周期量子图中的退化带边
Degenerate band edges in periodic quantum graphs
摘要：
周期结构的连续谱的带的边缘作为它们的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。本文构造了一类$\mathbb{Z}^3$ -周期量子图的例子，其中非简并假设失效：第一个带的最大值沿着余维为2的代数曲线达到。该示例对于边长、顶点条件和边势的扰动是鲁棒的。这个构造背后的简单思想允许推广到更复杂的图和格维数。沿其实现极值的曲线具有作为平面多边形的模空间的自然解释。
英文摘要：
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 $\mathbb{Z}^3$-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.
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