An inverse problem for the relativistic Schrödinger 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 Schrödinger 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 $n\ge 3$ and $T>\mbox{diam}(Ω)$ 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.

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References in corpus (2)

• The Inverse Problem for the Dirichlet-to-Neumann map on Lorentzian manifolds
• Unique determination of a time-dependent potential for wave equations from partial data

Cited by in corpus (4)

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• Stability estimates for relativistic Schrödinger equation from partial boundary data
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• Stability estimate for a partial data inverse problem for the convection-diffusion equation