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Published online by Cambridge University Press: 26 June 2025
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(ƛ, ω, θ), related to the wave equation in the exterior of the obstacle with Dirichlet boundary condition. It turns out that all singularities of the Fourier transform of a(ƛ, ω, θ), 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.
1. Introduction The scattering operator S(ƛ) presents a mathematical model for the data observed experimentally in many branches of physics, chemistry and mathematics. The operator S(.ƛ) is related to behavior as the time t → ±∞ of the solutions of an unperturbed operator Lo and to its perturbation L. The kernel of S(.ƛ) - I, the so called scattering amplitude a ƛ, ω, θ), contains the information related to the perturbation of Lo and this kernel is the leading term of the asymptotic of an outgoing solution of. Obstacle scattering problems arise in many physical phenomena and concern the perturbation caused by a bounded obstacle K with connected complement Ω. In general the inverse scattering problems deal with recovering geometric properties of K from information related to the scattering amplitude.
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