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

TL;DR

This paper studies the long-term behavior of quantum N-body systems with mixed long- and short-range interactions, introducing scattering spaces and proving asymptotic completeness of wave operators for certain potentials.

Contribution

It introduces the concept of scattering spaces for classifying initial states and proves asymptotic completeness for long-range potentials with decay rate greater than 1/2.

Findings

Decomposition theorem for the continuous spectral subspace.

Asymptotic completeness of wave operators for specific long-range potentials.

Classification of initial states based on asymptotic behavior.

Abstract

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.