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 $\mathcal{A}=\left(A_0\right(t,x),A_1(t,x),\dots ,A_n(t,x\left)\right)$ and $q(t,x)$, respectively, in the relativistic Schrödinger equation $\left({\left({\mathrm{\partial }}_t+A_0(t,x)\right)}^2-\sum _{j=1}^n{\left({\mathrm{\partial }}_j+A_j(t,x)\right)}^2+q(t,x)\right)u(t,x)=0$ in the region $Q=(0,T)\times \mathrm{\Omega }$, where Ω is a $C^2$ bounded domain in ${\mathbb{R}}^n$ for $n\ge 3$ and $T>\text{diam}\left(\mathrm{\Omega }\right)$ 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.

Keywords:

• Inverse problems
• relativistic Schrödinger equation
• Carleman estimates
• partial boundary data

2010 Mathematics Subject Classifications:

• 35L05
• 35L20
• 35R30

Acknowledgments

The authors thank Plamen Stefanov and Siamak RabieniaHaratbar for very useful discussions regarding the light ray transform. MV thanks Gen Nakamura and Guang-Hui Hu for discussions on this problem.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 We thank Plamen Stefanov for drawing our attention to the results of this paper.

Additional information

Funding

VK was supported in part by National Science Foundation (NSF) grant DMS 1616564. Both authors benefited from the support of Airbus Corporate Foundation Chair grant titled ‘Mathematics of Complex Systems’ established at TIFR CAM and TIFR ICTS, Bangalore, India.