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On the gap property of a linearized NLS operator

Dong Li, Kai Yang

Abstract

We consider a pair of linear operators corresponding to the linearization around the ground state soliton of the cubic nonlinear Schrödinger equation in dimension three. We introduce a new comparison-based approach and rigorously prove that the interval $(0, 1]$ does not contain any eigenvalues of these operators. Furthermore we show the absence of resonances at the bottom of the essential spectrum. All the obtained results are for the fully non-radial case. The method can be adapted to many other spectral problems.

On the gap property of a linearized NLS operator

Abstract

We consider a pair of linear operators corresponding to the linearization around the ground state soliton of the cubic nonlinear Schrödinger equation in dimension three. We introduce a new comparison-based approach and rigorously prove that the interval does not contain any eigenvalues of these operators. Furthermore we show the absence of resonances at the bottom of the essential spectrum. All the obtained results are for the fully non-radial case. The method can be adapted to many other spectral problems.

Paper Structure

This paper contains 12 sections, 199 equations.

Table of Contents

  1. Introduction
  2. Sturm comparison and properties of the ground state $Q$
  3. Proof of Theorem \ref{['thMain']} and Corollary \ref{['corT1']}
  4. when $0<\lambda\le 1$, the solution must change sign
  5. After the first positive zero
  6. The case $l=0$ for the eigenvalue case of $L_+$
  7. $L_-$ has no eigenvalues in $(0,1]$
  8. Some auxiliary estimates
  9. Absence of embedded eigenvalues
  10. The approximate ground state $\widetilde{Q}$
  11. Analysis of the equation $u^{\prime\prime}=-3Q^2 u$
  12. Analysis of the equation $v^{\prime\prime}=-Q^2 v$

Theorems & Definitions (25)

  • proof
  • proof : Proof of Lemma \ref{['lem_Qa1']}
  • proof : Proof of Theorem \ref{['thMain']}
  • proof : Proof of Corollary \ref{['corT1']}
  • proof
  • proof
  • proof : Proof of Proposition \ref{['prop3.2a']}
  • proof
  • proof : Proof of Lemma \ref{['lem6.1a']}
  • proof
  • ...and 15 more