发表机构
University of Jyväskylä（于韦斯屈莱大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探讨了从波观测确定流形的反问题，证明了在次临界谱下界条件下唯一性成立，并构造了临界端点处唯一性失效的反例。
AI 中文摘要
我们研究了从波观测确定一个维数至少为二的连通闭黎曼流形的反问题。源被支撑在开集 $S$ 中，产生的波在开接收集 $R$ 上被观测，$R$ 可能与 $S$ 有正距离。我们在满足次临界谱下界：$\forall$ 特征函数 $\phi$，$\\\\\\\\|\u03d5|_R\\\\|_{L^2(R)} \ge ce^{-C\lambda^\theta}\\\\\\\\|\u03d5\\\\|_{L^2(M)}$（其中 $\lambda$ 是对应特征值，$\theta\in[0,1/2)$，常数 $c,C>0$ 不依赖于 $\phi$）的流形类内证明了等距唯一性。若该类型的界在 $S$ 上也成立，则唯一性在所有连通闭流形类内成立。由谱不等式，该界在 $\theta=1/2$ 时总是成立。然而，唯一性在此端点失效：在每一维 $n\ge 2$ 中，我们构造了一对非等距的连通闭黎曼 $n$-流形，它们产生相同的波观测。
英文摘要
We study the inverse problem of determining a connected, closed Riemannian manifold of dimension at least two from wave observations. The sources are supported in an open set $S$, and the resulting waves are observed on an open receiver set $R$, which may be at a positive distance from $S$. We prove uniqueness up to isometry within the class of manifolds that satisfy a subcritical spectral lower bound: $\|ϕ|_R\|_{L^2(R)} \ge ce^{-Cλ^θ}\|ϕ\|_{L^2(M)}$ for every eigenfunction $ϕ$, where $λ$ is the corresponding eigenvalue, $θ\in[0,1/2)$, and the constants $c,C>0$ do not depend on $ϕ$. If a bound of this type holds also on $S$, then uniqueness holds within the class of all connected, closed manifolds. By the spectral inequality, the bound always holds with $θ=1/2$. Uniqueness nevertheless fails at this endpoint: in every dimension $n\ge 2$, we construct a pair of nonisometric connected, closed Riemannian $n$-manifolds that produce identical wave observations.
Comments62 pages, 3 figures
