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Abstract
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This paper is a mathematical analysis of conduction effects at interfaces between
insulators. Motivated by work of Haldane and Raghu (2008), we continue the study
of a linear PDE initiated by Fefferman, Lee-Thorp, and Weinstein (2016). This PDE
is induced by a continuous honeycomb Schrödinger 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, and Weinstein constructed
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 envelopes 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 Schrödinger operator to the emerging Dirac equation. We discuss
topological implications via the computation of the spectral flow, or edge
index.
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Keywords
edge states, graphene, Dirac points, Schrödinger operators
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Mathematical Subject Classification 2010
Primary: 35P15
Secondary: 35P25, 35Q40, 35Q41
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Milestones
Received: 21 December 2018
Revised: 26 February 2019
Accepted: 4 April 2019
Published: 17 July 2019
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