Nondegeneracy and Stability of Antiperiodic Bound States for Fractional Nonlinear Schr\"odinger Equations

TL;DR

This paper investigates the existence, stability, and spectral properties of antiperiodic standing wave solutions in fractional nonlinear Schr"odinger equations, revealing nondegeneracy and symmetry features crucial for understanding their dynamics.

Contribution

It establishes the nondegeneracy of the linearized operator for antiperiodic perturbations and characterizes eigenfunctions, advancing the spectral theory of fractional Schr"odinger operators.

Findings

Linearized operator is nondegenerate for antiperiodic perturbations.

Characterization of antiperiodic ground state eigenfunctions.

Development of Sturm-Liouville oscillation theory for fractional operators.

Abstract

We consider the existence and stability of real-valued, spatially antiperiodic standing wave solutions to a family of nonlinear Schr\"odinger equations with fractional dispersion and power-law nonlinearity. As a key technical result, we demonstrate that the associated linearized operator is nondegenerate when restricted to antiperiodic perturbations, i.e. that its kernel is generated by the translational and gauge symmetries of the governing evolution equation. In the process, we provide a characterization of the antiperiodic ground state eigenfunctions for linear fractional Schr\"odinger operators on $\mathbb{R}$ with real-valued, periodic potentials as well as a Sturm-Liouville type oscillation theory for the higher antiperiodic eigenfunctions.