Inverse Scattering Problem with Underdetermined Data

Abstract

Consider the Schrödinger operator −∇ 2 + q with a smooth compactly supported potential q, q = q(x), x ∈ R 3 . Let A(β, α, k) be the corresponding scattering amplitude, k 2 be the energy, α ∈ S 2 be the incident direction, β ∈ S 2 be the direction of scattered wave, S 2 be the unit sphere in R 3 . Assume that k = k 0 > 0 is fixed, and α = α 0 is fixed. Then the scattering data are A(β) = A(β, α 0 , k 0 ) = A q (β) is a function on S 2 . The following inverse scattering problem is studied: IP: Given an arbitrary f ∈ L 2 (S 2 ) and an arbitrary small number > 0, can one find q ∈ C ∞ 0 (D), where D ∈ R 3 is an arbitrary fixed domain, such that ||A q (β) − f (β)|| L 2 (S 2 ) < ? A positive answer to this question is given. A method for constructing such a q is proposed. There are infinitely many such q, not necessarily real-valued.