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

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.

44 pages, 2 figures. PH's undergraduate research project at University of Minnesota supervised by ABW