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

Abstract

We generalize the recent result of Erdoğan, 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 $(-\Delta + V(x))u(x) = 0$ 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 $1 < p \leq 2$ and unbounded for $2 < p < \infty$. We present also a new proof for the known part of the result.