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Scattering matrix for manifolds with conical ends
journal contribution
posted on 2023-06-07, 21:39 authored by L ParnovskiLet M be a manifold with conical ends. (For precise definitions see the next section; we only mention here that the cross-section K can have a nonempty boundary.) We study the scattering for the Laplace operator on M. The first question that we are interested in is the structure of the absolute scattering matrix S(s). If M is a compact perturbation of Rn, then it is well-known that S(s) is a smooth perturbation of the antipodal map on a sphere, that is, S(s)f(·)=f(-·) (mod C8) On the other hand, if M is a manifold with a scattering metric (see [8] for the exact definition), it has been proved in [9] that S(s) is a Fourier integral operator on K, of order 0, associated to the canonical diffeomorphism given by the geodesic flow at distance p. In our case it is possible to prove that S(s) is in fact equal to the wave operator at a time t = p plus C8 terms. See Theorem 3.1 for the precise formulation. This result is not too difficult and is obtained using only the separation of variables and the asymptotics of the Bessel functions. Our second result is deeper and concerns the scattering phase p(s) (the logarithm of the determinant of the (relative) scattering matrix).
History
Publication status
- Published
Journal
Journal of the London Mathematical SocietyISSN
0024-6107Publisher
Oxford University PressExternal DOI
Issue
2Volume
61Page range
555-567ISBN
0024-6107Department affiliated with
- Mathematics Publications
Full text available
- No
Peer reviewed?
- Yes