Phaseless Inverse Scattering with Background Information

We consider phaseless inverse scattering for the multidimensional Schrodinger 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).