Abstract
In the scattering of time-harmonic acoustic or electromagnetic waves, whether an impenetrable sound-soft obstacle Ω can be completely determined by its scattering amplitude (or the far field pattern) AΩ(ξ, k) given for all ξ^ in the unit sphere Sn−1 at one wave number |k| and one incident direction k^ is still unknown. In this paper, we show that any ball in R n(n ≥ 3) can be uniquely determined by its scattering amplitude AΩ(·, k) given at two linearly independent incident directions k^1 and k^2 with one wave number |k|. We also show that two balls in R n(n ≥ 2) with the same scattering amplitude AΩ(·, k) at one direction k^ ε S n−1 and one wave number |k| must coincide.
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Liu, C. (1997). An Inverse Obstacle Problem: a Uniqueness Theorem for Balls. In: Chavent, G., Sacks, P., Papanicolaou, G., Symes, W.W. (eds) Inverse Problems in Wave Propagation. The IMA Volumes in Mathematics and its Applications, vol 90. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1878-4_16
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DOI: https://doi.org/10.1007/978-1-4612-1878-4_16
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