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Boundedness of energy for N-body Schrödinger equations with time dependent small potentials

Kenji Yajima

Abstract

We prove that Sobolev norms of solutions to time dependent Schrödinger equations for $d$-dimensional $N$-partcles interacting via time dependent two body potentials are bounded in time if certain Lebesgue norms of the potentials are small uniformly in time. The proof uses the scattering theory in the extended phase space which proves that all particles scatter freely in the remote past and far future.

Boundedness of energy for N-body Schrödinger equations with time dependent small potentials

Abstract

We prove that Sobolev norms of solutions to time dependent Schrödinger equations for -dimensional -partcles interacting via time dependent two body potentials are bounded in time if certain Lebesgue norms of the potentials are small uniformly in time. The proof uses the scattering theory in the extended phase space which proves that all particles scatter freely in the remote past and far future.
Paper Structure (8 sections, 11 theorems, 90 equations)

This paper contains 8 sections, 11 theorems, 90 equations.

Table of Contents

  1. Introduction, Results
  2. Existence and regularity of propagators
  3. Howland scheme
  4. Wave operators
  5. Proof of Theorem \ref{['th:1']}
  6. Proof of Theorem \ref{['th:1']} for the case $m=0$ and $n=1,2, \dots$
  7. Proof of Theorem \ref{['th:1']} for general $m=1,2, \dots,$
  8. Proof of Theorem \ref{['th:2']}

Key Result

Theorem 1.1

Let $n=1,2, \dots$ and $m\in {\mathbb N} \cup \{0\}$. Suppose that $V_{jk}(t,y)\in M(n)$, ${1\leq j<k\leq N}$, are factorized by $A_{jk}(t,y)\in M(n)$ and $B_{jk}(t,y)\in M(n)$, which satisfy the following conditions for any $|\alpha|\leq m$: Then, for $|\kappa|<\kappa_m$, $\kappa_m>0$ being a small constant, there exits a $C_{\kappa,m}<\infty$ such that

Theorems & Definitions (20)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Lemma 3.1: Howland
  • Remark 3.2
  • proof
  • Theorem 4.1
  • ...and 10 more