Determining a magnetic Schrödinger equation by a single far-field measurement

Chaohua Duan; Zhen Xue

doi:10.3934/ipi.2026018

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2026, Volume 25: 113-132. Doi: 10.3934/ipi.2026018

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Determining a magnetic Schrödinger equation by a single far-field measurement

Chaohua Duan^1, , and
Zhen Xue^{1, 2,}

1.
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China
2.
School of Mathematics, North University of China, Taiyuan, China

^*Corresponding author: Chaohua Duan
^*Corresponding author: Chaohua Duan

Received: July 28, 2025

Revised: December 26, 2025

Early access: February 13, 2026

Published online: February 13, 2026

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Abstract

We study the inverse scattering problem for the magnetic Schrödinger equation. Assuming the well-posedness of the direct problem under natural conditions on the potentials (established in the Appendix), we prove that a single far-field measurement can uniquely determine the polyhedral support $ \Omega $ of the potentials when the scatterer has a polyhedral structure.

A key theoretical ingredient is the proof that transmission eigenfunctions must vanish at corners in two dimensions and edges in three dimensions, provided the angle is not $ \pi $. This extends previous results for the non-magnetic case. The proof combines complex geometric optics solutions with careful asymptotic analysis near singular points.

Our work thus demonstrates that unique shape reconstruction is possible with minimal data for polyhedral scatterers, advancing the inverse problem theory for magnetic Schrödinger equations. The results have potential applications in material characterization and nondestructive testing involving magnetic fields.

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Mathematics Subject Classification: Primary: 58J05, 35P25; Secondary: 35Q60, 78A05.

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Figure 1. Illustration of the geometry in 2D

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Figure 2. Illustration of the geometry in 3D

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Figure 3. Illustration of $ V_{2h} $

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Figure 4. Schematic illustration

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