Inverse scattering by obstacles and Santalo's formula

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Abstract

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.
Original languageEnglish
Pages (from-to)185-200
Number of pages17
JournalPliska Studia Mathematica
Volume30
Publication statusPublished - 2019
EventFifth International Conference on New Trends of Differential Equations in Sciences: NTADES 2018 - Institute of Mechanics, Bulgarian Academy of Sciences, Sofia, Bulgaria
Duration: 18 Jun 201821 Jun 2018
Conference number: 5
http://math.bas.bg/en/event/ntades-2018-new-trends-in-the-applications-of-differential-equations-in-sciences/ (Conference website)

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