A Partial Data Result for the Magnetic Schrodinger Inverse Problem

TL;DR

This paper demonstrates that partial boundary data can determine a magnetic Schrödinger operator, improving previous results by reducing the required measurement and input subsets through advanced Carleman estimate techniques.

Contribution

It introduces a novel method using conjugated pseudodifferential operators to enhance partial data results in inverse boundary value problems.

Findings

Determines magnetic Schrödinger operators from partial boundary data.

Reduces the size of boundary subsets needed for measurement and input.

Improves upon previous partial data inverse problem results.

Abstract

This article shows that knowledge of the Dirichlet-Neumann map on certain subsets of the boundary for input functions supported roughly on the rest of the boundary can be used to determine a magnetic Schr\"{o}dinger operator. With some geometric conditions on the domain, either the subset on which the DN map is measured or the subset on which the input functions have support may be made arbitrarily small. This is an improvement on the partial data result in a paper by Dos Santos Ferreira, Kenig, Sj\"{o}strand, and Uhlmann. The method involves modifying the Carleman estimate in that paper by conjugation with operators built from pseudodifferential pieces.