We construct two types of multi-layer quantum graphs (Schrödinger operators on metric graphs) for which the dispersion function of wave vector and energy is proved to be a polynomial in the dispersion function of the single layer. This leads to the reducibility of the algebraic Fermi surface, at any energy, into several components. Each component contributes a set of bands to the spectrum of the graph operator. When the layers are graphene, AA-, AB-, and ABC-stacking are allowed within the same multi-layer structure. One of the tools we introduce is a surgery-type calculus for obtaining the dispersion function for a periodic quantum graph by joining two graphs together. Reducibility of the Fermi surface allows for the construction of local defects that engender bound states at energies embedded in the radiation continuum.

我们构造了两种类型的多层量子图（度量图上的薛定谔算子），证明波矢量和能量的色散函数是单层色散函数中的多项式。这导致代数费米面在任何能量下都可还原为几个分量。每个组件为图算子的频谱贡献一组波段。当层是石墨烯时，AA-、AB-和 ABC-堆叠允许在同一多层结构中。我们引入的工具之一是手术型微积分，用于通过将两个图连接在一起来获得周期量子图的色散函数。费米表面的可还原性允许构建局部缺陷，从而在嵌入辐射连续谱的能量下产生束缚态。