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：不可公度结构源于将低维材料的单层以错位方式（例如面内扭转取向）堆叠而成。尽管这些结构具有重要的物理意义，但由于周期性丧失，它们带来了诸多理论挑战。本文在弱意义下刻画了不可公度体系中薛定谔算子的态密度，并发展了新的数值逼近方法。具体而言，我们：(i) 在实空间表述中严格证明了态密度的热力学极限；(ii) 提出了基于平面波近似和倒易空间采样的高效数值格式用于计算态密度。我们通过严谨的理论分析与数值模拟共同验证了所提数值算法的可靠性与高效性。