SCATTERING MATRIX FOR MANIFOLDS WITH CONICAL ENDS

Abstract

Let M be a manifold with conical ends. (For precise definitions see the next section; we only mention here that the cross-section [Kscr ] 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 [Sscr ](s). If M is a compact perturbation of ℝⁿ, then it is well-known that [Sscr ](s) is a smooth perturbation of the antipodal map on a sphere, that is,

formula here

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 [Sscr ](s) is a Fourier integral operator on [Kscr ], of order 0, associated to the canonical diffeomorphism given by the geodesic flow at distance π. In our case it is possible to prove that [Sscr ](s) is in fact equal to the wave operator at a time t = π plus C^∞ 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).