Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map

Mourad Bellassoued; Zouhour Rezig

doi:10.3934/dcdss.2021158

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2022, Volume 15, Issue 5: 1061-1084. Doi: 10.3934/dcdss.2021158

This issue Previous Article Identifying the heat sink Next Article Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and

$ 1-d $

heat diffusion equation

Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map

Mourad Bellassoued^1, , and
Zouhour Rezig^2,

1.
University of Tunis El Manar, National Engineering School of Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia
2.
University of Tunis El Manar, Faculty of Sciences of Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia

^*Corresponding author: Mourad Bellassoued
^*Corresponding author: Mourad Bellassoued

Received: September 2021

Early access: December 2021

Published: May 2022

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Abstract

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 $ n\geq 2 $ 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.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 35J10.

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