## The $W^{k,p}$-continuity of wave operators for SchrÃ¶dinger operators III, even dimensional cases $m\geq4$

J. Math. Sci. Univ. Tokyo
Vol. 2 (1995), No. 2, Page 311--346.

Yajima, Kenji
The $W^{k,p}$-continuity of wave operators for SchrÃ¶dinger operators III, even dimensional cases $m\geq4$
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Abstract:
Let $H=-Î+V(x)$ be the SchrÃ¶dinger operator on ${\bf R}^m$, $m\ge 3$. We show that the wave operators $W_\pm=\lim_{t\to\pm\infty}e^{itH}\cdot e^{-itH_0}$, $H_0=-Î$, are bounded in Sobolev spaces $W^{k, p}({\bf R}^m)$, $1\le p\le\infty$, $k=0, 1, \ldots, \ell$, if $V$ satisfies $\|D^Î± V(y)\|_{L^{p_0}(|x-y|\le 1)}\le C(1+|x|)^{-Î´}$ for $Î´>(3m/2)+1$, $p_0>m/2$ and $|Î±|\le\ell+\ell_0$, where $\ell_0=0$ if $m=3$ and $\ell_0=[(m-1)/2]$ if $m\ge 4$, $[Ï]$ is the integral part of $Ï$. 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 SchrÃ¶dinger equations and wave equations with potentials, and the $L^p$-boundedness of Fourier multiplier in generalized eigenfunction expansions are given.

Mathematics Subject Classification (1991): Primary 47A40; Secondary 35P25, 81Uxx
Mathematical Reviews Number: MR1366561