The $L^p$ boundedness of wave operators for Schr\"odinger operators with threshold singularities II. Even dimensional case

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Let $H_0=-\lap$ and $H=-\lap +V(x)$ be Schr\"odinger operators on $\R^m$ and $m \geq 6$ be even. We assume that $\Fg(\ax^{-2\s}V) \in L^{m_\ast}(\R^m)$ for some $\s>\frac{1}{m_\ast}$, $m_\ast=\frac{m-1}{m-2}$ and $V(x)\leq C \ax^{-\d}$ for some $\d>m+2$, so that the wave operators $W_\pm=\lim_{t\to \pm \infty} e^{itH}e^{-itH_0}$ exist. We show the following mapping properties of $W_\pm$: (1) If $0$ is not an eigenvalue of $H$, $W_\pm$ are bounded in Sobolev spaces $W^{k,p}(\R^m)$ for all $0 \leq k \leq 2$ and $1<p<\infty$ and also in $L^1(\R^m)$ and $L^\infty(\R^m)$; (2) if $0$ is an eigenvalue of $H$ and if $V$ satisfies stronger decay condition $V(x)\leq C\ax^{-\d}$, $\d>m+4$ if $m=6$ and $\d>m+3$ if $m\geq 8$, $W_\pm$ are bounded in $W^{k,p}(\R^m)$ for all $0 \leq k \leq 2$ and $\frac{m}{m-2}<p<\frac{m}2$; (3) the same holds in Sobolev spaces of higher orders if derivatives of $V(x)$ satisfy suitable boundedness conditions. This paper is a continuation of the one with the same title, part one, where odd dimensional cases $m \geq 3$ are treated, however, it can mostly be read independently.
Graduate School of Mathematical Sciences, The University of Tokyoの論文
2006-12-27