## Abstract

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 L^{2}(R^{2}) : Hedgeλ=-Δ+λ2V♯, with a potential V_{♯} given by a sum of translates an atomic potential well, V, of depth λ^{2}, 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 in the strong binding regime (λ large). In particular, we prove scaled resolvent convergence of Hedgeλ acting on L^{2}(R^{2}) , to the (appropriately conjugated) resolvent of a limiting discrete tight-binding Hamiltonian acting in l^{2}(N; C^{2}). We also prove the existence of edge states: solutions of the eigenvalue problem for Hedgeλ 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.

Original language | English (US) |
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Pages (from-to) | 853-945 |

Number of pages | 93 |

Journal | Communications In Mathematical Physics |

Volume | 380 |

Issue number | 2 |

DOIs | |

State | Published - Dec 1 2020 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics