Periodic quantum graphs from the Bethe–Sommerfeld perspective
Periodic quantum graphs from the Bethe–Sommerfeld perspective
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贝特-索末菲视角下的周期量子图
DOI：
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发表时间：
2017
期刊：
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通讯作者：
Ondřej Turek
中科院分区：
文献类型：
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作者：
P. Exner;Ondřej Turek
本文研究了周期量子图的谱中的开间隙数。Bethe和Sommerfeld(1933)的著名猜想说，一个系统在一个以上方向上的周期开放光谱带隙的数量是有限的。到目前为止，它的有效性被建立在许多系统中，然而，我们知道量子图不符合这一定律，因为它们的光谱通常有无限多的间隙，或者根本没有间隙。这些事实引出了关于量子图是否存在的问题，这些量子图具有“Bethe-Sommerfeld性质”，即光谱中具有非零的有限数目的间隙。在这篇文章中，我们证明了对于具有标度不变或以特定方式与标度不变的顶点耦合的图，上述性质是不可能的。另一方面，我们证明了具有有限个开放间隙的量子图确实存在。我们以一个矩形格子为例说明了这一现象，该格子的顶点具有δ耦合，边的比例是适当的无理。我们的结果允许人们显式地找到具有任意指定精确间隙数目的量子图，这是迄今为止第一个这样的例子。
The paper is concerned with the number of open gaps in spectra of periodic quantum graphs. The well-known conjecture by Bethe and Sommerfeld (1933) says that the number of open spectral gaps for a system periodic in more than one direction is finite. To date, its validity is established for numerous systems, however, it is known that quantum graphs do not comply with this law as their spectra have typically infinitely many gaps, or no gaps at all. These facts gave rise to the question about the existence of quantum graphs with the ‘Bethe–Sommerfeld property’, that is, featuring a nonzero finite number of gaps in the spectrum. In this paper we prove that the said property is impossible for graphs with vertex couplings which are either scale-invariant or associated to scale-invariant ones in a particular way. On the other hand, we demonstrate that quantum graphs with a finite number of open gaps do indeed exist. We illustrate this phenomenon on an example of a rectangular lattice with a δ coupling at the vertices and a suitable irrational ratio of the edges. Our result allows one to find explicitly a quantum graph with any prescribed exact number of gaps, which is the first such example to date.