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 L-2(R-2): H-edge(lambda) = -Delta + lambda V-2(not equal), with a potential V-not equal 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-edge(lambda) in the strong binding regime (lambda large). In particular, we prove scaled resolvent convergence of H-edge(lambda) acting on L-2(R-2), to the (appropriately conjugated) resolvent of a limiting discrete tight-binding Hamiltonian acting in l(2)(N-0; C-2). We also prove the existence of edge states: solutions of the eigenvalue problem for H-edge(lambda) 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.