Phaseless inverse scattering with background information

We consider phaseless inverse scattering for the multidimensional Schrödinger equation with unknown potential v using the method of known background scatterers. In particular, in dimension d 2, we show that |f 1|2 at high energies uniquely determines v via explicit formulas, where f 1 is the scattering amplitude for v + w 1, w 1 is an a priori known nonzero background scatterer, under the condition that supp v and supp w 1 are sufficiently disjoint. If this condition is relaxed, then we give similar formulas for finding v from |f|2, |f 1|2, where f is the scattering amplitude for v. In particular, we continue studies of Novikov (2016 J. Geom. Anal. 26 346-59) and Leshem et al (2016 Nat. Commun. 7 1-6).