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

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.