| 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 ∈ S2 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 ϵ> 0 , can one find q ∈ C0∞( D) , where D ∈ R3 is an arbitrary fixed domain, such that || Aq( β) − f( β)|| L2( S2) < ϵ? 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.
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