Local uniqueness for the Dirichlet-to-Neumann map via the two-plane transform

TL;DR

This paper demonstrates that the integral of a potential over a two-plane can be uniquely determined from the Dirichlet-to-Neumann map using special solutions, advancing inverse boundary value problem theory.

Contribution

It introduces a novel method to recover two-plane integrals of potentials from boundary data in higher dimensions, extending previous uniqueness results.

Findings

Potential integrals over two-planes are uniquely determined by boundary measurements.

The method applies to Schrödinger equations in Lipschitz domains in three or more dimensions.

It uses exponentially growing solutions to connect boundary data with interior potential integrals.

Abstract

We consider the Dirichlet-to-Neumann map associated to the Schr\"odinger equation with a potential in a bounded Lipschitz domain in three or more dimensions. We show that the integral of the potential over a two-plane is determined by the Cauchy data of certain exponentially growing solutions on any neighborhood of the intersection of the two-plane with the boundary.