# Reconstruction of the magnetic field for a Schr\"odinger operator in a cylindrical setting

## TL;DR

This paper develops a constructive method to recover magnetic fields in a cylindrical domain from boundary data for the magnetic Schrödinger operator, utilizing new global Carleman estimates and pseudodifferential conjugation techniques.

## Contribution

It introduces the first global Carleman estimates and reconstruction algorithms for magnetic Schrödinger operators in cylindrical geometries, advancing inverse boundary value problem solutions.

## Key findings

- Established a global Carleman estimate for the magnetic Schrödinger operator in a cylindrical setting.
- Developed a pseudodifferential conjugation method to relate the magnetic operator to the Laplacian.
- Provided a constructive boundary measurement-based reconstruction procedure for magnetic fields.

## Abstract

In this thesis we consider a magnetic Schr\"odinger inverse problem over a compact domain contained in an infinite cylindrical manifold. We show that, under certain conditions on the electromagnetic potentials, we can recover the magnetic field from boundary measurements in a constructive way. A fundamental tool for this procedure is a global Carleman estimate for the magnetic Schr\"odinger operator. We prove this by conjugating the magnetic operator essentially into the Laplacian, and using the Carleman estimates for it proven by Kenig-Salo-Uhlmann in the anisotropic setting, see [KSU11a]. The conjugation is achieved through pseudodifferential operators over the cylinder, for which we develop the necessary results. The main motivations to attempt this question are the following results concerning the magnetic Schr\"odinger operator: first, the solution to the uniqueness problem in the cylindrical setting in [DSFKSU09], and, second, the reconstruction algorithm in the Euclidean setting from [Sal06]. We will also borrow ideas from the reconstruction of the electric potential in the cylindrical setting from [KSU11b]. These two new results answer partially the Carleman estimate problem (Question 4.3.) proposed in [Sal13] and the reconstruction for the magnetic Schr\"odinger operator mentioned in the introduction of [KSU11b]. To our knowledge, these are the first global Carleman estimates and reconstruction procedure for the magnetic Schr\"odinger operator available in the cylindrical setting.