Topologically protected states in one-dimensional continuous systems and Dirac points

We study a class of periodic Schrödinger operators on R that have Dirac points. The introduction of an "edge" via adiabatic modulation of a periodic potential by a domain wall results in the bifurcation of spatially localized "edge states," associated with the topologically protected zero-energymode of an asymptotic one-dimensionalDirac operator. The bound states we construct can be realized as highly robust transverse-magnetic electromagnetic modes for a class of photonic waveguideswith a phase defect. Ourmodel captures many aspects of the phenomenon of topologically protected edge states for 2D bulk structures such as the honeycomb structure of graphene.