Incommensurate structures arise from stacking single layers of low-dimensional materials on top of one another with misalignment such as an in-plane twist in orientation. While these structures are of significant physical interest, they pose many theoretical challenges due to the loss of periodicity. In this paper, we characterize the density of states of Schr\"{o}dinger operators in the weak sense for the incommensurate system and develop novel numerical methods to approximate them. In particular, we (i) justify the thermodynamic limit of the density of states in the real space formulation; and (ii) propose efficient numerical schemes to evaluate the density of states based on planewave approximations and reciprocal space sampling. We present both rigorous analysis and numerical simulations to support the reliability and efficiency of our numerical algorithms.

翻译：非公度结构源于低维材料单层以面内扭转等错位方式堆叠而成。此类结构具有重要物理意义，但因周期性缺失带来诸多理论挑战。本文从弱意义角度刻画非公度系统中薛定谔算子的态密度，并发展新型数值方法对其进行逼近。具体而言，我们：(i) 证明了实空间表述中态密度热力学极限的合理性；(ii) 提出了基于平面波近似与倒空间采样的高效数值格式来评估态密度。我们通过严谨的数学分析与数值模拟，验证了所提算法的可靠性与高效性。