Second-order homogenization of periodic Schrödinger operators with highly oscillating potentials

We consider the homogenization at second-order in $\varepsilon$ of $\mathbb{L}$-periodic Schrödinger 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