Multidimensional inverse scattering for the Schrödinger equation

We give a short review of old and recent results on the multidimensional inverse scattering problem for the Schrödinger equation.A special attention is paid to efficient reconstructions of the potential from scattering data which can be measured in practice.In this connection our considerations include reconstructions from non-overdetermined monochromatic scattering dataand formulas for phase recovering from phaseless scattering data.Potential applications include phaseless inverse X-ray scattering, acoustic tomography and tomographies using elementary particles.This paper is based, in particular, on results going back to M. Born (1926), L. Faddeev (1956, 1974), S. Manakov (1981), R.Beals, R. Coifman (1985), G. Henkin, R. Novikov (1987), and on more recent results of R. Novikov ( 1998 - 2019), A. Agaltsov, T. Hohage, R. Novikov (2019).This paper is an extended version of the talk given at the 12th ISAAC Congress, Aveiro, Portugal, 29 July - 2 August, 2019