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Asymptotic completeness for short-range N-body systems revisited

Erik Skibsted

Abstract

We review Yafaev's approach to asymptotic completeness for systems of particles mutually interacting with short-range potentials. The theory is based on computation of commutators with time-independent (mostly bounded) observables yielding a sufficient supply of Kato smoothness bounds.

Asymptotic completeness for short-range N-body systems revisited

Abstract

We review Yafaev's approach to asymptotic completeness for systems of particles mutually interacting with short-range potentials. The theory is based on computation of commutators with time-independent (mostly bounded) observables yielding a sufficient supply of Kato smoothness bounds.

Paper Structure

This paper contains 16 sections, 10 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and organization of paper
  3. $N$-body Hamiltonians, assumptions and notation
  4. Generalized $N$-body short-range Hamiltonians
  5. General notation
  6. More notation
  7. Basic $N$-body scattering theory
  8. Yafaev's constructions
  9. Yafaev's homogeneous functions
  10. Yafaev's observables and their commutator properties
  11. Controlling the commutators $\mathrm{i}[ H,M_a]$
  12. Localization and more calculus properties
  13. Asymptotic completeness
  14. Partition of unity
  15. Proof of Lemma \ref{['lemma:asympt-compl']}
  16. ...and 1 more sections

Key Result

Lemma 3.1

For any $a\in{\mathcal{A}}'$ and $\varepsilon\in (0,1)$

Theorems & Definitions (16)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3: Ya1Sk2
  • Theorem 4.1
  • Lemma 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • ...and 6 more