The $L^p$-boundedness of wave operators for two dimensional Schrödinger operators with threshold singularities

Abstract

We generalize the recent result of Erdo{\u g}an, Goldberg and Green on the $L^p$-boundedness of wave operators for two dimensional Schrödinger operators and prove that they are bounded in $L^p({\mathbb{R}}^2)$ for all $1<p<\infty$ if and only if the Schrödinger operator possesses no $p$-wave threshold resonances, viz. Schrödinger equation $(-\lap + V(x))u(x)=0$ possesses no solutions which satisfy $u(x)=(a_1{x}_1+a_2{x}_2)|x{|}^{-2}+o(|x{|}^{-1})$ as $|x|\to \infty$ for an $(a_1,a_2)\in {\mathbb{R}}^2\setminus \{(0,0)\}$ and, otherwise, they are bounded in $L^p({\mathbb{R}}^2)$ for $1<p\le 2$ and unbounded for $2<p<\infty$. We present also a new proof for the known part of the result.

References in corpus (2)

• The $L^p$ boundedness of wave operators for Schrödinger Operators with threshold singularities
• $L^p$-boundedness of wave operators for 2D Schrödinger operators with point interactions