Reconstruction in the Calderón problem on conformally transversally anisotropic manifolds
Reconstruction in the Calderón problem on conformally transversally anisotropic manifolds
复制标题
共形横向各向异性流形上 Calderón 问题的重构
DOI：
10.1016/j.jfa.2021.109191
复制
发表时间：
2021
影响因子：
1.7
通讯作者：
Uhlmann, Gunther
中科院分区：
文献类型：
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作者：
Feizmohammadi, Ali;Krupchyk, Katya;Oksanen, Lauri;Uhlmann, Gunther
我们证明了在维数≥ 3的共形横观各向异性流形上，薛定谔算子− Δ g+ q的Dirichlet-to-Neumann映射可以构造性地确定连续势q，只要横观流形上的测地线射线变换是构造可逆的.这是[12]的唯一性结果的构造性对应。在我们的重建过程中起着至关重要的作用是建设性地确定合适的复杂的几何光学解决方案的基础上高斯光束准模集中沿着非切向测地线的横向流形，享有独特的属性的边界轨迹。这是通过将[33]的方法的简化版本应用于我们的设置来实现的。我们还将[33]中引入的主空间与流形边界上的标准Sobolev空间等同起来。在证明我们的结果的另一个成分是一个重建公式的边界跟踪连续潜在的知识的狄利克雷诺依曼映射。
We show that a continuous potential q can be constructively determined from the knowledge of the Dirichlet–to–Neumann map for the Schrödinger operator− Δ g+ q on a conformally transversally anisotropic manifold of dimension≥ 3, provided that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [12]. A crucial role in our reconstruction procedure is played by a constructive determination of the boundary traces of suitable complex geometric optics solutions based on Gaussian beams quasimodes concentrated along non-tangential geodesics on the transversal manifold, which enjoy uniqueness properties. This is achieved by applying the simplified version of the approach of [33] to our setting. We also identify the main space introduced in [33] with a standard Sobolev space on the boundary of the manifold. Another ingredient in the proof of our result is a reconstruction formula for the boundary trace of a continuous potential from the knowledge of the Dirichlet–to–Neumann map.