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Sojourn times, singularities of the scattering kernel and inverse problems. (English) Zbl 1086.35146
Uhlmann, Gunther (ed.), Inside out: Inverse problems and applications. Cambridge: Cambridge University Press (ISBN 0-521-82469-9/hbk). Math. Sci. Res. Inst. Publ. 47, 297-332 (2003).
Summary: We study inverse problems in the scattering by obstacles in odd-dimensional Euclidean spaces. In general, such problems concern the recovery of the geometric properties of the obstacle from the information related to the scattering amplitude \(a(\lambda,\omega,\theta)\), related to the wave equation in the exterior of the obstacle with Dirichiet boundary condition. It turns out that all singularities of the Fourier transform of \(a(\lambda,\omega, \theta)\), the so-called scattering kernel, are given by the sojourn (traveling) times of scattering rays in the exterior of the obstacle. Apart from that these sojourn times are a naturally observable data. The purpose of this survey is to describe several results in obstacle scattering obtained in the last twenty years concerning sojourn times of scattering rays, and to motivate further study of related inverse scattering problems.
For the entire collection see [Zbl 1034.78003].

MSC:
35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
35L05 Wave equation
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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