Continuum Schroedinger operators for sharply terminated graphene-like structures

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\(mathbb{R}²\): H^λ_{rm edge}=-Δ+λ² V_sharp, with a potential V_sharp given by a sum of translates an atomic potential well, V₀, of depth λ², 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^λ_{rm edge} in the strong binding regime (λ large). In particular, we prove scaled resolvent convergence of H^λ_{rm edge} acting on L²\(mathbb{R}²\), to the (appropriately conjugated) resolvent of a limiting discrete tight-binding Hamiltonian acting in l²\(mathbb{N}₀;mathbb{C}²\). We also prove the existence of {\it edge states}: solutions of the eigenvalue problem for H^λ_{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.