Inverse Problems for Magnetic Schrödinger Operators in Transversally Anisotropic Geometries

We study inverse boundary problems for magnetic Schrödinger operators on a compact Riemannian manifold with boundary of dimension ≥  3. In the first part of the paper, we are concerned with the case of admissible geometries, i.e. compact Riemannian manifolds with boundary that are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Cauchy data on the boundary of the manifold for the magnetic Schrödinger operator with L∞ magnetic and electric potentials, determines the magnetic field and electric potential uniquely. In the second part of the paper, we address the case of more general conformally transversally anisotropic geometries, i.e. compact Riemannian manifolds with boundary that are conformally embedded in a product of the Euclidean line and a compact manifold, which need not be simple. Here, under the assumption that the geodesic ray transform on the transversal manifold is injective, we prove that the knowledge of the Cauchy data on the boundary of the manifold for a magnetic Schrödinger operator with continuous potentials, determines the magnetic field uniquely. Assuming that the electric potential is known, we show that the Cauchy data determines the magnetic potential up to a gauge equivalence.