An inverse problem for the relativistic Schrödinger equation with partial boundary data

摘要

We study the inverse problem of determining the time-dependent vector andscalar potentials$mathcal{A}=left(A_{0}(t,x),A_{1}(t,x),....,A_{n}(t,x)ight)$ and $q(t,x)$,respectively, in the wave equation egin{align*} egin{aligned}left[left(partial_{t}+A_{0}(t,x)ight)^{2}-sum_{j=1}^{n}left(partial_{j}+A_{j}(t,x)ight)^{2}+q(t,x)ight]u(t,x)=0,ext{in} Q end{aligned} end{align*} where $Q=(0,T)imesOmega$, with $T>0$and $Omega$ is a $C^{2}$ bounded domain in $mathbb{R}^{n}$, from the partialknowledge of solutions $u(t,x)$ on $partial Q$. We prove the uniquedetermination of these potentials modulo a natural gauge-invariance for thevector term.