Fixed-distance multipoint formulas for the scattering amplitude from phaseless measurements. (5th January 2022)

• Record Type: Journal Article
• Authors: Novikov, R G; Sivkin, V N
• Is Part Of: Inverse problems. Volume 38:Number 2(2022)
• Publication Date: 2022-01-05
• Subjects: Schrödinger equation -- Helmholtz equation -- monochromatic scattering data -- phase retrieval -- phaseless inverse scattering; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ac44db
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We give new formulas for finding the complex (phased) scattering amplitude at fixed frequency and angles from absolute values of the scattering wave function at several points x 1, …, x m . In dimension d ⩾ 2, for m > 2, we significantly improve previous results in the following two respects. First, geometrical constraints on the points needed in previous results are significantly simplified. Essentially, the measurement points x j are assumed to be on a ray from the origin with fixed distance τ = | x j +1 − x j |, and high order convergence (linearly related to m ) is achieved as the points move to infinity with fixed τ . Second, our new asymptotic reconstruction formulas are significantly simpler than previous ones. In particular, we continue studies going back to Novikov (2015 Bull. Sci. Math. 139 923–936).