On wave operators for Schr\"odinger operators with threshold singuralities in three dimensions

We show that wave operators for three dimensional Schr\"odinger operators $H=-\Delta + V$ 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 $\phi(x)$ have vanishing first three moments: $\int_{{\mathbb R}^3} x^\alpha V(x)\phi(x)dx=0$, $|\alpha|=0,1,2$.