-body problem and classes of singular method on manifolds with corners.
For the entire collection see [Zbl 0798.00016].

Reviewer: S.Carmi (Philadelphia)

MSC:

35P25	Scattering theory for PDEs
47A40	Scattering theory of linear operators
34L25	Scattering theory, inverse scattering involving ordinary differential operators

Keywords:

scattering metric; spectral theory; Euclidean Laplacian; stereographic projection