Abstract:

In his paper of 1949, V.Bargmann suggested several classes of isospectral Schroedinger operators on the line having the same negative bound states and absolutely continuous spectrum coinciding with the positive half-line. Several years later I.Gelfand, B.Levitan, V.Marchenko and M.Krein shaped up the inverse scattering theory, and it became clear that most of the Bargmann examples are given by reflectionless potentials. After C.S.Gardner, J.M.Greene, M.D.Kruskal and R.M.Miura discovered in 1967 the inverse scattering method of solving the famous Korteweg-de Vries (KdV) equation, the Bargmann potentials re-appeared as n-soliton solutions of KdV.

V.Marchenko (1991) and F.Gesztesy, W.Karwowski and Z.Zhao (1992) suggested several ways to generalize the notions of reflectionless Schroedinger operators and soliton solutions of the KdV equation. The aim of the talk is to report on our recent results in this direction obtained jointly with Ya.Mykytyuk (Lviv).