The $L^{p}$-boundedness of wave operators for two dimensional Schrödinger operators with threshold singularities
The $L^{p}$-boundedness of wave operators for two dimensional Schrödinger operators with threshold singularities
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DOI：
10.2969/jmsj/85418541
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发表时间：
2020-08
影响因子：
0.7
通讯作者：
K. Yajima
中科院分区：
文献类型：
--
作者：
K. Yajima
我们概括了 Erdo{\u g}an、Goldberg 和 Green 关于二维薛定谔算子波算子 $L^p$ 有界性的最新结果，并证明对于所有 $1<p<\infty$，当且仅当薛定谔算子不具有 $p$ 波阈值共振时，它们在 $L^p(\R^2)$ 中有界，即薛定谔方程 $(-\lap + V(x))u(x)=0$ 不存在满足 $u(x)= (a_1x_1+a_2 x_2)|x|^{-2}+ o(|x|^{-1})$ 作为 $|x|\to \infty$ 对于 $(a_1, a_2) \in \R^2\setminus \{(0,0)\}$ 的解，并且，否则，对于 $1<p\leq 2$，它们在 $L^p(\R^2)$ 内有界，对于 $2<p<\infty$ 则无界。我们还为结果的已知部分提供了新的证明。
We generalize the recent result of Erdo{\u g}an, Goldberg and Green on the $L^p$-boundedness of wave operators for two dimensional Schrodinger operators and prove that they are bounded in $L^p(\R^2)$ for all $1<p<\infty$ if and only if the Schrodinger operator possesses no $p$-wave threshold resonances, viz. Schrodinger equation $(-\lap + V(x))u(x)=0$ possesses no solutions which satisfy $u(x)= (a_1x_1+a_2 x_2)|x|^{-2}+ o(|x|^{-1})$ as $|x|\to \infty$ for an $(a_1, a_2) \in \R^2\setminus \{(0,0)\}$ and, otherwise, they are bounded in $L^p(\R^2)$ for $1<p\leq 2$ and unbounded for $2<p<\infty$. We present also a new proof for the known part of the result.