Boundary determination for the Schrödinger equation with unknown embedded obstacles by local data

TL;DR

This paper addresses the inverse boundary value problem for the Schrödinger operator $\Delta+q$ in a bounded domain with an unknown embedded obstacle. It introduces a local data approach using only the local Cauchy data of the fundamental solution and a Green's representation to prove boundary determination of $q$ to infinite order on $\partial\Omega$ and, for analytic $q$, global uniqueness for $(q,D,\mathcal{B})$. The method hinges on singularity analysis of singular solutions and explicit expansions of the volume potential of the fundamental solution, showing that boundary information suffices despite the unknown obstacle. The results advance partial-data inverse problems by reducing reliance on obstacle knowledge and hint at extensions to Maxwell equations.

Abstract

In this paper, we consider the inverse boundary value problem of the elliptic operator $Δ+q$ in a fixed region $Ω\subset\mathbb{R}^3$ with unknown embedded obstacles $D$. In particular, we give a new and simple proof to uniquely determine $q$ and all of its derivatives at the boundary from the knowledge of the local Dirichlet-to-Neumann map on $\partialΩ$, disregarding the unknown obstacle, where in fact only the local Cauchy data of the fundamental solution is used. Our proof mainly depends on the rigorous singularity analysis on certain singular solutions and the volume potentials of fundamental solution, which is easy to extend to many other cases.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof