Recovery of a time-dependent Hermitian connection and potential appearing in the dynamic Schr\"odinger equation

TL;DR

This paper proves that the Dirichlet-to-Neumann map uniquely determines the time- and space-dependent connection and potential in a vector-valued Schr"odinger equation on certain Riemannian manifolds.

Contribution

It establishes unique recovery of both connection form and potential in a dynamic Schr"odinger equation from boundary measurements on specific classes of manifolds.

Findings

Unique determination of connection and potential from D-to-N map

Results apply to 2D simple manifolds and higher-dimensional manifolds with convex boundary

Advances inverse problems for time-dependent Schr"odinger equations

Abstract

We consider, on a trivial vector bundle over a Riemannian manifold with boundary, the inverse problem of uniquely recovering time- and space-dependent coefficients of the dynamic, vector-valued Schr\"odinger equation from the knowledge of the Dirichlet-to-Neumann map. We show that the D-to-N map uniquely determines both the connection form and the potential appearing in the Schr\"odinger equation, under the assumption that the manifold is either a) two-dimensional and simple, or b) of higher dimension with strictly convex boundary and admits a smooth, strictly convex function.