Inverse scattering problem with underdetermined data

Abstract

Consider the Schrödinger operator-2 + q with a smooth compactly supported potential q, q = q(x),x R3. Let A(β,α,k) be the corresponding scattering amplitude, k2 be the energy, α S2 be the incident direction, β S 2 be the direction of scattered wave, S2 be the unit sphere in R3. Assume that k = k0> 0 is fixed, and α = α0 is fixed. Then the scattering data are A(β) = A(β,α0,k0) = Aq(β) is a function on S2. The following inverse scattering problem is studied: IP: Given an arbitrary f L2(S2) and an arbitrary small number Ïμ > 0, can one find q C0(D) q C 0 (D), where D R3 is an arbitrary fixed domain, such that ||Aq(β)- f(β)|| L2(S2)