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. $\Ng={u \in \ax^{s} L^{2}(\R^{m}) \colon
(1+ (-\lap)^{-1}V)u=0 }\not={0}$ for some $1/2<s<\delta-1/2$. It has
recently been proved by Goldberg and Green for $m≥ 5$ 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 $\int_{\Rm} V\f dx=0$ and, for $1≤ p<\infty$
if $\int_{\Rm} x_{i} V\f dx=0$, $i=1, \dots, m$ 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 $L^{p}(\R^{3})$ for $1<p<3$ and that
the same holds for $1<p<\infty$ if and only if all $\f\in \Ng$ satisfy
$\int_{\R3}V\f dx=0$ 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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