Inverse scattering problem with data at fixed energy and fixed incident direction
نویسنده
چکیده
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 an arbitrary large 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 ∈ S 2, an arbitrary fixed k > 0, and an arbitrary small ε > 0, there exists a potential q(x) ∈ L2(D) where D ⊂ R3 is a bounded domain such that ‖Aq (α , α0, k)− f (α ′)‖L2(S2) < ε. (∗) The potential q , for which (∗) holds, is not unique. We give a method for finding such a q , and a formula for this q. c © 2007 Elsevier Ltd. All rights reserved. MSC: 35J05; 35J10; 35R30; 74J25; 81U40; 81V05 PACS: 03.04.Kf
منابع مشابه
Uniqueness of the solution to inverse scattering problem with scattering data at a fixed direction of the incident wave
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Let A q (α ′ , α, k) be the scattering amplitude, corresponding to a local potential q(x), x ∈ R 3 , q(x) = 0 for |x| > a, where a > 0 is a fixed number, α ′ , α ∈ S 2 are unit vectors, S 2 is the unit sphere in R 3 , α is the direction of the incident wave, k 2 > 0 is the energy. We prove that given an arbitrary function f (α ′) ∈ L 2 (S 2), an arbitrary fixed α 0 ∈ S 2 , an arbitrary fixed k ...
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