Convergence of the Planewave Approximations for Quantum Incommensurate Systems

Explore a lecture on the convergence of planewave approximations for quantum incommensurate systems. Delve into the numerical approximations of spectrum distribution for Schrödinger operators in incommensurate systems, focusing on the density of states. Examine the thermodynamic limit justification, planewave approximation methods with novel energy cutoffs, and convergence analysis with error estimates. Discover an efficient algorithm for evaluating density of states through reciprocal space sampling. Follow the progression from periodic structures to incommensurate systems, exploring Schrödinger-type eigenvalue problems, band structures, and matrix structures. Investigate supercell approximations, planewave discretizations, and higher-dimensional formulations. Conclude with practical examples demonstrating the convergence of planewave approximations and a comprehensive summary of the topic.

Intro
Periodic structure
Incommensurate structure: definition
Incommensurate structure: 1D example
2D materials: periodic vs. incommensurate structures
Schrödinger-type eigerwalue problems
Example of periodic systems: Spectrum on supercels
Band structure and density of States
Matrix structure: Periodic vs. Incommensurate
Thermodynamic limit in real space
Supercell approximations for incommensurate systems
Plane wave discretizations
Higher dimensional formulations
Plane wave cutols
Convergence of the plane wave approximations
Accelerate the convergence by sampling
Example 1: Convergence of plane wave approximations
Summary