The phase retrieval problem for solutions of the Helmholtz equation

In this paper we consider the phase retrieval problem for Herglotz functions, that is, solutions of the Helmholtz equation $\Delta u+\lambda^2u=0$ on domains $\Omega\subset\mathbb{R}^d$, $d\geq2$. In dimension $d=2$, if $u,v$ are two such solutions then $|u|=|v|$ implies that either $u=cv$ or $u=c\bar v$ for some $c\in\mathbb{C}$ with $|c|=1$. In dimension $d\geq3$, the same conclusion holds under some restriction on $u$ and $v$: either they are real valued or zonal functions or have non vanishing mean.