An Inverse Boundary Value Problem for the Magnetic Schr\"odinger Operator on a Half Space

TL;DR

This thesis investigates an inverse boundary value problem for the magnetic Schrödinger operator in a half space, demonstrating that certain potentials are uniquely recoverable from boundary measurements.

Contribution

It establishes uniqueness results for recovering magnetic and electric potentials from boundary data in a half space setting.

Findings

Unique determination of the electric potential q from boundary measurements.

Unique determination of the curl of the magnetic potential A.

Existence and uniqueness of the direct boundary value problem.

Abstract

This licentiate thesis is concerned with an inverse boundary value problem for the magnetic Schr\"odinger equation in a half space, for compactly supported potentials $A\in W^{1,\infty}(\bar{\mathbb{R}^3_{-}},\R^3)$ and $q \in L^{\infty}(\bar{\mathbb{R}^3_{-}},\C)$. We prove that $q$ and the curl of $A$ are uniquely determined by the knowledge of the Dirichlet-to-Neumann map on parts of the boundary of the half space. The existence and uniqueness of the corresponding direct problem are also considered.