On reconstruction from imaginary part for radiation solutions in two dimensions

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.