  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.
