Defect modes for dislocated periodic media
Abstract
We study defect modes in a onedimensional 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 2dimensional honeycomb analog) Fefferman, LeeThorp 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 selfcontained 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 bulkedge correspondence for continuous dislocated systems.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 arXiv:
 arXiv:1810.05875
 Bibcode:
 2018arXiv181005875D
 Keywords:

 Mathematics  Analysis of PDEs;
 35P15;
 35P25;
 35Q40;
 35Q41
 EPrint:
 48 pages, 4 figures