2006/06/21 by A. G. Ramm, А. Г. Рамм, Ramm, A. G.
Engineering · Mathematics · Physics and Astronomy · #35J05 #35J10 #35R30 #74J25 #81U40 #81V05 #Electromagnetic Scattering and Analysis #FOS: Physical sciences #Mathematical Physics (math-ph) #Microwave Imaging and Scattering Analysis #Numerical methods in inverse problems #math-ph #math.MP #msc:35J05 #msc:35J10 #msc:35R30 #msc:74J25 #msc:81U40 #msc:81V05
paper · pdf · doi:10.48550/arxiv.math-ph/0606055
openalex publication_date 2006/06/21 · arxiv created 2006/10/03 · arxiv updated 2016/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Aq(α',α,k) be the scattering amplitude, corresponding to a local potential q(x), x∈\R3, q(x)=0 for |x|>a, where a>0 is a fixed number, α',α∈ S2 are unit vectors, S2 is the unit sphere in \R3, α is the direction of the incident wave, k2>0 is the energy. We prove that given an arbitrary function f(α')∈ L2(S2), an arbitrary fixed α0∈ S2, an arbitrary fixed k>0, and an arbitrary small \ve>0, there exists a potential q(x)∈ L2(D), where D⊂ R3 is a bounded domain such that \bee ‖Aq(α',α0,k)-f(α')‖L2(S2)<\ve. \eee The potential q, for which (∗) holds, is nonunique. We give an method for finding q, and a formula for such a q.