Scattering Spaces and a Decomposition of Continuous Spectral Subspace

We introduce the notion of scattering space $S_b^r$ for $N$-body quantum mechanical systems, where $b$ is a cluster decomposition with $2\le \mathrm{\mid }b\mathrm{\mid }\le N$ and $r$ is a real number $0\le r\le 1$. Utilizing these spaces, we give a decomposition of continuous spectral subspace by $S_b^1$ for $N$-body quantum systems with long-range pair potentials $V_{\alpha }^L(x_{\alpha })=O(\mathrm{\mid }{x}_{\text{al}}{\mathrm{\mid }}^{-\text{ep}})$. This is extended to a decomposition by $S_b^r$ with $0\le r\le 1$ for some long-range case. We also prove a characterization of ranges of wave operators by $S_b^0$.