Embedded Eigenvalues and the Nonlinear Schrodinger Equation

TL;DR

This paper proves the absence of embedded eigenvalues and certain resonances in the spectrum of linearized operators for various nonlinear Schrödinger equations, aiding the stability analysis of solitary waves.

Contribution

It extends previous work by establishing spectral properties for a broader class of nonlinear Schrödinger equations, including supercritical and cubic-quintic cases, using computer-assisted proofs.

Findings

No embedded eigenvalues in the spectrum for the considered equations.

Nonzero eigenvalues within the spectral gap are ruled out.

Endpoint resonances are excluded in 3D cases.

Abstract

A common challenge to proving asymptotic stability of solitary waves is understanding the spectrum of the operator associated with the linearized flow. The existence of eigenvalues can inhibit the dispersive estimates key to proving stability. Following the work of Marzuola & Simpson, we prove the absence of embedded eigenvalues for a collection of nonlinear Schrodinger equations, including some one and three dimensional supercritical equations, and the three dimensional cubic-quintic equation. Our results also rule out nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonances. The proof is computer assisted as it depends on the sign of certain inner products which do not readily admit analytic representations. Our source code is available for verification at http://www.math.toronto.edu/simpson/files/spec_prop_asad_simpson_code.zip.