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

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

2010-10-13

arxiv.org/abs/1010.2694v2

Physics

Mathematical Physics

To appear in SIAM Multiscale Modeling, Analysis and Simulation (2011)

Scientific paper

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].

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