Phaseless inverse scattering problems in 3 - d

Consider the Schrödinger equation in R with the compactly supported potential q (x) , x ∈ R. The problem of the reconstruction of the function q (x) from measurements of the solution of that equation on a certain set is called “inverse scattering problem”. In this paper we prove uniqueness theorems for some 3-d inverse scattering problems in the case when only the modulus of the complex valued wave field is measured, while the phase is unknown. This is the phaseless case. In the past, phaseless inverse scattering problems were studied only in the 1-d case (section 1.2). As to the 3-d inverse scattering problems in the frequency domain, it was assumed in all studies so far that both the modulus and the phase of the complex valued wave field are measured, see, e.g. [2] for uniqueness results in the case of a piecewise analytic potential and [26, 27] for global uniqueness results and reconstruction methods. Below C are Hölder spaces, where s ≥ 0 is an integer and α ∈ (0, 1) . Let Ω, G ⊂ R be two bounded domains, Ω ⊂ G. For an arbitrary point y ∈ R and for an arbitrary number ω ∈ (0, 1) denote Bω (y) = {x : |x− y| < ω} and Pω (y) = R Bω (y) . For any two sets M,N ⊂ R let dist (M,N) be the Hausdorff distance between them. Let G1 ⊂ R be a convex bounded domain with its boundary S ∈ C. Let ε ∈ (0, 1) be a number. We assume that Ω ⊂ G1 ⊂ G, dist (S, ∂G) > 2ε and dist (S, ∂Ω) > 2ε. Hence,