发表机构
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对二维亥姆霍兹方程的后向散射数据，提出两种测量模式下的内部逆散射唯一性结果，通过谱比较和实解析性证明域的唯一确定。
AI 中文摘要
我们研究了二维亥姆霍兹方程在源与接收器重合的后向散射数据下的内部逆散射问题。此类数据在应用中很常见，然而现有的唯一性结果几乎完全针对源与接收器分离的全多静态数据建立。我们考虑两种主要测量模式。在第一种模式中，数据在未知域内部的一条闭合曲线上收集。利用后向散射场对波数的亚纯依赖性，我们证明极点处的留数决定了该曲线上谱投影核的对角线，从而决定了狄利克雷谱的“可见”部分。这产生了一个无条件的谱比较：两个狄利克雷谱的对称差包含在该曲线所包围的内部域的谱中。此外，如果共同的可见特征值具有匹配的重数或匹配的限制秩，我们得到一个面积估计，将两个域面积的差异限制为该内部域的面积。在第二种模式中，数据在域的一个非空开子集上给出，且单个非共振频率已经唯一确定一个有界连通的利普希茨域。证明结合了亥姆霍兹格林函数正则部分的内部实解析性及其边界爆破。
英文摘要
We study an interior inverse scattering problem for the two-dimensional Helmholtz equation with backscattering data, where the source and receiver coincide. Such data are common in applications, yet existing uniqueness results are almost exclusively established for full multistatic data with separated sources and receivers. We consider two main measurement regimes. In the first regime, data are collected on a closed curve inside the unknown domain. Exploiting the meromorphic dependence of the backscattering field on the wavenumber, we show that the residues at the poles determine the diagonal of the spectral projection kernel on the curve, and hence the \emph{visible} part of the Dirichlet spectrum. This yields an unconditional spectral comparison: the symmetric difference of the two Dirichlet spectra is contained in the spectrum of the interior domain enclosed by the curve. If, in addition, the common visible eigenvalues have either matching multiplicities or matching restriction ranks, we obtain an area estimate bounding the discrepancy of the two domain areas by the area of this interior domain. In the second regime, data are given on a nonempty open subset of the domain, and a single non-resonant frequency already uniquely determines a bounded connected Lipschitz domain. The proof combines interior real-analyticity of the regular part of the Helmholtz Green function with its boundary blow-up.
Comments29 pages
