AI 中文总结
研究双曲空间中全测地缺陷的共形无穷远逆散射，通过规定边界标签得到远场模式，证明全测地缺陷可由其唯一确定，狄利克雷型缺陷用单个标签，诺伊曼型用\(n + 1\)个标签，还给出定量稳定性估计。
AI 中文摘要
我们研究双曲空间\(\mathbb H^n\)（\(n \geq 2\)）中相互作用表面为全测地的不可穿透拓扑缺陷的共形无穷远逆散射。对于固定的谱参数\(\lambda_0 > 0\)，规定输入为边界标签\(\xi \in \partial_\infty \mathbb H^n\)。给定缺陷\(\mathcal P \Subset \mathbb H^n\)，该模式通常不满足相互作用表面上的齐次迹条件，从而产生出射修正。此修正在共形无穷远的首项系数定义了测量的远场模式。核心问题是\(\mathcal P\)能否从对应于一个或有限多个规定边界标签的远场模式中恢复。我们的第一个主要结果确立了全测地缺陷的唯一确定性，包括体分量和超曲面支撑的缺陷。对于狄利克雷型缺陷，单个边界标签就足够；对于诺伊曼型缺陷，\(n + 1\)个边界标签足够且通常必要。我们的第二个主要结果提供了同一框架内的定量稳定性估计。两个缺陷之间的双曲豪斯多夫距离由它们在共形无穷远的远场模式差异控制。这些结果共同产生了基于形式确定数据的共形无穷远定性和定量远场逆散射理论。
英文摘要
We study inverse scattering from conformal infinity for impenetrable topological defects whose interaction surfaces are totally geodesic in hyperbolic space \(\mathbb H^n\), \(n \geq 2\). For a fixed spectral parameter \(λ_0 > 0\), the prescribed inputs are boundary labels \(ξ\in \partial_\infty \mathbb H^n\). Each label selects an incoming Helgason mode in the hyperbolic interior. Given a defect \(\mathcal P \Subset \mathbb H^n\), this mode generally fails to satisfy the homogeneous trace condition on the interaction surface and hence generates an outgoing correction. The leading coefficient of this correction at conformal infinity defines the measured far-field pattern. The central question is whether \(\mathcal P\) can be recovered from the far-field patterns corresponding to one or finitely many prescribed boundary labels. This gives a formally determined inverse problem at one fixed spectral parameter. Our first main result establishes the unique determination of totally geodesic defects, an admissible class that includes both bulk components and hypersurface-supported ones. For Dirichlet-type defects, a single boundary label suffices. For Neumann-type defects, \(n+1\) boundary labels are sufficient and in general necessary. These labels are required to satisfy the natural affine-independence condition at conformal infinity. Our second main result provides quantitative stability estimates within the same framework. The hyperbolic Hausdorff distance between two defects is controlled by the discrepancy of their far-field patterns at conformal infinity. The proof combines continuation from conformal infinity with quantitative geodesic reflection across totally geodesic hypersurfaces. Taken together, these results yield a qualitative and quantitative far-field inverse scattering theory at conformal infinity, based on formally determined data.
Comments85 pages
