Paper detail

Structure of wave operators in R^3

We prove a structure formula for the wave operators in R^3 and their adjoints for a scaling-invariant class of scalar potentials V, under the assumption that zero is neither an eigenvalue, nor a resonance for -Δ+V. The formula implies the boundedness of wave operators on L^p spaces, 1 \leq p \leq \infty, on weighted L^p spaces, and on Sobolev spaces, as well as multilinear estimates for e^{itH} P_c. When V decreases rapidly at infinity, we obtain an asymptotic expansion of the wave operators. The first term of the expansion is of order < y >^{-4}, commutes with the Laplacian, and exists when V \in <x >^{-3/2-ε} L^2. We also prove that the scattering operator S = W_-^* W_+ is an integrable combination of isometries. The proof is based on an abstract version of Wiener's theorem, applied in a new function space.