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

Abstract

We study one-dimensional scattering for a decaying potential with rapid periodic oscillations and strong localized singularities. In particular, we consider the Schrödinger equation \[ H_εψ:= (-\partial_x^2 + V_0(x) + q(x,x/ε)) ψ= k^2 ψ, \] 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: $e^{\pm ikx}+u_{\pm }^s(x;k)$, $u_{\pm }^s$ 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].

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

References in corpus (1)

• Wave operator bounds for 1-dimensional Schrödinger operators with singular potentials and applications

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