Remarks on L^p-Boundedness of Wave Operators for Schrödinger Operators with Threshold Singularities

We consider the continuity property in Lebesgue spaces $L^p(\R^m)$ of the wave operators $W_\pm$ of scattering theory for Schrödinger operators $H=-\lap + V$ on $\R^m$, $|V(x)|≤ C\ax^-\delta$ for some $\delta>2$ when $H$ is of exceptional type, i.e. $\Ng={u \in \ax^s L²(\R^m) \colon (1+ (-\lap)^-1V)u=0 }\not={0}$ for some $1/2<s<\delta-1/2$. It has recently been proved by Goldberg and Green for $m≥ 5$ that $W_\pm$ are in general bounded in $L^p(\R^m)$ for $1≤ p<m/2$, for $1≤ p<m$ if all $\f\in \Ng$ satisfy $\int_\R^m V\f dx=0$ and, for $1≤ p<\infty$ if $\int_\R^m x_i V\f dx=0$, $i=1, \dots, m$ in addition. 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 $L^p(\R³)$ for $1<p<3$ and that the same holds for $1<p<\infty$ if and only if all $\f\in \Ng$ satisfy $\int_\R³V\f dx=0$ and $\int_\R³ x_i V\f dx=0$, $i=1, 2, 3$, simultaneously.

2010 Mathematics Subject Classification: 35P25, 81U05, 47A40.

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