An inverse problem for the relativistic Schr\"odinger equation with partial boundary data

TL;DR

This paper addresses the inverse problem of uniquely determining vector and scalar potentials in a relativistic Schrödinger equation from partial boundary data, accounting for gauge invariance, in a bounded domain.

Contribution

It establishes the unique recovery of potentials in the relativistic Schrödinger equation using partial boundary measurements, considering gauge invariance.

Findings

Unique determination of potentials from partial boundary data

Results valid for dimensions n ≥ 3

Addresses gauge invariance in the inverse problem

Abstract

We study the inverse problem of determining the vector and scalar potentials $\mathcal{A}(t,x)=\left(A_{0},A_{1},\cdots,A_{n}\right)$ and $q(t,x)$, respectively, in the relativistic Schr\"odinger equation \begin{equation*} \Big{(}\left(\partial_{t}+A_{0}(t,x)\right)^{2}-\sum_{j=1}^{n}\left(\partial_{j}+A_{j}(t,x)\right)^{2}+q(t,x)\Big{)}u(t,x)=0 \end{equation*} in the region $Q=(0,T)\times\Omega$, where $\Omega$ is a $C^{2}$ bounded domain in $\mathbb{R}^{n}$ for $n\geq 3$ and $T>\mbox{diam}(\Omega)$ from partial data on the boundary $\partial Q$. We prove the unique determination of these potentials modulo a natural gauge invariance for the vector field term.