Edges of bands of continuous spectrum of periodic structures arise as maxima and minima of the dispersion relation of their Floquet–Bloch transform. It is often assumed that the extrema generating the band edges are non-degenerate. This paper constructs a family of examples of $${\mathbb {Z}}^3$$ Z 3 -periodic quantum graphs where the non-degeneracy assumption fails: the maximum of the first band is achieved along an algebraic curve of co-dimension 2. The example is robust with respect to perturbations of edge lengths, vertex conditions and edge potentials. The simple idea behind the construction allows generalizations to more complicated graphs and lattice dimensions. The curves along which extrema are achieved have a natural interpretation as moduli spaces of planar polygons.

中文翻译：

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周期性量子图中的简并带边

周期性结构的连续谱带的边缘作为其 Floquet-Bloch 变换的色散关系的最大值和最小值出现。通常假设产生带边缘的极值是非退化的。本文构建了一系列 $${\mathbb {Z}}^3$$ Z 3 -周期量子图的例子，其中非简并假设失败：沿着 co-维度 2。该示例在边长、顶点条件和边势的扰动方面是稳健的。构造背后的简单想法允许推广到更复杂的图和格维度。沿其达到极值的曲线具有作为平面多边形的模空间的自然解释。