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.