Commutators and Local Decay

Abstract

A typical problem in mathematical scattering theory is, given a pair of differential equations involving space and time which are asymptotically the same at large values of the space variable, to show that pairs of solutions become asymptotically the same in the distant future or past. For this it is necessary to prove local decay: the solutions eventually leave the region where the difference between the equations is large. For quantum mechanical two-body problems with potentials approaching zero without too much oscillation at infinity, local decay can be proved by a method, to be explained below, whose essential element is closely related to the generator of the group of dilations of space. The results of Balslev and Combes [1] and Van Winter [2] have already made it clear that this group is a powerful tool in scattering theory.