A time-dependent,two Hilbert space approach to long-range quantum scattering

Scattering of a quantum mechanical particle by a long-range potential is studied using Enss's time-dependent method. More precisely, a simple and natural extension of the Enss method to a two Hilbert space setting is established. Applied to the consideration of long-range scattering, this extended Enss method reduces the problem of proving existence and completeness of the wave operators to the problem of solving an eikonal equation on a cone in phase space. Relying on a solution of the eikonal equation constructed by Isozaki and Kitada, it is shown that the wave operators exist and are complete whenever the potential is in the long-range class introduced by Hörmander.