A uniqueness theorem for an inverse scattering problem in an exterior domain
A uniqueness theorem for an inverse scattering problem in an exterior domain
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DOI：
10.1137/s0036141097318614
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发表时间：
1998-09
影响因子：
2
通讯作者：
P. Hähner
中科院分区：
文献类型：
--
作者：
P. Hähner
R2 中考虑了非均匀介质中简单连接的软声障碍物对时谐声波的散射。我们证明，对于所有入射波和波数区间$\kappa$，大圆上的散射波的柯西数据唯一地确定了障碍物和不均匀性。为此，我们证明了在环形平面域的内部边界上满足狄利克雷条件的调和函数的乘积是完整的。然后，我们使用 $\kappa \ 为 0$ 的限制，并从 Schiffer 的唯一性结果中获得障碍物的唯一性。不均匀性的唯一性也遵循从 $\kappa \ 到 0$ 以及调和函数乘积的线性跨度的稠密结果。
The scattering of time-harmonic acoustic waves by a simply connected, sound-soft obstacle in an inhomogeneous medium in R2 is considered. We prove that the Cauchy data of the scattered waves on a large circle for all incident waves and for an interval of wave numbers $\kappa$ uniquely determine the obstacle and the inhomogeneity. To this end we show that products of harmonic functions satisfying a Dirichlet condition on the interior boundary of an annular plane domain are complete. Then, we use the limit $\kappa \to 0$ and obtain uniqueness of the obstacle from Schiffer's uniqueness result. Uniqueness of the inhomogeneity also follows from $\kappa \to 0$ together with the denseness result for the linear span of products of harmonic functions.