$L^1$ and $L^\infty$-boundedness of Wave Operators for Three Dimensional Schrödinger Operators with Threshold Singularities

Abstract

It is known that wave operators for three dimensional Schrödinger operators $-\Delta + V$ with threshold singularities are bounded in $L^p(\mathbb{R}^3)$ for $1<p<3$ in general and, for $1<p<\infty$ if and only if zero energy resonances are absent and all zero energy eigenfunctions $\phi$ of $-\Delta + V$ satisfy $\int V(x)x^\alpha \phi(x) dx=0$ for $|\alpha|\leq 1$. We prove here that they are bounded in $L^1(\mathbb{R}^3)$ if and only if zero energy resonances are absent. We also show that they are bounded in $L^\infty(\mathbb{R}^3)$ if no resonances are present and all zero energy eigenfunctions $\phi(x)$ satisfy $\int_{\mathbb{R}^3} x^\alpha V(x)\phi(x)dx=0$ for $0\leq |\alpha|\leq 2$. This fills the unknown parts of the $L^p$-boundedness problem for wave operators of three dimensional Schrödinger operators.