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Dispersive Estimates for Nonlinear Schrödinger Equations with External Potentials

Charlotte Dietze

Abstract

We consider the long time dynamics of nonlinear Schrödinger equations with an external potential. More precisely, we look at Hartree type equations in three or higher dimensions with small initial data. We prove an optimal decay estimate, which is comparable to the decay of free solutions. Our proof relies on good control on a high Sobolev norm of the solution to estimate the terms in Duhamel's formula.

Dispersive Estimates for Nonlinear Schrödinger Equations with External Potentials

Abstract

We consider the long time dynamics of nonlinear Schrödinger equations with an external potential. More precisely, we look at Hartree type equations in three or higher dimensions with small initial data. We prove an optimal decay estimate, which is comparable to the decay of free solutions. Our proof relies on good control on a high Sobolev norm of the solution to estimate the terms in Duhamel's formula.

Paper Structure

This paper contains 10 sections, 15 theorems, 174 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof strategy
  4. Organisation
  5. Notations
  6. Preliminaries
  7. Dispersive estimates for $e^{-\mathrm{i} tH}$
  8. The Hartree type equation
  9. Various estimates
  10. Conclusion of the main theorem

Key Result

Theorem \oldthetheorem

Let $d\ge 3$ and let $k\in\mathbb{N}$ be the smallest even number with $k>\frac{d}{2}$. Let $V\in W^{k,\infty}(\mathbb{R}^d)$ be a real-valued function and satisfy for every $f\in L^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)$ and some constant $C^V\ge 1$. Let the interaction potential $w\in L^1(\mathbb{R}^d)\cap L^{\frac{d}{2}}(\mathbb{R}^d)$ be an even, real-valued function. Let $u_0\in H^k(\mathbb{R

Figures (1)

  • Figure 1: This graph shows the function $f:[0,\infty), f(x):=\varepsilon+Cx^3-x$ for $\varepsilon=0.1$ and $C=7$. Note that the set $\{f\ge 0\}$ consists of two disjoint closed intervals.

Theorems & Definitions (45)

  • Theorem \oldthetheorem: Dispersive estimates for the Hartree type equation in $d\ge 3$ for small initial data
  • Remark \oldthetheorem
  • Remark \oldthetheorem: Application in many-body quantum mechanics
  • Remark \oldthetheorem: Dispersive estimates for $e^{-\mathrm{i} t (-\Delta+V)}$
  • Remark \oldthetheorem: Extensions of Theorem \ref{['th:dispersive d>=3 small']}
  • Remark \oldthetheorem: Further questions
  • Theorem \oldthetheorem: Dispersive estimates for the cubic nonlinear Schrödinger equation for small initial data
  • Remark \oldthetheorem: The cubic nonlinear Schrödinger equation as a limit of Hartree type equations
  • Theorem \oldthetheorem: Dispersive estimate for $e^{-\mathrm{i} tH}$ in $d\ge 3$
  • Remark \oldthetheorem
  • ...and 35 more