## Communications in Mathematical Analysis

- Commun. Math. Anal.
- Volume 19, Number 1 (2016), 6-66.

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

#### 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.

#### Article information

**Source**

Commun. Math. Anal., Volume 19, Number 1 (2016), 6-66.

**Dates**

First available in Project Euclid: 17 February 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.cma/1455715853

**Mathematical Reviews number (MathSciNet)**

MR3439529

**Zentralblatt MATH identifier**

1333.81423

**Subjects**

Primary: 81U10: $n$-body potential scattering theory 35J10: Schrödinger operator [See also 35Pxx] 35P25: Scattering theory [See also 47A40] 47A40: Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx]

**Keywords**

wave operators similarity N-body Schrödinger operator long-range scattering asymptotic behavior scattering space extended micro-local analysis

#### Citation

Kitada, Hitoshi. Wave Operators and Similarity for Long Range $N$-body Schrödinger Operators. Commun. Math. Anal. 19 (2016), no. 1, 6--66. https://projecteuclid.org/euclid.cma/1455715853