Remarks on $L^p$-boundedness of wave operators for Schr\"odinger operators with threshold singularities

TL;DR

This paper investigates the boundedness of wave operators for Schrödinger operators with threshold singularities in Lebesgue spaces, providing necessary and sufficient conditions for their boundedness in various dimensions.

Contribution

It refines and extends previous results on $L^p$-boundedness of wave operators, establishing necessary and sufficient conditions for different dimensions and $p$ ranges.

Findings

For $m=3$, $W_$ are bounded in $L^p( ^3)$ for $1<p<3$.

Boundedness in $L^p$ for all $1<p<$ occurs if and only if certain integral conditions on $V\f$ hold.

Conditions on $V\f$ are necessary for the boundedness of wave operators in the exceptional case.

Abstract

We consider the continuity property in Lebesgue spaces $L^p(\R^m)$ of wave operators $W_\pm$ of scattering theory for Schr\"odinger operator $H=-\lap + V$ on $\R^m$, $|V(x)|\leq C\ax^{-\delta}$ 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 $m\geq 5$ 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.