On the gap property of a linearized NLS operator

TL;DR

This paper proves that certain linearized operators around solitons in the 3D cubic NLS have no eigenvalues in (0,1] and no resonances at the spectrum's bottom, using a new comparison-based spectral method.

Contribution

Introduces a novel comparison-based approach to analyze the spectral properties of linearized NLS operators in non-radial cases, establishing eigenvalue absence in a key interval.

Findings

No eigenvalues in (0,1] for the operators

Absence of resonances at the spectrum's bottom

Method applicable to other spectral problems

Abstract

We consider a pair of linear operators corresponding to the linearization around the ground state soliton of the cubic nonlinear Schr\"odinger 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.