Open AccessRecovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map
Author(s) -
Mourad Bellassoued,
Zouhour Rezig
Publication year - 2022
Publication title -
discrete and continuous dynamical systems. series s
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.481
H-Index - 34
eISSN - 1937-1632
pISSN - 1937-1179
DOI - 10.3934/dcdss.2021158
Subject(s) - mathematics , dirichlet distribution , metric (unit) , riemannian manifold , boundary (topology) , dimension (graph theory) , neumann boundary condition , manifold (fluid mechanics) , mathematical analysis , transversal (combinatorics) , simple (philosophy) , schrödinger's cat , boundary value problem , pure mathematics , mechanical engineering , operations management , engineering , economics , philosophy , epistemology
In this paper, we deal with the inverse problem of determining simple metrics on a compact Riemannian manifold from boundary measurements. We take this information in the dynamical Dirichlet-to-Neumann map associated to the Schrödinger equation. We prove in dimension \begin{document}$ n\geq 2 $\end{document} that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the simple metric (up to an admissible set). We also prove a Hölder-type stability estimate by the construction of geometrical optics solutions of the Schrödinger equation and the direct use of the invertibility of the geodesical X-ray transform.