Discrete-to-continuum convergence of the density of states for Mathieu's equation

2025/12/12 by Peter Hofhansel, Hofhansel, Peter, Alexander B. Watson +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Numerical Analysis (math.NA) #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2512.12039

openalex publication_date 2025/12/12 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28

Abstract

The density of states of a self-adjoint operator generalizes the eigenvalue distribution of a Hermitian matrix. We prove convergence of the density of states for a tight-binding model with a slowly-varying periodic potential to the density of states of its continuum approximation, a Mathieu-type equation.

Citations

• Higher-order continuum models for twisted bilayer graphene
• Semiclassical analysis of two-scale electronic Hamiltonians for twisted bilayer graphene
• An extended plane wave framework for the electronic structure calculations of twisted bilayer material systems
• Convergence of the Planewave Approximations for Quantum Incommensurate Systems
• Second-order homogenization of periodic Schrödinger operators with highly oscillating potentials
• Incommensurate Heterostructures in Momentum Space
• Electronic Density of States for Incommensurate Layers
• QM/MM methods for crystalline defects. Part 1: Locality of the tight binding model

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