Fixed-distance multipoint formulas for the scattering amplitude from phaseless measurements

We give new formulas for finding the complex (phased) scattering amplitude at fixed frequency and angles from absolute values of the scattering wave function at several points $x_1,..., x_m$. In dimension $d\geq 2$, for $m>2$, we significantly improve previous results in the following two respects. First, geometrical constraints on the points needed in previous results are significantly simplified. Essentially, the measurement points $x_j$ are assumed to be on a ray from the origin with fixed distance $\tau=|x_{j+1}- x_j|$, and high order convergence (linearly related to $m$) is achieved as the points move to infinity with fixed $\tau$. Second, our new asymptotic reconstruction formulas are significantly simpler than previous ones. In particular, we continue studies going back to [Novikov, Bull. Sci. Math. 139(8), 923-936, 2015].