An inverse problem for the magnetic Schr\"odinger equation in infinite cylindrical domains

TL;DR

This paper investigates the inverse problem of uniquely determining magnetic and electric potentials in an infinite cylindrical domain using boundary measurements, with stability results and extensions to limited boundary data.

Contribution

It establishes unique and Hölder-stable recovery of magnetic and electric potentials from Dirichlet-to-Neumann maps in infinite cylindrical domains, including cases with limited boundary observations.

Findings

Unique determination of magnetic and electric potentials

Hölder stability estimates for the inverse problem

Extension to finite boundary observations

Abstract

We study the inverse problem of determining the magnetic field and the electric potential entering the Schr\"odinger equation in an infinite 3D cylindrical domain, by Dirichlet-to-Neumann map. The cylindrical domain we consider is a closed waveguide in the sense that the cross section is a bounded domain of the plane. We prove that the knowledge of the Dirichlet-to-Neumann map determines uniquely, and even H\"older-stably, the magnetic field induced by the magnetic potential and the electric potential. Moreover, if the maximal strength of both the magnetic field and the electric potential, is attained in a fixed bounded subset of the domain, we extend the above results by taking finitely extended boundary observations of the solution, only.