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2016 Wave Operators and Similarity for Long Range $N$-body Schrödinger Operators
Hitoshi Kitada
Commun. Math. Anal. 19(1): 6-66 (2016).

Abstract

We consider asymptotic behavior of $e^{-itH}f$ for $N$-body Schrödigner operator $H=H_{0}+\sum_{1 \leq i < j \leq N } V_{ij}(x)$ 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.

Citation

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Hitoshi Kitada. "Wave Operators and Similarity for Long Range $N$-body Schrödinger Operators." Commun. Math. Anal. 19 (1) 6 - 66, 2016.

Information

Published: 2016
First available in Project Euclid: 17 February 2016

zbMATH: 1333.81423
MathSciNet: MR3439529

Subjects:
Primary: 35J10, 35P25, 47A40, 81U10

Rights: Copyright © 2016 Mathematical Research Publishers

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Vol.19 • No. 1 • 2016
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