On the Reconstruction from the Imaginary Part for Radiation Solutions in Two Dimensions

Abstract

We consider a radiation solution \(\psi\) for the Helmholtz equation in an exterior domain in \(\mathbb{R}^2\). We show that \(\psi\) in the exterior domain is uniquely determined by its imaginary part \(\operatorname{Im}(\psi)\) on an interval of a line \(L\) lying in the exterior domain. This result has a holographic prototype in the recent paper by Nair and Novikov (2025, J. Geom. Anal. 35, 4, 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.

DOI 10.1134/S1061920825601077