The back-scattering problem in three dimensions

In this article we study the (inverse) back-scattering problem for the Schrodinger operator in R-3. We introduce the back-scattering transform B(upsilon) of a real-valued potential upsilon is an element of C-0(infinity) (R-3), and prove that the back-scattering data associated to v determine B(upsilon). Under the assumption that the Schrodinger operator H-upsilon = -Delta+v has no eigenvectors in L-2(R-3) it is shown that B(upsilon) may be expressed in terms of the wave group K-upsilon(t) = sin(t root H-upsilon)/root H-upsilon. We prove also that the mapping v bar right arrow B(upsilon) is a homeomorphism in a neighbourhood of the origin in the Banach space X-0(r), which is the completion of C-0(infinity) (R-3; R) w.r.t. the norm f bar right arrow Sigma(vertical bar a vertical bar=1) parallel to partial derivative(a) f parallel to L-1.