Inverse scattering problem with data at fixed energy and fixed incident direction

✍ Scribed by A.G. Ramm

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 233 KB
• Volume: 69
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2007.06.047

✦ Synopsis

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 an arbitrary large 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 > 0, and an arbitrary small ε > 0, there exists a potential q(x) ∈ L 2 (D) where D ⊂ R 3 is a bounded domain such that

The potential q, for which ( * ) holds, is not unique. We give a method for finding such a q, and a formula for this q.