Stable determination of coefficients in the dynamical anisotropic Schr{\"o}dinger equation from the Dirichlet-to-Neumann map

TL;DR

This paper proves that the electric potential and conformal factor in a Schrödinger equation on a Riemannian manifold can be uniquely determined and stably estimated from boundary measurements, extending inverse problem results to higher dimensions.

Contribution

It establishes uniqueness and Hölder stability estimates for inverse boundary problems for the Schrödinger equation on Riemannian manifolds in dimensions greater than two.

Findings

Unique determination of electric potential from Dirichlet-to-Neumann map.

Hölder stability estimates for potential reconstruction.

Results also apply to conformal factors near 1.

Abstract

In this paper we are interested in establishing stability estimates in the inverse problem of determining on a compact Riemannian manifold the electric potential or the conformal factor in a Schr\"odinger equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the Schr\"odinger equation. We prove in dimension n bigger than 2 that the knowledge of the Dirichlet-to-Neumann map for the Schr\"odinger equation uniquely determines the electric potential and we establish H\"older-type stability estimates in determining the potential. We prove similar results for the determination of a conformal factor close to 1.