# Inverse problems for magnetic Schr\"odinger operators in transversally anisotropic geometries

## TL;DR

This paper proves the unique determination of magnetic fields and electric potentials in magnetic Schrödinger operators on certain Riemannian manifolds from boundary measurements, advancing inverse boundary problem theory in geometric settings.

## Contribution

It extends inverse boundary problem results to more general transversally anisotropic geometries, including non-simple manifolds, under injectivity assumptions of the geodesic ray transform.

## Key findings

- Unique determination of magnetic and electric potentials in admissible geometries.
- Extension of uniqueness results to non-simple conformally transversally anisotropic manifolds.
- Magnetic potential determined up to gauge when electric potential is known.

## Abstract

We study inverse boundary problems for magnetic Schr\"odinger operators on a compact Riemannian manifold with boundary of dimension $\ge 3$. In the first part of the paper we are concerned with the case of admissible geometries, i.e. compact Riemannian manifolds with boundary which 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 Schr\"odinger operator with $L^\infty$ 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 which 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 Schr\"odinger 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.