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

TL;DR

This paper proves that the density of states for a discrete tight-binding model with a slowly-varying periodic potential converges to that of its continuum limit, modeled by a Mathieu-type equation, bridging discrete and continuous spectral analysis.

Contribution

It establishes the convergence of the density of states from a discrete model to a continuum Mathieu-type equation, providing a rigorous link between discrete and continuous spectral theories.

Findings

Density of states converges from discrete to continuum model

Mathematical proof of convergence for slowly-varying potentials

Connects tight-binding models with Mathieu equations

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.