Inverse Boundary Problems for Systems in Two Dimensions

TL;DR

This paper proves that boundary measurements uniquely determine coefficients and gauge transformations for elliptic systems on two-dimensional surfaces and domains, extending inverse boundary problem results to geometric and complex settings.

Contribution

It establishes the unique identification of connection and potential up to gauge from boundary data for elliptic systems on Riemann surfaces and complex domains.

Findings

Cauchy data determines connection and potential up to gauge.

Results apply to Dirac-type and Schrödinger operators on surfaces.

Recovery of zeroth order terms in complex domains.

Abstract

We prove identification of coefficients up to gauge by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of $\mathbb{C}$. In the geometric setting, we fix a Riemann surface with boundary, and consider both a Dirac-type operator plus potential acting on sections of a Clifford bundle and a connection Laplacian plus potential (i.e. Schr\"odinger Laplacian with external Yang-Mills field) acting on sections of a Hermitian bundle. In either case we show that the Cauchy data determines both the connection and the potential up to a natural gauge transformation: conjugation by an endomorphism of the bundle which is the identity at the boundary. For domains of $\mathbb{C}$, we recover zeroth order terms up to gauge from Cauchy data at the boundary in first order elliptic systems.