Luchezar Stoyanov

Some dynamical problems in inverse scattering by obstacles.

An obstacle in \mathbb{R}^n is a compact subset of \mathbb{R}^n with smooth boundary \partial K and connected complement \Omega_K = \overline{\mathbb{R}^n\setminus K}. By a scattering trajectory in \Omega_K we mean the trajectory of a point moving with unit speed in the interior of \Omega and reflecting at \partial K following the usual law of geometrical optics. These are just trajectories of the billiard flow in the co-sphere bundle S^*(\Omega_K) of the exterior domain \Omega_K. In scattering theory physicists consider as “observables” the singularities of the so called scattering matrix which are related to sojourn times of scattering trajectories in the exterior of the obstacle. A natural problem is then to recover information about the obstacle K from sojourn times. An equivalent and easier to state problem concerns travelling times of scattering trajectories. Let S_0 be a large sphere in \mathbb{R}^n containing K in its interior. For any x = (p,\xi), where p \in S_0 and \xi is a unit vector pointing “inside” the sphere, let \gamma^+(x) be the scattering trajectory in \Omega_K issued from p in direction \xi. Let t(x) be the travelling time of x, i.e. the time it spends in S_0 “before going to infinity”. It may happen that t(x) = \infty, this defines a “trapped trajectory”. This talk is concerned with the following General Inverse Problem: Obtain information about the obstacle K from measurements of travelling times t(x) of scattering trajectories \gamma^+(x).

It turns out that some obstacles are completely recoverable from scattering data, however there are examples of obstacles that cannot be recovered. As one would expect, the dynamical system generated by the billiard flow in \Omega_K plays an important role in this study. We will also discuss an analogue of the well-known Santalo’s formula in geometry for billiard flows that turns out to be useful in dealing with the inverse scattering problem.