We show that a continuous potential q can be constructively determined from the knowledge of the Dirichlet–to–Neumann map for the Schrödinger operator $-{\mathrm{\Delta }}_g+q$ 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.

我们表明，可以从薛定谔算子的狄利克雷到诺依曼映射的知识建设性地确定连续势q$-{\mathrm{\Delta }}_G+q$在维度≥3 的共形横向各向异性流形上，假设横向流形上的测地线变换是建设性可逆的。这是[12]的唯一性结果的建设性对应物。在我们的重建过程中，一个至关重要的作用是通过基于沿横向流形上的非切线测地线集中的高斯光束准模对​​合适的复杂几何光学解的边界轨迹进行建设性确定而发挥作用，这些准模具有独特性。这是通过将 [33] 方法的简化版本应用于我们的设置来实现的。我们还使用流形边界上的标准 Sobolev 空间识别 [33] 中引入的主要空间。