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Scattering matrix for asymptotically Euclidean manifolds (joint work with Richard Melrose). (English) Zbl 0878.58056

From the text: “The purpose of this exposé is to present the result of [R. B. Melrose and M. Zworski, ‘Scattering matrix for asymptotically Euclidean manifolds’, in preparation] and to indicate the methods used there.
We consider scattering in a setting generalizing the Euclidean one and introduced by R. B. Melrose in [Lect. Notes Pure Appl. Math. 161, 85-130 (1994; Zbl 0837.35107), for short [4]]. The main purpose there was to obtain a systematic framework for the study of scattering theory without relying on the symmetries of the Euclidean situation. Roughly speaking, the sphere at infinity was replaced by an arbitrary Riemannian manifold, which constituted in some sense a ‘smooth’ deformation of infinity. In the future, one can envision allowing also ‘singular’ infinities such as arise in the \(N\)-body problem or in scattering by non-compact obstacles.
In the Euclidean case the absolute scattering matrix acts on functions on the sphere at infinity and is essentially the pull-back by the antipodal map. From the microlocal point of view it is a Fourier integral operator associated to the geodesic flow, on the sphere at time \(\pi\). We show that in the general situation, the scattering matrix has the same property with the geodesic flow now on the boundary at infinity, proving a statement conjectured in [4]”.

MSC:

58J40 Pseudodifferential and Fourier integral operators on manifolds
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35P25 Scattering theory for PDEs
53C22 Geodesics in global differential geometry
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry

Citations:

Zbl 0837.35107
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