The first solution of a long standing problem: reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation (Q2517188)

The paper deals with a three-dimensional inverse scattering problem for the Schrödinger equation in the frequency domain \(\nabla_x u +k^2 u -q(x)u = -\delta(x-x^0)\) with a nonnegative, compactly supported unknown potential. The data for the problem are measurements of the absolute value of the scattered wave field. The main result is the reconstruction formula for a smooth potential \(q(x)\in C^4({\mathbb R}^3)\), which in fact is a one-parameter collection of inversions of the two-dimensional Radon transforms of the potential. This information can be obtained from the large-\(k\) asymptotics of the data, as a function of \(x\) and \(x^0\) varying along the boundary of a slice of the ball supporting the potential. The main technical result, which is used in establishing the asymptotics, is the \(C^2\)-smoothness of the regular part of the fundamental solution of the associated hyperbolic PDE.