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

Wave Operators and Similarity for Long Range $N$-body Schr\"odinger Operators