On wave operators for Schrödinger operators with threshold singuralities in three dimensions

Abstract

We show that wave operators for three dimensional Schr\"odinger operators H=−Δ+V with threshold singularities are bounded in L1(R3) if and only if zero energy resonances are absent from H and the existence of zero energy eigenfunctions does not destroy the L1-boundedness of wave operators for H with the regular threshold behavior. We also show in this case that they are bounded in Lp(R3) for all 1≤p≤∞ if all zero energy eigenfunctions ϕ(x) have vanishing first three moments: ∫R3xαV(x)ϕ(x)dx=0, |α|=0,1,2.