billiard trajectory scattering by obstacles travelling time Santalo’s formula
Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences
Pliska Studia Mathematica Bulgarica, Vol. 30, No 1, (2019), 185p-200p
We consider some problems related to recovering information about an obstacle K in an Euclidean space from certain measurements of lengths of generalized geodesics in the exterior of the obstacle – e.g. sojourn times of scattering rays in the exterior of the obstacle, or simply, travelling times of geodesics within a certain large ball containing the obstacle. It is well-known in scattering theory that this scattering data is related to the singularities of the scattering kernel of the scattering operator for the wave equation in the exterior of K with Dirichlet boundary condition on the boundary. For some classes of obstacles, K can be completely recovered from the scattering data. On the other hand, for some obstacles the set of trapped points is too large and this makes it impossible to recover them from scattering data. We discuss also a certain stability property of the trapping set, which is obtained from a generalisation of Santalo’s formula to integrals over billiard trajectories in the exterior of an obstacle. Some other applications of this formula to scattering by obstacles are discussed as well. 2010 Mathematics Subject Classification: 37D40, 37D50, 53D25, 58J50.