标题： 周期弹性梁晶格的频谱：单层图 显示英文标题

标题： On the Spectra of Periodic Elastic Beam Lattices: Single-Layer Graph

摘要： 我们给出了定义在周期性六边形弹性格点上的哈密顿量的谱的完整描述。这些连续介质由欧拉-伯努利梁构成，每根梁都服从一个标量自伴算子的控制，该算子也被称为四阶薛定谔算子，并配备了一个实值周期对称势。与量子图文献中常用的二阶薛定谔算子相比，在这里顶点匹配条件通过边相遇时的角度依赖性来编码潜在图的几何结构。我们证明了对于一种特殊的等角晶格（即石墨烯），色散关系具有与六边形晶格上周期性二阶薛定谔算子所报告的类似结构。这一特性随后被进一步利用来证明奇异狄拉克点的存在。我们还讨论了这种特殊晶格在无穷多个低能级下费米面的可约性。通过摄动分析，我们将发展出的理论扩展到推导出在石墨烯附近的几何邻域内，六边形晶格角度扰动哈密顿量的色散关系。在这些图中，与石墨烯不同，色散关系并非完全分为纯能量和准动量相关的项，但存在类似于石墨烯情况下的奇异狄拉克点。 显示英文摘要

摘要： We present full description of spectra for a Hamiltonian defined on periodic hexagonal elastic lattices. These continua are constructed out of Euler-Bernoulli beams, each governed by a scalar-valued self-adjoint operator, which is also known as the fourth order Schr\"{o}dinger operator, equipped with a real periodic symmetric potential. In contrast to the second order Schr\"{o}dinger operator commonly applied in quantum graph literature, here vertex matching conditions encode geometry of the underlying graph by their dependence on angles at which edges are met. We show that for a special equal angle lattice, known as graphene, dispersion relation has a similar structure as reported for the periodic second order Schr\"{o}dinger operator on hexagonal lattices. This property is then further utilized to prove existence of singular Dirac points. We further discuss reducibility of Fermi surface at uncountably many low-energy levels for this special lattice. Applying perturbation analysis, we extend the developed theory to derive dispersion relation for angle-perturbed Hamiltonian of hexagonal lattices in a geometric neighborhood of graphene. In these graphs, unlike graphene, dispersion relation is not splitted into purely energy and quasimomentum dependent terms, however singular Dirac points exist similar to the graphene case.