Scattering, Homogenization, and Interface Effects for Oscillatory Potentials with Strong Singularities

We study one-dimensional scattering for a decaying potential with rapid periodic oscillations and strong localized singularities. In particular, we consider the Schrödinger equation Hǫ ψ ≡ ` −∂ x + V0(x) + q (x, x/ǫ) ́ ψ = kψ for k ∈ R and ǫ ≪ 1. Here, q(·, y + 1) = q(·, y), has mean zero and |V0(x) + q(x, ·)| → 0 as |x| → ∞. The distorted plane waves of Hǫ are solutions of the form: eV ǫ±(x; k) = e±ikx + u±(x; k), u s ± outgoing as |x| → ∞. We derive their ǫ small asymptotic behavior, from which the asymptotic behavior of scattering quantities such as the transmission coefficient, tǫ(k), follow. Let thom 0 (k) denote the homogenized transmission coefficient associated with the average potential V0. If the potential is smooth, then classical homogenization theory gives asymptotic expansions of, for example, distorted plane waves, and transmission and reflection coefficients. Singularities of V0 or discontinuities of qǫ are “interfaces” across which a solution must satisfy interface conditions (continuity or jump conditions). To satisfy these conditions it is necessary to introduce interface correctors, which are highly oscillatory in ǫ. Our theory admits potentials which have discontinuities in the microstructure, qǫ(x) as well as strong singularities in the background potential, V0(x). A consequence of our main results is that tǫ(k)− thom 0 (k), the error in the homogenized transmission coefficient is (i) O(ǫ) if qǫ is continuous and (ii) O(ǫ) if qǫ has discontinuities. Moreover, in the discontinuous case the correctors are highly oscillatory in ǫ, i.e. ∼ exp(2πi ν ǫ ), for ǫ ≪ 1. Thus a first order corrector is not well-defined since ǫ−1 ` tǫ(k)− thom 0 (k) ́ does not have a limit as ǫ → 0. This expression may have limits which depend on the particular sequence through which ǫ tends to zero. The analysis is based on a (pre-conditioned) Lippman-Schwinger equation, introduced in [9].