A new proof of the invariance principle in scattering theory

Abstract

For free and interacting Hamiltonians, H 0 and H = H 0 + V( r) acting in L 2(R 3, dx) with V( r) a radial potential satisfying certain technical conditions, and for ϕ a real function on R with ϕ′ > 0 except on a discrete set, we prove that the Moller wave operators Ω± = strong limit e itϕ(H) e −itϕ(H 0) exist and are independent of ϕ. The scattering operator S = (Ω +)∗Ω − is shown to be unitary. Our proof utilizes time independent methods (eigenfunction expansions) and is effective in cases not previously analyzed, e.g. V(r) = sin r r and many others.