We study the single electron model of a semi-infinite graphene sheet interfaced with the vacuum and terminated along a zigzag edge. The model is a Schroedinger operator acting on L2(R2)\documentclass[12pt]{minimal}
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\begin{document}$$L^2(\mathbb {R}^2)$$\end{document}: Hedgeλ=-Δ+λ2V♯\documentclass[12pt]{minimal}
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\begin{document}$$H^\lambda _{\mathrm{edge}}=-\Delta +\lambda ^2 V_\sharp $$\end{document}, with a potential V♯\documentclass[12pt]{minimal}
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\begin{document}$$V_\sharp $$\end{document} given by a sum of translates an atomic potential well, V0\documentclass[12pt]{minimal}
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\begin{document}$$V_0$$\end{document}, of depth λ2\documentclass[12pt]{minimal}
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\begin{document}$$\lambda ^2$$\end{document}, centered on a subset of the vertices of a discrete honeycomb structure with a zigzag edge. We give a complete analysis of the low-lying energy spectrum of Hedgeλ\documentclass[12pt]{minimal}
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\begin{document}$$H^\lambda _{\mathrm{edge}}$$\end{document} in the strong binding regime (λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} large). In particular, we prove scaled resolvent convergence of Hedgeλ\documentclass[12pt]{minimal}
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\begin{document}$$H^\lambda _{\mathrm{edge}}$$\end{document} acting on L2(R2)\documentclass[12pt]{minimal}
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\begin{document}$$L^2(\mathbb {R}^2)$$\end{document}, to the (appropriately conjugated) resolvent of a limiting discrete tight-binding Hamiltonian acting in l2(N0;C2)\documentclass[12pt]{minimal}
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\begin{document}$$l^2(\mathbb {N}_0;\mathbb {C}^2)$$\end{document}. We also prove the existence of edge states: solutions of the eigenvalue problem for Hedgeλ\documentclass[12pt]{minimal}
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\begin{document}$$H^\lambda _{\mathrm{edge}}$$\end{document} which are localized transverse to the edge and pseudo-periodic plane-wave like parallel to the edge. These edge states arise from a “flat-band” of eigenstates of the tight-binding model.
