Defect modes for dislocated periodic media

TL;DR

This paper analyzes defect modes in dislocated one-dimensional periodic media, showing all such modes originate from eigenmodes of an effective Dirac operator and providing detailed eigenpair expansions.

Contribution

It proves that all defect modes in dislocated periodic systems derive from the Dirac operator eigenmodes, enhancing understanding of topological stability and bulk-edge correspondence.

Findings

All defect modes arise from Dirac eigenmodes.

Eigenpairs are expanded in powers of dislocation amplitude δ.

The work advances topological stability analysis of defect states.

Abstract

We study defect modes in a one-dimensional periodic medium with a dislocation. The model is a periodic Schrodinger operator on $\mathbb{R}$, perturbed by an adiabatic dislocation of amplitude $\delta\ll 1$. If the periodic background admits a Dirac point $-$ a linear crossing of dispersion curves $-$ then the dislocated operator acquires a gap in its essential spectrum. For this model (and its 2-dimensional honeycomb analog) Fefferman, Lee-Thorp and Weinstein constructed in previous work defect modes with energies within the gap. The bifurcation of defect modes is associated with the discrete eigenmodes of an effective Dirac operator. We improve upon this result: we show that all the defect modes of the dislocated operator arise from the eigenmodes of the Dirac operator. As a byproduct, we derive full expansions of the eigenpairs in powers of $\delta$. The self-contained proof relies on (a) resolvent estimates for the bulk operators; (b) scattering theory for highly oscillatory potentials developed by the first author. This work significantly advances the understanding of the topological stability of certain defect states, particularly the bulk-edge correspondence for continuous dislocated systems.