Stationary Scattering Theory

Abstract

We present a complementary approach to the time dependent scattering theory described by V. Enss for one-body Schrodinger operators. Roughly speaking the stationary theory is concerned with those objects you read about in textbooks on quantum theory like scattering waves and amplitudes. The starting point of the old theory is not here an asymptotic condition for large times, as for wave-operators of V. Enss lecture, but rather for large distances. It is known as “Sommerfeld radiation condition” and leads, when incorporated in the Schrodinger equation as a “Cauchy condition at infinity”, to the celebrated Lippman-Schwinger equation. In the more recent abstract stationary theory some generalized form of the Lippman-Schwinger equation plays the basic role; solving this equation leads to a linear map between generalized eigenfunctions of the perturbed and unperturbed operators. This map is the “section” at fixed energy of the wave-operator from the time dependent theory. Although the radiation condition does not appear explicitly in this formulation it can be shown to hold a posteriori in a variety of situations, thus restoring the link with physical theories. A general approach to the radiation condition for a large class of Partial Differential operators is described in the work of S. Agmon and L. Hormander1; we will mention some of their results here.