Inverse problems with partial data for a magnetic Schr\"odinger operator in an infinite slab and on a bounded domain

TL;DR

This paper proves unique determination of magnetic fields and electric potentials in inverse boundary value problems for magnetic Schrödinger operators with partial data, extending previous results to infinite slabs and bounded domains.

Contribution

It generalizes existing uniqueness results for magnetic Schrödinger inverse problems to cases with partial boundary data on infinite slabs and bounded domains.

Findings

Unique determination of magnetic field and electric potential in infinite slabs with partial data.

Extension of uniqueness results to bounded domains with data on two boundary subsets.

Results applicable when inaccessible boundary parts are hyperplanes.

Abstract

In this paper we study inverse boundary value problems with partial data for the magnetic Schr\"odinger operator. In the case of an infinite slab in $R^n$, $n\ge 3$, 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 [41], obtained for the Schr\"odinger operator without magnetic potentials. In the case of a bounded domain in $R^n$, $n\ge 3$, extending the results of [2], 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 [31], we also obtain uniqueness results for the magnetic Schr\"odinger 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.