Second-order homogenization of periodic Schr\"odinger operators with highly oscillating potentials

TL;DR

This paper develops a second-order homogenization theory for periodic Schrödinger operators with highly oscillating potentials, improving approximation accuracy for solutions and physical observables like the density of states.

Contribution

It introduces a second-order homogenization framework for Schrödinger operators with oscillating potentials, including numerical validation of correction effectiveness.

Findings

Second-order corrections significantly improve approximation accuracy.

Corrections are effective even for moderate oscillation scales.

The approach applies to both eigenvalue problems and physical observables.

Abstract

We consider the homogenization at second-order in $\varepsilon$ of $\mathbb{L}$-periodic Schr\"odinger operators with rapidly oscillating potentials of the form $H^\varepsilon =-\Delta + \varepsilon^{-1} v(x,\varepsilon^{-1}x ) + W(x)$ on $L^2(\mathbb{R}^d)$, where $\mathbb{L}$ is a Bravais lattice of $\mathbb{R}^d$, $v$ is $(\mathbb{L} \times \mathbb{L})$-periodic, $W$ is $\mathbb{L}$-periodic, and $\varepsilon \in \mathbb{N}^{-1}$. We treat both the linear equation with fixed right-hand side and the eigenvalue problem, as well as the case of physical observables such as the integrated density of states. We illustrate numerically that these corrections to the homogenized solution can significantly improve the first-order ones, even when $\varepsilon$ is not small.