具有量子混沌界的流形上的不相交数据反问题
Disjoint data inverse problem on manifolds with quantum chaos bounds
摘要：
我们考虑了由流形上的波动方程的源-解算子$Λ_{\mathcal{S,R}}$的一个限制确定光滑紧致黎曼流形$(M,g)$的反问题。这里，$\mathcal{S}$和$\mathcal{R}$是$M$上的开集，而$Λ_{\mathcal{S,R}}$表示由$\mathcal{S}$上支持并在$\mathcal{R}$上观察到的平滑源产生的波的测量值。我们强调，$\overline{\mathcal{S}}$和$\overline{\mathcal{R}}$可能不相交。我们证明了$Λ_{\mathcal{S,R}}$确定的歧管$(M,g)$唯一的集合$\mathcal{S}$的谱界条件下：存在一个常数$C>0$，使任何规范化的本征函数$ϕ_k$的拉普拉斯-贝尔特拉米算子$(M,g)$满足\begin{equation*} 1\leq C\|ϕ_k\|_{L^2(\mathcal{S})}. \end{equation*}我们注意到，对于Anosov表面，这个谱界条件是满足任何非空的开子集$\mathcal{S}$。我们的方法是基于文件[18]和谱约束条件是一个类似的Hassell-Tao条件。
英文摘要：
We consider the inverse problem to determine a smooth compact Riemannian manifold $(M,g)$ from a restriction of the source-to-solution operator, $Λ_{\mathcal{S,R}}$, for the wave equation on the manifold. Here, $\mathcal{S}$ and $\mathcal{R}$ are open sets on $M$, and $Λ_{\mathcal{S,R}}$ represents the measurements of waves produced by smooth sources supported on $\mathcal{S}$ and observed on $\mathcal{R}$. We emphasise that $\overline{\mathcal{S}}$ and $\overline{\mathcal{R}}$ could be disjoint. We demonstrate that $Λ_{\mathcal{S,R}}$ determines the manifold $(M,g)$ uniquely under the following spectral bound condition for the set $\mathcal{S}$: There exists a constant $C>0$ such that any normalized eigenfunction $ϕ_k$ of the Laplace-Beltrami operator on $(M,g)$ satisfies \begin{equation*} 1\leq C\|ϕ_k\|_{L^2(\mathcal{S})}. \end{equation*} We note that, for the Anosov surface, this spectral bound condition is fulfilled for any non-empty open subset $\mathcal{S}$. Our approach is based on the paper [18] and the spectral bound condition above is an analogue of the Hassell-Tao condition there.
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