For the KdV equation a complete asymptotic expansion of the "dispersive tail" for large times is described, and generalized wave operators are introduced. The asymptotics for large times of the spectral Schrödinger equation with a potential of the type of a solution of the KdV equation is studied. It is shown that the KdV equation is connected in a specific manner with the structure of the asymptotics of solutions of the spectral equation. As a corollary, known explicit formulas for the leading terms of the asymptotics of solutions of the KdV equation in terms of spectral data corresponding to the initial conditions are obtained. A plan for justifying the results listed is outlined. © 1986 Plenum Publishing Corporation.