  This paper is a mathematical analysis of conduction effects at interfaces
between insulators. Motivated by work of Haldane-Raghu , we continue the study
of a linear PDE initiated in papers of Fefferman-Lee-Thorp-Weinstein. This PDE
is induced by a continuous honeycomb Schrodinger operator with a line defect.
  This operator exhibits remarkable connections between topology and spectral
theory. It has essential spectral gaps about the Dirac point energies of the
honeycomb background. In a perturbative regime, Fefferman-Lee-Thorp-Weinstein
construct edge states: time-harmonic waves propagating along the interface,
localized transversely. At leading order, these edge states are adiabatic
modulations of the Dirac point Bloch modes. Their envelops solve a Dirac
equation that emerges from a multiscale procedure.
  We develop a scattering-oriented approach that derives all possible edge
states, at arbitrary precision. The key component is a resolvent estimate
connecting the Schrodinger operator to the emerging Dirac equation. We discuss
topological implications via the computation of the spectral flow, or edge
index.
