  We consider asymptotic behavior of $e^{-itH}f$ for $N$-body Schr\"odinger
operator $H=H_0+\sum_{1\le i<j\le N}V_{ij}(x)$ with long- and short-range pair
potentials $V_{ij}(x)=V_{ij}^L(x)+V_{ij}^S(x)$ $(x\in {\mathbb R}^\nu)$ such
that $\partial_x^\alpha V_{ij}^L(x)=O(|x|^{-\delta-|\alpha|})$ and
$V_{ij}^S(x)=O(|x|^{-1-\delta})$ $(|x|\to\infty)$ with $\delta>0$. Introducing
the concept of scattering spaces which classify the initial states $f$
according to the asymptotic behavior of the evolution $e^{-itH}f$, we give a
generalized decomposition theorem of the continuous spectral subspace
${\mathcal{H}}_c(H)$ of $H$. The asymptotic completeness of wave operators is
proved for some long-range pair potentials with $\delta>1/2$ by using this
decomposition theorem under some assumption on subsystem eigenfunctions.
