On reconstruction from imaginary part for radiation solutions in two dimensions

TL;DR

This paper proves that in two dimensions, the entire radiation solution to the Helmholtz equation can be uniquely reconstructed from its imaginary part measured along a line outside the source region, with implications for inverse problems and imaging.

Contribution

It establishes the uniqueness of reconstructing Helmholtz solutions from imaginary part data on a line, extending holographic principles and applications to inverse spectral problems.

Findings

Unique determination of solutions from imaginary part on a line

Extension to other measurement curves

Applications to inverse spectral and passive imaging

Abstract

We consider a radiation solution $\psi$ for the Helmholtz equation in an exterior region in $\mathbb R^2$. We show that $\psi$ in the exterior region is uniquely determined by its imaginary part $Im(\psi)$ 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.