Let $H=-Δ+V(x)$ be the Schrodinger operator on ${\bf R}^m$, $m\ge 3$. We show that the wave operators $W_\pm=\lim_{t\to\pm\infty}e^{itH}\cdot e^{-itH_0}$, $H_0=-Δ$, are bounded in Sobolev spaces $W^{k, p}({\bf R}^m)$, $1\le p\le\infty$, $k=0, 1, \ldots, \ell$, if $V$ satisfies $\|D^α V(y)\|_{L^{p_0}(|x-y|\le 1)}\le C(1+|x|)^{-δ}$ for $δ>(3m/2)+1$, $p_0>m/2$ and $|α|\le\ell+\ell_0$, where $\ell_0=0$ if $m=3$ and $\ell_0=[(m-1)/2]$ if $m\ge 4$, $[σ]$ is the integral part of $σ$. This result generalizes the author's previous result which appears in J. Math.\ Soc.\ Japan 47, where the theorem is proved for the odd dimensional cases $m\ge 3$ and several applications such as $L^p$-decay of solutions of the Cauchy problems for time-dependent Schrodinger equations and wave equations with potentials, and the $L^p$-boundedness of Fourier multiplier in generalized eigenfunction expansions are given.
雑誌名
Journal of mathematical sciences, the University of Tokyo
巻
2
号
2
ページ
311 - 346
発行年
1995
ISSN
13405705
書誌レコードID
AA11021653
フォーマット
application/pdf
日本十進分類法
415
Mathematical Reviews Number
MR1366561
Mathmatical Subject Classification
47A40(MSC1991)
35P25(MSC1991)
81Uxx(MSC1991)
出版者
Graduate School of Mathematical Sciences, The University of Tokyo
原稿受領日
1994-09-19
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The $W^{k,p}$-continuity of wave operators for Schrödinger operators III, even dimensional cases $m\geq4$
jms020204.pdf
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