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

J. Math. Sci. Univ. Tokyo
Vol. 13 (2006), No. 3, Page 277--346.

Finco, Domenico; Yajima, Kenji
The $L^p$ boundedness of wave operators for Schr\"odinger operators with threshold singularities II. Even dimensional case
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Abstract:
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 $1m+4$ 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}
Mathematics Subject Classification (2000): Primary 35P25; Secondary 35J10, 47A40, 47F05,47N50, 81U50
Mathematical Reviews Number: MR2284406

Received: 2005-05-25