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

Abstract

We consider the continuity property in Lebesgue spaces $L^p({\mathbb{R}}^m)$ of wave operators $W_{\pm }$ of scattering theory for Schrödinger operator $H=-\lap + V$ on ${\mathbb{R}}^m$, $|V(x)|\leq C\ax^{-δ}$ for some $\delta >2$ 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 $1/2<s<\delta -1/2$. It has recently been proved by Goldberg and Green for $m\ge 5$ that $W_{\pm }$ are bounded in $L^p({\mathbb{R}}^m)$ for $1\le p<m/2$, the same holds for $1\le p<m$ if all $\f\in \Ng$ satisfy $\int_{\R^m} V\f dx=0$ and, for $1\le p<\infty$ 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 $L^p({\mathbb{R}}^3)$ 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.

58 pages

References in corpus (2)

• The $L^p$ boundedness of wave operators for Schrödinger Operators with threshold singularities
• Wave Operators for Schrödinger Operators with Threshold Singuralities, Revisited

Cited by in corpus (1)

• Remarks on $L^p$-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems