We show that the knowledge of the set of the Cauchy data on the boundary of a bounded open set in R n , n ≥ 3, for the magnetic Schrödinger operator with L ∞ magnetic and electric potentials determines the magnetic field and electric potential inside the set uniquely. The proof is based on a Carleman estimate for the magnetic Schrödinger operator with a gain of two derivatives.

In this paper we study inverse boundary value problems with partial data for the magnetic Schrödinger operator. In the case of an infinite slab in R n , n ≥ 3, we establish that the magnetic field and the electric potential can be determined uniquely, when the Dirichlet and Neumann data are given either on the different boundary hyperplanes of the slab or on the same hyperplane. This is a generalization of the results of Li and Uhlmann (Inverse Probl Imaging 4(3):449-462, 2010), obtained for the Schrödinger operator without magnetic potentials.In the case of a bounded domain in R n , n ≥ 3, extending the results of Ammari and Uhlmann (Indiana Univ Math J 53(1):169-183, 2004), we show the unique determination of the magnetic field and electric potential from the Dirichlet and Neumann data, given on two arbitrary open subsets of the boundary, provided that the magnetic and electric potentials are known in a neighborhood of the boundary. Generalizing the results of Isakov (Inverse Probl Imaging 1(1): [95][96][97][98][99][100][101][102][103][104][105] 2007), we also obtain uniqueness results for the magnetic Schrödinger operator, when the Dirichlet and Neumann data are known on the same part of the boundary, assuming that the inaccessible part of the boundary is a part of a hyperplane.

Abstract. We show that a first order perturbation A(x) · D + q(x) of the polyharmonic operator (−∆) m , m ≥ 2, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in R n , n ≥ 3. Notice that the corresponding result does not hold in general when m = 1.