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

Abstract

We show that wave operators for three dimensional Schrödinger 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 $1\le p\le \infty$ if all zero energy eigenfunctions $\varphi (x)$ have vanishing first three moments: ${\int }_{{\mathbb{R}}^3}{x}^{\alpha }V(x)\varphi (x)dx=0$, $|\alpha |=0,1,2$.

21 pages

Cited by in corpus (3)

• On the $L^p$ boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions
• Dispersive estimates for Schrödinger operators with point interactions in ${\mathbb{R}}^3$
• On real resonances for three-dimensional Schrödinger operators with point interactions