Recovery from phase-free Fourier transform and scattering data with reference information

We consider the problem of finding a potential with a compact carrier in the multidimensional Schrodinger equation by the differential scattering cross section (the square of the scattering amplitude modulus) at a fixed energy. In the Born approximation, this problem is simplified to the problem of restoring the potential by the absolute value of its Fourier transform on the ball. To compensate for the missing phase information, we use the method of a priori known reference diffusers. In particular, we propose an iterative scheme for finding the potential from measurements of only one differential scattering cross section corresponding to the sum of an unknown potential and a known reference potential sufficiently distant from each other. A numerical implementation of the proposed recovery algorithms is obtained. The results of the report are based, in particular, on joint work with R.G. Novikov and T. Khokhage.