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

Chaohua Duan, Zhen Xue

TL;DR

This work addresses recovering the shape of a polyhedral magnetic scatterer from a single far-field measurement for the magnetic Schrödinger operator $L_{oldsymbol{A}, q}$. It establishes forward problem well-posedness via a variational framework and a DtN formulation, then proves a novel corner-vanishing property for transmission eigenfunctions using CGO solutions. Leveraging these vanishing results, the authors derive uniqueness of shape recovery from one far-field pattern for polyhedral geometries in 2D and 3D, with corollaries for convex polygons and rectangular boxes; this advances shape reconstruction in quantum settings with magnetic effects. The findings have potential implications for quantum imaging, material characterization, and nondestructive testing where magnetic potentials are relevant, by showing that minimal data can determine object geometry under suitable assumptions. All mathematical notation is preserved in $...$ to clarify the involved spectral and geometric conditions.

Abstract

This paper investigates the inverse scattering problem for the magnetic Schrödinger equation. We first establish the well-posedness of the direct problem through a variational approach under physically meaningful assumptions on the magnetic and electric potentials. Our main results demonstrate that a single far-field measurement uniquely determines the support of the potential functions when the scatterer has polyhedral structures. A significant theoretical byproduct of our analysis reveals that transmission eigenfunctions must vanish at corners in two dimensions and edge corners in three dimensions, provided the angle is not $π$. This geometric property of eigenfunctions extends previous results for the non-magnetic case and provides new insights into the interaction between quantum effects and singular geometries. The proof combines complex geometric optics solutions with careful asymptotic analysis near singular points. From an inverse problems perspective, our work shows that minimal measurement data suffices for shape reconstruction in important practical cases, advancing the theoretical understanding of inverse scattering with magnetic potentials. The results have potential applications in quantum imaging, material characterization, and nondestructive testing where magnetic fields play a crucial role.

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

TL;DR

This work addresses recovering the shape of a polyhedral magnetic scatterer from a single far-field measurement for the magnetic Schrödinger operator . It establishes forward problem well-posedness via a variational framework and a DtN formulation, then proves a novel corner-vanishing property for transmission eigenfunctions using CGO solutions. Leveraging these vanishing results, the authors derive uniqueness of shape recovery from one far-field pattern for polyhedral geometries in 2D and 3D, with corollaries for convex polygons and rectangular boxes; this advances shape reconstruction in quantum settings with magnetic effects. The findings have potential implications for quantum imaging, material characterization, and nondestructive testing where magnetic potentials are relevant, by showing that minimal data can determine object geometry under suitable assumptions. All mathematical notation is preserved in to clarify the involved spectral and geometric conditions.

Abstract

This paper investigates the inverse scattering problem for the magnetic Schrödinger equation. We first establish the well-posedness of the direct problem through a variational approach under physically meaningful assumptions on the magnetic and electric potentials. Our main results demonstrate that a single far-field measurement uniquely determines the support of the potential functions when the scatterer has polyhedral structures. A significant theoretical byproduct of our analysis reveals that transmission eigenfunctions must vanish at corners in two dimensions and edge corners in three dimensions, provided the angle is not . This geometric property of eigenfunctions extends previous results for the non-magnetic case and provides new insights into the interaction between quantum effects and singular geometries. The proof combines complex geometric optics solutions with careful asymptotic analysis near singular points. From an inverse problems perspective, our work shows that minimal measurement data suffices for shape reconstruction in important practical cases, advancing the theoretical understanding of inverse scattering with magnetic potentials. The results have potential applications in quantum imaging, material characterization, and nondestructive testing where magnetic fields play a crucial role.

Paper Structure

This paper contains 8 sections, 13 theorems, 88 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Statement of the main results and discussions
  4. well-posedness of the forward problem
  5. Vanishing Properties
  6. Vanishing properties in 2D
  7. Vanishing properties in 3D
  8. Unique recovery results

Key Result

Theorem 1.1

Consider the scattering problem described by equation main:eqs associated with two scatterers $\left(\Omega_j ; k, d, q_j, \mathbf{A}_j\right)$ for $j=1,2$, in $\mathbb{R}^n$ with $n=2,3$. If the far-field patterns corresponding to the two scatterers are the same, i.e. $u_{\infty}^1\left(\hat{\mathb

Figures (4)

  • Figure 1: Illustration of the geometry in 2D
  • Figure 2: Illustration of the geometry in 3D
  • Figure 3: Illustration of $V_{2h}$
  • Figure 4: Schematic illustration

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • ...and 16 more