Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces.

*(English)*Zbl 0837.35107
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].

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)