Completeness of the set of scattering amplitudes

Let f ∈ L2(S2) be an arbitrary fixed function on the unit sphere S2, and D ⊂R3 be an arbitrary fixed bounded domain. Let k > 0 and α ∈ S2 be fixed. It is proved that there exists a potential q ∈ L2(D) such that the corresponding scattering amplitude A(α′) = Aq(α′) = Aq(α′, α, k) approximates f (α′) with arbitrary high accuracy: ‖f (α′)−Aq(α′)L2(S2)‖ ε where ε > 0 is an arbitrarily small fixed number. This means that the set {Aq(α′)}∀q∈L2(D) is complete in L2(S2). The results can be used for constructing nanotechnologically “smart materials”. © 2006 Elsevier B.V. All rights reserved. MSC: 35J10; 35P25; 35R30; 74J25; 81V05 PACS: 02.30; 03.40.K; 03.65