Configuration space properties of S- matrix in scattering (*)

Abstract

We prove estimates on time decay for products of Schrodinger evolution groups between weighted L ²-spaces by using commutator techniques. These estimates are used to show that, for potentials V(x̲) decaying more rapidly than |x̲| ^-2, the scattering operator leaves a certain dense subset ~ of the domain of definition of Q̲² invariant. This implies the existence of the global time delay (defined in terms of sojourn times) on ~ if V(x̲) = O(|x̲|^-2-ε) at infinity, ε>0.