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

Abstract

We consider the homogenization at second-order in ε of Lperiodic Schrödinger operators with rapidly oscillating potentials of the formWe 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 ε is not small.

Linear elliptic homogenization for a class of highly oscillating non-periodic potentials

“…In particular, this prevents us from using classical techniques (the Lax-Milgram lemma for instance) to solve this equation. A natural approach to show the existence of a solution to (12) consists in finding a solution of the form w = G * V , where G denotes the Green function associated with ∆ on R d . For d ≥ 3, it is well-known that G is of the form G(x) = C(d)…”

“…To address the question related to the existence of a solution w to (12), our approach first consists in using the specific structure of V , that is a perturbation of the periodic potential ( 8) by (9) in order to find a corrector of the form…”

“…Our twofold purpose here is to motivate our assumptions (A1) through (A3) and to emphasize the difficulties related to the corrector equation (12). We first address a particular illustrative case in dimension d = 1 and we next introduce our approach to solve (12) for higher dimensions.…”

“…In this specific case, the derivative of the solution w to (12), which reads here as w = V , is explicitly given by…”

“…The sequence of remainders R ε = u ε − u * − εu * w per (./ε) is shown to strongly converge to 0 in H 1 (Ω). The convergence of the eigenvalues of −∆+ 1 ε V (./ε)+ν is also studied in [25,12]. Respectively denoting by λ ε l and λ * l the lowest l th eigenvalue (counting multiplicities) of −∆ + 1 ε V (./ε) + ν and −∆ + w per V per + ν, the results of [25,Theorem 1.4] show the existence of a constant C > 0 such that |λ ε l − λ * l | ≤ C|λ * l | 3/2 ε.…”