论文标题
Calderón问题中的重建在同型横向各向异性歧管上
Reconstruction in the Calderón problem on conformally transversally anisotropic manifolds
论文作者
论文摘要
我们表明，可以从Schrödinger操作员的Dirichlet到Neumann地图的知识中建设性地确定连续的潜在$ Q $，前提是，在横向上的横向跨性别歧管上，只要横向射线在横向歧管上转变为横向歧管上的横向变化。这是dos santos ferreira-kurylev-lassas-salo的独特性结果的建设性对应物。通过建设性确定基于高斯梁的合适的复杂几何光学溶液的边界痕迹，沿横向横向上的非区域地理学浓缩的合适的复杂几何光学溶液的边界痕迹，在我们的重建过程中起着关键作用。这是通过将Nachman-Street方法的简化版本应用于我们的设置来实现的。我们还标识了Nachman-Street引入的主要空间，并在歧管边界上具有标准的Sobolev空间。我们结果证明的另一个成分是从Dirichlet到Neumann映射的知识的连续电势边界轨迹的重建公式。
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 $\geq 3$, provided that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of Dos Santos Ferreira-Kurylev-Lassas-Salo. 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 Nachman-Street to our setting. We also identify the main space introduced by Nachman-Street 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.
