Second-Order Homogenization of Periodic Schrödinger Operators with Highly Oscillating Potentials
Éric Cancès, Louis Garrigue, David Gontier · SIAM Journal on Mathematical Analysis · 2023
Abstract. We consider a family of [Formula: see text]-periodic Schrödinger operators with rapidly oscillating potentials [Formula: see text] on [Formula: see text], where [Formula: see text] is a Bravais lattice of [Formula: see text], the two-scale potential [Formula: see text] is [Formula: see text]-periodic with zero mean with respect to its second variable, [Formula: see text] is [Formula: see text]-periodic, and [Formula: see text]. We construct correctors at second-order in [Formula: see text] for both the linear equation with fixed right-hand side and the eigenvalue problem and use these correctors to obtain accurate estimates on physical observables such as the integrated density of states. We illustrate numerically that second-order corrections to the homogenized solution can significantly improve the first-order ones, even when [Formula: see text] is not small.