发表机构
Beijing Institute of Technology; Shenzhen MSU-BIT University; City University of Hong Kong（北京理工大学; 深圳大学; 香港城市大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明有限时间区间的完整Dirichlet-to-Neumann映射可全局确定洛伦兹度量，适用于空间维数≥3、波速可分的情况，还构造例子验证唯一性条件的尖锐性。
AI 中文摘要
我们证明，有限时间区间上的完整Dirichlet-to-Neumann映射可全局确定由波算子主符号编码的洛伦兹度量\\(\mathbf G_{g,c}=-c(x,t)^2\mathrm{d}t^2+g(x)\\)。该结果适用于空间维数\\(n\geq3\\)、波速可分形式\\(c(x,t)=a(x)b(t)\\)的情况，前提是实验期间的累积有效时间超过从边界到内部再返回的最大传播时间。数据可确定观测柱体内未知的黎曼度量\\(g(x)\\)和波速\\(c(x,t)\\)，其不确定性源于固定边界且保持测量时间不变的空间坐标变换。特别地，时间因子\\(b\\)无需预先设定，可从相同测量中恢复。我们还构造了多个例子，证明唯一性的维数和可见性条件是尖锐的。
英文摘要
We prove that the full Dirichlet-to-Neumann map on a finite time interval globally determines the Lorentzian metric \(\mathbf G_{g,c}=-c(x,t)^2\mathrm{d}t^2+g(x)\) encoded by the principal symbol of the wave operator. The result holds in spatial dimensions \(n\geq3\) for multiplicatively separable wave speeds \(c(x,t)=a(x)b(t)\), provided the accumulated effective time during the experiment exceeds the maximal travel time from the boundary to the interior and back. The data determine both the unknown Riemannian metric \(g(x)\) and the wave speed \(c(x,t)\) throughout the observation cylinder, up to a spatial change of coordinates that fixes the boundary and leaves the measured time unchanged. In particular, the temporal factor \(b\) is not prescribed but is recovered from the same measurements. We also construct several examples demonstrating that the dimensional and visibility conditions for uniqueness are sharp.
Comments48 pages
