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On reconstruction from imaginary part for radiation solutions in two dimensions

Arjun Nair, Roman Novikov

TL;DR

This work establishes that in two dimensions, a radiation solution $\psi$ to the exterior Helmholtz problem is globally recoverable from the imaginary part $Im(\psi)$ measured on a line segment. By employing the Karp expansion, a two-point reconstruction formula, and a Green-type boundary integral representation, the authors show that $Im(\psi)$ on an interval of a line uniquely determines $\psi$ in the exterior domain, via recovery of the angular coefficients $f_j(\phi)$ and their recurrence. The results yield holographic-type reconstruction and enable a reduction of the Gelfand-Krein-Levitan inverse problem at fixed energy in $\mathbb{R}^2$ to an inverse boundary-value problem for the potential $v$, with uniqueness demonstrated for data on intervals of a measurement line and, more generally, on curves or cross-interval data. The methods also connect to passive imaging and phaseless inverse scattering in 2D, offering global uniqueness results under realistic measurement geometries and showing how to extend to multi-component boundaries. Overall, the paper advances 2D inverse scattering theory by proving boundary-imaginary-part-based reconstruction and linking it to GK-Levitan-type inference.

Abstract

We consider a radiation solution $ψ$ for the Helmholtz equation in an exterior region in $\mathbb R^2$. We show that $ψ$ in the exterior region is uniquely determined by its imaginary part $Im(ψ)$ on an interval of a line $L$ lying in the exterior region. This result has holographic prototype in the recent work Nair, Novikov (2025, J. Geom. Anal. 35, 123). Some other curves for measurements instead of the lines $L$ are also considered. Applications to the Gelfand-Krein-Levitan inverse problem (from boundary values of the spectral measure in $\mathbb R^2$) and to passive imaging are also indicated.

On reconstruction from imaginary part for radiation solutions in two dimensions

TL;DR

This work establishes that in two dimensions, a radiation solution to the exterior Helmholtz problem is globally recoverable from the imaginary part measured on a line segment. By employing the Karp expansion, a two-point reconstruction formula, and a Green-type boundary integral representation, the authors show that on an interval of a line uniquely determines in the exterior domain, via recovery of the angular coefficients and their recurrence. The results yield holographic-type reconstruction and enable a reduction of the Gelfand-Krein-Levitan inverse problem at fixed energy in to an inverse boundary-value problem for the potential , with uniqueness demonstrated for data on intervals of a measurement line and, more generally, on curves or cross-interval data. The methods also connect to passive imaging and phaseless inverse scattering in 2D, offering global uniqueness results under realistic measurement geometries and showing how to extend to multi-component boundaries. Overall, the paper advances 2D inverse scattering theory by proving boundary-imaginary-part-based reconstruction and linking it to GK-Levitan-type inference.

Abstract

We consider a radiation solution for the Helmholtz equation in an exterior region in . We show that in the exterior region is uniquely determined by its imaginary part on an interval of a line lying in the exterior region. This result has holographic prototype in the recent work Nair, Novikov (2025, J. Geom. Anal. 35, 123). Some other curves for measurements instead of the lines are also considered. Applications to the Gelfand-Krein-Levitan inverse problem (from boundary values of the spectral measure in ) and to passive imaging are also indicated.

Paper Structure

This paper contains 12 sections, 7 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Determination of radiation solutions $\psi$ from Im$(\psi)$
  4. Applications to Gelfand-Krein-Levitan problem
  5. Preliminaries
  6. The Karp Expansion
  7. A two-point approximation for $\psi$
  8. A Green type formula
  9. Some facts of direct scattering
  10. Proof of Theorem 1
  11. Case $L^+_{x_1,\theta} \cup L^-_{x_2,\theta} \subseteq L_{0,\theta}$
  12. General Case

Key Result

Theorem 1

Let $\psi$ be a radiation solution of equation (eq:1.1) as in (eq:1.3). Let $L$, $L^+$ and $L^-$ be as given in (eq:1.3) such that $L^+=L^+_{x_1, \theta}\subset {\cal U}$ and $L^-=L^-_{x_2, \theta}\subset {\cal U}$, $x_1,x_2 \in L$, where ${\cal U}$ is the region in (eq:1.1). Then $\psi$ on $L^+$$\c

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Example 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Proposition 1
  • Remark 2
  • ...and 1 more