The inverse problem for the one-dimensional Schroedinger equation with an energy-dependent potential. II

The one-dimensional Schroedinger equation y+''+ ) 7k2-V+(k,x){y+=0, x belonging to R, was previously considered when the potential V+(k,x) depends on the energy k2 in the following way: V+(k,x)=U(x)+2kQ(x), (U(x), Q(x)) belonging to a large class of pairs of real potentials admitting no bound state). The two systems of differential and integral equations then introduced are solved. Then, investigating the inverse scattering problem it is found that a necessary and sufficient condition for one of the functions S+(k) and Ssub(-1)sup(+)(k) to be the scattering matrix associated with a pair (U(x), Q(x)) is that S+(k) (or equivalently Ssub(-1)sup(+)(k) belongs to the class S introduced. This pair is the only one admitting this function as its scattering matrix. Investigating the inverse reflection problem, it is found that a necessary and sufficient condition for a function S21+(k) to be the reflection coefficient to the right associated with a pair (U(x), Q(x)) is that S21+(k) belongs to the class R introduced. This pair is the only one admitting this function as