An Inverse Problem for the Magnetic Schrödinger Equation in Infinite Cylindrical Domains

Abstract

We study the inverse problem of determining the magnetic field and the electric potential entering the Schrödinger equation in an infinite 3D cylindrical domain, by the Dirichletto–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 knowledge of the Dirichlet-to-Neumann map determines uniquely and even Hölder-stably the magnetic field 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 only finitely extended boundary observations of the solution.