On solutions of the Schrödinger equation with radiation conditions at infinity: the long-range case
The paper gives a complete description of solutions of the homogeneous Schrödinger equation with a long range potential. It is shown that this solution is asymptotically a sum of incoming and outgoing distorted spherical waves, where the phase must be a suitable chosen approximate solution of the eikonal equation to the potential. So the norm of the incoming and outgoing waves are equal and there exists for any incoming function a outgoing one and a asymptotic solution of the Schrödinger equation, also the roles of incoming and outgoing functions can be interchanged. Finally it is shown that the time dependent scattering matrix can be constructed by this one of the stationary case.
scientific article; zbMATH DE number 1355992
scientific article