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 $L^2(\mathbb{R}^2)$: $H^\lambda_{\rm edge}=-\Delta+\lambda^2 V_\sharp$, with a potential $V_\sharp$ given by a sum of translates an atomic potential well, $V_0$, of depth $\lambda^2$, 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 $H^\lambda_{\rm edge}$ in the strong binding regime ($\lambda$ large). In particular, we prove scaled resolvent convergence of $H^\lambda_{\rm edge}$ acting on $L^2(\mathbb{R}^2)$, to the (appropriately conjugated) resolvent of a limiting discrete tight-binding Hamiltonian acting in $l^2(\mathbb{N}_0;\mathbb{C}^2)$. We also prove the existence of {\it edge states}: solutions of the eigenvalue problem for $H^\lambda_{\rm edge}$ which are localized transverse to the edge and pseudo-periodic (propagating or plane-wave like) parallel to the edge. These edge states arise from a "flat-band" of eigenstates the tight-binding Hamiltonian.

中文翻译：

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尖锐终止类石墨烯结构的连续薛定谔算子

我们研究了与真空接口并沿锯齿形边缘终止的半无限石墨烯片的单电子模型。该模型是作用于 $L^2(\mathbb{R}^2)$ 的 Schroedinger 算子：$H^\lambda_{\rm edge}=-\Delta+\lambda^2 V_\sharp$，具有潜在的 $由总和给出的 V_\sharp$ 平移原子势阱 $V_0$，深度为 $\lambda^2$，以具有锯齿形边缘的离散蜂窝结构的顶点子集为中心。我们对强结合区（$\lambda$ 大）中$H^\lambda_{\rm edge}$ 的低能谱进行了完整分析。特别是，我们证明了 $H^\lambda_{\rm edge}$ 作用于 $L^2(\mathbb{R}^2)$ 的缩放解析解收敛到有限离散紧的解析解（适当共轭）约束哈密顿量作用于 $l^2(\mathbb{N}_0;\mathbb{C}^2)$。我们还证明了 {\it 边缘状态}的存在：$H^\lambda_{\rm edge}$ 的特征值问题的解决方案，其横向于边缘和伪周期（传播或平面波状）平行到边缘。这些边缘态来自紧束缚哈密顿量本征态的“平带”。