Global uniqueness and reconstruction for the multi-channel Gel'fand-Calder\'on inverse problem in two dimensions

TL;DR

This paper presents a global reconstruction method and proves uniqueness for the multi-channel Gel'fand-Calderón inverse problem in two dimensions, enabling the recovery of matrix-valued potentials from boundary measurements.

Contribution

It introduces an exact global reconstruction technique and establishes uniqueness results for the inverse boundary value problem involving matrix potentials in 2D.

Findings

Successful global reconstruction of matrix-valued potentials from boundary data.

Proof of uniqueness: identical boundary data implies identical potentials.

Applicable to smooth potentials on bounded planar domains.

Abstract

We study the multi-channel Gel'fand-Calder\'on inverse problem in two dimensions, i.e. the inverse boundary value problem for the equation $-\Delta \psi + v(x) \psi = 0$, $x\in D$, where $v$ is a smooth matrix-valued potential defined on a bounded planar domain $D$. We give an exact global reconstruction method for finding $v$ from the associated Dirichlet-to-Neumann operator. This also yields a global uniqueness results: if two smooth matrix-valued potentials defined on a bounded planar domain have the same Dirichlet-to-Neumann operator then they coincide.