论文标题
二维Schrödinger操作员带有阈值奇异性
The $L^p$-boundedness of wave operators for two dimensional Schrödinger operators with threshold singularities
论文作者
论文摘要
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 $L^p(\R^2)$ for all $1<p<\infty$ if and only if the Schrödinger operator possesses no $p$-wave threshold resonances, viz. schrödinger方程$（ - \ lap+ v（x））u（x）= 0 $没有满足$ u（x）=（a_1x_1+ a_1+ a_2 x_2）| x | x |^{ - 2}+ o（| x |^{ - 1}）$ as $ | x | x | x | x | x | a_1的$（a_1x_1+ a_2 x_2） \ r^2 \ setMinus \ {（0,0）\} $，否则，它们在$ l^p（\ r^2）$中限制为$ 1 <p \ leq 2 $，并以$ 2 <p <\ p <\ infty $而无限。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 Schrödinger operators and prove that they are bounded in $L^p(\R^2)$ 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 $u(x)= (a_1x_1+a_2 x_2)|x|^{-2}+ o(|x|^{-1})$ 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.
