SCATTERING MATRICES WITH FINITE PHASE-SHIFT AND THE INVERSE SCATTERING PROBLEM

The inverse scattering problem for the Schrodinger operator on the hal f-axis is studied. It is shown that this problem can be solved for the scattering matrices with arbitrary finite phase shift on the real axi s if one admits potentials with long-range oscillating tails at infini ty. The solution of the problem is constructed with the help of the Ge lfand-Levitan-Marchenko procedure. The inverse problem has no unique s olution for the standard set of scattering data which includes the sca ttering matrix, energies of the bound states and corresponding normali zing constants. This fact is related to zeros of the spectral density on the real axis. It is proven that the inverse problem has a unique s olution in the defined class of potentials if the zeros of the spectra l density are added to the set of scattering data.