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

Abstract

We show that wave operators for three dimensional Schrödinger operators $H=-Δ+ V$ with threshold singularities are bounded in $L^1({\mathbb R}^3)$ if and only if zero energy resonances are absent from $H$ and the existence of zero energy eigenfunctions does not destroy the $L^1$-boundedness of wave operators for $H$ with the regular threshold behavior. We also show in this case that they are bounded in $L^p({\mathbb R}^3)$ for all $1\leq p \leq \infty$ if all zero energy eigenfunctions $ϕ(x)$ have vanishing first three moments: $\int_{{\mathbb R}^3} x^αV(x)ϕ(x)dx=0$, $|α|=0,1,2$.