Unique determination of a Kato class potential from boundary data

TL;DR

This work proves that a Kato class potential $V$ in $U\subset\mathbb{R}^3$ is uniquely determined by the Dirichlet-to-Neumann map for the Schrödinger equation $-\Delta u+Vu=0$. The authors develop Complex Geometrical Optics (CGO) solutions for $V\in\mathcal{K}_3$, establishing existence of $u_z=e^{iz\cdot x}(1+r_z)$ with $|z|\to\infty$ and $\|r_z\|_{L^2(U)}\to0$, using a right inverse $G_z$ to the conjugated Laplacian and Stein interpolation to obtain key operator bounds. They formulate the Dirichlet-to-Neumann map via closed quadratic forms, enabling a density-type identity that relates boundary data to the interior potential. The main result follows by inserting CGO solutions into the bilinear identity derived from equal DN maps, which yields the vanishing of the Fourier transform of $V_1-V_2$, hence $V_1=V_2$. This extends Calderón-type uniqueness to Kato class potentials beyond the $L^{n/2}$ framework and enhances understanding of inverse boundary value problems for nonsmooth, singular potentials.

Abstract

We prove that a Kato class potential $V$ defined on an open, bounded set in $\mathbb{R}^3$ with Lipschitz boundary is uniquely determined by the Dirichlet-to-Neumann operator associated to the equation $-Δu + Vu = 0\,.$

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Example 2.4
• Lemma 3.1
• Lemma 3.2
• proof