Wave Operators and Similarity for Long Range $N$-body Schrödinger Operators

Abstract

We consider asymptotic behavior of $e^{-itH}f$ for $N$-body Schrödinger 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.

69 pages