$L^p$-boundedness of wave operators for 2D Schr\"odinger operators with point interactions

Abstract

For two dimensional Schr\"odinger operator $H$ with point interactions, We prove that wave operators of scattering for the pair $(H,H_0)$, $H_0$ being the free Schr\"odinger operator, are bounded in the Lebesgue space $L^p(\R^2)$ for $1<p<\infty$ if and only if there are no generalized eigenfunctions of $Hu(x)=0$ which satisfy $u(x)= C|x|^{-1}+ o(|x|^{-1})$ as $|x|\to \infty$, $C\not=0$. Otherwise they are bounded for $1<p\leq 2$ and unbounded for $2<p<\infty$.