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

We study the inverse problem of determining the vector and scalar potentials $\mathcal{A}(t,x)=(A_0,A_1,\cdots ,A_n)$ 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 $Q=(0,T)\times \mathrm{\Omega }$, where $\mathrm{\Omega }$ is a $C^2$ bounded domain in ${\mathbb{R}}^n$ for $n\ge 3$ and $T>diam(\mathrm{\Omega })$ from partial data on the boundary $\mathrm{\partial }Q$. We prove the unique determination of these potentials modulo a natural gauge invariance for the vector field term.