arXiv 1602.070372016-02-23math-phmath.MP

关于阈值奇异Schr dinger算子的波算子的$L^p$ -有界性的注记

Remarks on $L^p$-boundedness of wave operators for Schrödinger operators with threshold singularities

Kenji Yajima

备注：: 58 pages

摘要：

本文研究了散射理论中的波算子$W_\pm$在Lebesgue空间$L^p(\R^m)$中的连续性，当$H$是例外型时，即$\Ng=\{u \in \ax^{-s} L^2(\R^m) \colon (1+ (-\lap)^{-1}V)u=0 \}\not=\{0\}$对某些$1/2<s<δ-1/2$，薛定谔算子$H=-\lap + V$在$\R^m$上，$|V(x)|\leq C\ax^{-δ}$对某些$δ>2$上. Goldberg和Green最近证明了$m\geq 5$的$W_\pm$在$L^p(\R^m)$中有界，如果所有$\f\in \Ng$满足$\int_{\R^m} V\f dx=0$，则$1\leq p<m$也是如此，如果除$\int_{\R^m} x_i V\f dx=0$外，还有$i=1, \dots, m$，则$1\leq p<\infty$也是如此。我们使$p>m/2$的结果更精确，并特别证明，这些条件也是必要的$W_\pm$的规定属性。我们还证明了，$m=3$，$W_\pm$有界$L^p(\R^3)$ $1<p<3$和$1<p<\infty$同样成立，当且仅当所有$\f\in \Ng$满足$\int_{\R^3}V\f dx=0$和$\int_{\R^3} x_i V\f dx=0$，$i=1, 2, 3$，同时。

英文摘要：

We consider the continuity property in Lebesgue spaces $L^p(\R^m)$ of wave operators $W_\pm$ of scattering theory for Schrödinger operator $H=-\lap + V$ on $\R^m$, $|V(x)|\leq C\ax^{-δ}$ for some $δ>2$ when $H$ is of exceptional type, i.e. $\Ng=\{u \in \ax^{-s} L^2(\R^m) \colon (1+ (-\lap)^{-1}V)u=0 \}\not=\{0\}$ for some $1/2<s<δ-1/2$. It has recently been proved by Goldberg and Green for $m\geq 5$ that $W_\pm$ are bounded in $L^p(\R^m)$ for $1\leq p<m/2$, the same holds for $1\leq p<m$ if all $\f\in \Ng$ satisfy $\int_{\R^m} V\f dx=0$ and, for $1\leq p<\infty$ if in addition $\int_{\R^m} x_i V\f dx=0$, $i=1, \dots, m$. We make the results for $p>m/2$ more precise and prove in particular that these conditions are also necessary for the stated properties of $W_\pm$. We also prove that, for $m=3$, $W_\pm$ are bounded in $L^p(\R^3)$ for $1<p<3$ and that the same holds for $1<p<\infty$ if and only if all $\f\in \Ng$ satisfy $\int_{\R^3}V\f dx=0$ and $\int_{\R^3} x_i V\f dx=0$, $i=1, 2, 3$, simultaneously.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

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