强奇异振荡势的散射、均匀化和界面效应
Scattering, homogenization and interface effects for oscillatory potentials with strong singularities
摘要：
研究了具有快速周期振荡和强局域奇点的衰减势场的一维散射。特别是，我们考虑薛定谔方程\[ H_εψ:= (-\partial_x^2 + V_0(x) + q(x,x/ε)) ψ= k^2 ψ, \]的$k\in\RR$和$ε<< 1$。这里，$q(.,y+1)=q(.,y)$的均值为零，当$|x|$趋于无穷大时，$|V_0(x)+q(x,.)|$趋于零。$H_ε$的扭曲平面波是以下形式的解：$e^{\pm ikx}+u^s_\pm(x;k)$，当$|x|$走向无穷大时，$u^s_\pm$向外输出。我们推导出它们的$ε$小的渐近行为，从该渐近行为的散射量，如透射系数，$t^ε(k)$，遵循。令$t_0^{hom}(k)$表示与平均电势$V_0$相关联的均匀化透射系数。如果势是光滑的，那么经典的均匀化理论给出了例如畸变平面波、透射系数和反射系数的渐近展开式。我们的理论承认，$V_0$的奇异性或$q_ε$的不连续性是一个解必须满足界面条件（连续性或跳跃条件）的“界面”。为了满足这些条件，有必要引入界面校正器，这是高度振荡的$ε$。我们的主要结果的一个后果是，$t^ε(k)-t_0^{hom}(k)$，在均匀化的透射系数的误差是（i）$O(ε^2)$，如果$q_ε$是连续的和（ii）$O(ε)$，如果$q_ε$有不连续。此外，在不连续的情况下，校正器在$ε$中是高度振荡的，使得一阶校正器不是很好定义的。分析是基于（预处理）李普曼-施温格方程，介绍了在[SIAM J. Mult.模块Sim. (3)，3（2005），pp. 477- 521]。
英文摘要：
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 $ε<< 1$. 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 $H_ε$ 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 $ε$ small asymptotic behavior, from which the asymptotic behavior of scattering quantities such as the transmission coefficient, $t^ε(k)$, 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 $q_ε$, 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 $ε$. A consequence of our main results is that $t^ε(k)-t_0^{hom}(k)$, the error in the homogenized transmission coefficient is (i) $O(ε^2)$ if $q_ε$ is continuous and (ii) $O(ε)$ if $q_ε$ has discontinuities. Moreover, in the discontinuous case the correctors are highly oscillatory in $ε$, 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].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
