Scattering, homogenization and interface effects for oscillatory potentials with strong singularities

TL;DR

This paper analyzes one-dimensional scattering for oscillatory potentials with singularities, deriving asymptotic behaviors of scattering quantities and establishing error bounds for homogenized transmission coefficients, especially in the presence of interface discontinuities.

Contribution

It develops a theory for scattering with singular potentials, including interface correctors and precise error estimates for homogenized transmission coefficients.

Findings

Error in transmission coefficient is O(ε^2) for continuous q_ε.

Error is O(ε) when q_ε has discontinuities.

Highly oscillatory correctors are needed for interface conditions.

Abstract

We study one-dimensional scattering for a decaying potential with rapid periodic oscillations and strong localized singularities. In particular, we consider the Schr\"odinger equation \[ H_\epsilon \psi := (-\partial_x^2 + V_0(x) + q(x,x/\epsilon)) \psi = k^2 \psi, \] for $k\in\RR$ and $\epsilon << 1$. Here, $q(.,y+1)=q(.,y)$, has mean zero and $|V_0(x)+q(x,.)|$ goes to zero as $|x|$ goes to infinity. The distorted plane waves of $H_\epsilon$ are solutions of the form: $e^{\pm ikx}+u^s_\pm(x;k)$, $u^s_\pm$ outgoing as $|x|$ goes to infinity. We derive their $\epsilon$ small asymptotic behavior, from which the asymptotic behavior of scattering quantities such as the transmission coefficient, $t^\epsilon (k)$, follow. Let $t_0^{hom}(k)$ denote the homogenized transmission coefficient associated with the average potential $V_0$. 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 $V_0$ or discontinuities of $q_\epsilon $, that our theory admits, 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 $\epsilon$. A consequence of our main results is that $t^\epsilon (k)-t_0^{hom}(k)$, the error in the homogenized transmission coefficient is (i) $O(\epsilon ^2)$ if $q_\epsilon $ is continuous and (ii) $O(\epsilon)$ if $q_\epsilon $ has discontinuities. Moreover, in the discontinuous case the correctors are highly oscillatory in $\epsilon$, so that a first order corrector is not well-defined. The analysis is based on a (pre-conditioned) Lippman-Schwinger equation, introduced in [SIAM J. Mult. Mod. Sim. (3), 3 (2005), pp. 477--521].