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On Strichartz estimates for many-body Schrödinger equation in the periodic setting

Xiaoqi Huang, Xueying Yu, Zehua Zhao, Jiqiang Zheng

Abstract

In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori $\mathbb{T}^d$, where $d\geq 3$. The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in Bourgain-Demeter \cite{BD}. As a comparison, this result can be regarded as a periodic analogue of Hong \cite{hong2017strichartz} though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate.

On Strichartz estimates for many-body Schrödinger equation in the periodic setting

Abstract

In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori , where . The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in Bourgain-Demeter \cite{BD}. As a comparison, this result can be regarded as a periodic analogue of Hong \cite{hong2017strichartz} though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate.
Paper Structure (10 sections, 7 theorems, 51 equations)

This paper contains 10 sections, 7 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Background and Motivations
  3. The statement of main results
  4. Structure of this paper
  5. Notations
  6. Preliminaries
  7. Why does the perturbation method fail for the periodic case?
  8. The proof for Theorem 1.1
  9. Discussions for the nonlinear applications
  10. Further remarks

Key Result

Theorem 1.1

Assuming $d\geq 3$, we consider maineq and fix a finite time interval $I$. There exists a small number $\epsilon>0$ such that if the interacting potential $V$ satisfies the condition condition, then for $q = \frac{2(d+2)}{d}$ and any $\alpha \in \{1,\ldots,N \}$, we have where $x_{\alpha}$ refers to the $\alpha$-th variable and $\hat{x}_{\alpha}$ denotes the remaining $N-1$ spatial variables othe

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1: Strichartz estimate in tori
  • Remark 2.2
  • Proposition 2.3: Strichartz estimate for the many-body case
  • proof : Proof of Proposition \ref{['mbStrichartz']}
  • Lemma 2.4: Transference principle
  • ...and 8 more