Inverse Problems With Partial Data for a Magnetic Schrödinger Operator in an Infinite Slab and on a Bounded Domain

Abstract

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.

Determination of non-compactly supported electromagnetic potentials in an unbounded closed waveguide

“…First, Theorem 1.1 is stated in a general unbounded domain subject only to condition (1.1). This makes an important difference with other related results which, to our best knowledge, have all been stated in specific unbounded domains like a slab, the half space or a cylindrical domain (see [33,37,14,15]). In particular, Theorem 1.1 holds true with domains having different types of geometrical deformations like bends or twisting, which are frequently used in problems of transmission for improving the propagation.…”

On the Calderón problem in periodic cylindrical domain with partial Dirichlet and Neumann data

“…Unique determination of a compactly supported electric potential of the Laplace equation in an infinite slab, by partial DN map, is established in . This result was extended to the magnetic case in and to bi‐harmonic operators in .…”

Inverse problems: seeing the unseen

“…Since γ 0 is tangent to ∂ M at x 0 , we have τ x n (x 0 ) = 0 by (117). The third term in the l.h.s.…”

“…The case of partial data on a slab was studied in [122]. The DN map with partial data for the magnetic Schrödinger operator was studied in [51,109,117,198]. The case of the polyharmonic operator was considered in [118].…”