Convergence of the Planewave Approximations for Quantum Incommensurate Systems

Incommensurate structures come from stacking the single layers of low dimensional materials on top of one another with misalignment such as a twist in orientation. While these structures are of significant physical interest, they pose many theoretical challenges due to the loss of periodicity. This paper studies the physical observables of continuum Schrodinger operators for incommensurate systems. We characterize the density of states and local density of states in real and reciprocal spaces, and develop novel numerical methods to approximate them. In particular, we (i) justify the thermodynamic limit of the density of states and local density of states via the operator kernel characterization; and (ii) propose efficient numerical schemes based on plane wave approximation and the trapezoidal rule in reciprocal space. We present both rigorous analysis and numerical simulations to support the reliability and efficiency of our numerical algorithms.