布里渊区内周期算子谱边的出现
On occurrence of spectral edges for periodic operators inside the Brillouin zone
摘要：
本文讨论了数学物理中周期算子谱结构中经常出现的下列问题（例如，Schroedinger、Maxwell、波导算子等）。人们是否可以通过仅使用（约化）布里渊区边界上的准动量值而不是整个布里渊区来获得正确的谱？或者，频谱的边缘必然出现在高对称点的“角”上吗？这在1D中是真实的，而在更高的维度中没有明显的原因。然而，在许多实际情况下，这似乎是正确的，这有时会导致声称这总是正确的。文献中似乎没有明确的答案，人们在社区中遇到了关于这个问题的不同意见。在本文中，从简单的离散图算子，我们构建了各种令人信服的多周期的例子表明，频谱边缘可能会发生在布里渊区的内部深处。另一方面，它也表明，在“一般”的情况下，出现在高对称点的谱边缘的情况下，小扰动是稳定的。这在某种程度上解释了为什么在许多（甚至可能是大多数）实际情况下，这一说法仍然成立。
英文摘要：
The article discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics (e.g., Schroedinger, Maxwell, waveguide operators, etc.). Is it true that one can obtain the correct spectrum by using the values of the quasimomentum running over the boundary of the (reduced) Brillouin zone only, rather than the whole zone? Or, do the edges of the spectrum occur necessarily at the set of ``corner'' high symmetry points? This is known to be true in 1D, while no apparent reasons exist for this to be happening in higher dimensions. In many practical cases, though, this appears to be correct, which sometimes leads to the claims that this is always true. There seems to be no definite answer in the literature, and one encounters different opinions about this problem in the community. In this paper, starting with simple discrete graph operators, we construct a variety of convincing multiply-periodic examples showing that the spectral edges might occur deeply inside the Brillouin zone. On the other hand, it is also shown that in a ``generic'' case, the situation of spectral edges appearing at high symmetry points is stable under small perturbations. This explains to some degree why in many (maybe even most) practical cases the statement still holds.
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