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).