Total positivity and spectral properties of linearized operators

TL;DR

This paper provides criteria based on the symbol and symmetry of solutions to verify spectral properties of linearized operators in semilinear elliptic equations, aiding stability analysis of solitary waves.

Contribution

It introduces verifiable criteria involving the symbol and symmetry that do not require explicit solutions, advancing spectral analysis methods.

Findings

Criteria for spectral assumptions based on the symbol and symmetry

Applicable to a class of semilinear elliptic equations

Facilitates stability analysis without explicit solutions

Abstract

For a class of semilinear elliptic equations, we establish criteria that guarantee that the linearized operator associated with a solution satisfies certain spectral assumptions that are widely used in the analysis of the stability of solitary waves. The criteria only involve the symbol of the linear operator and positivity and symmetry of the solution, and can therefore be verified without an explicit formula for the solution.