Scattering and Localization Properties of Highly Oscillatory Potentials
Scattering and Localization Properties of Highly Oscillatory Potentials
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高振荡势的散射和局域化特性
DOI：
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发表时间：
2012
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通讯作者：
M. Weinstein
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作者：
V. Duchêne;Iva Vuki'cevi'c;M. Weinstein
我们研究了具有快速振荡和空间局域化电位q ε = q （x, x / ε）的一维Schrödinger方程的散射、局域化和色散时间衰减特性，其中q （x, y）是周期性的，相对于y平均为零。这种电位模拟了微观结构介质。均匀化理论不能正确地捕捉低能量（k小）散射量的行为，例如，当传输系数t q ε (k)趋于零时。我们推导出一个有效势阱σeffε (x) = - ε 2 Λ eff (x)，使得t q ε (k) - t σeffε (k)很小，对于k∈R以及在合适复带的任何有界子集上都是一致的。在这样一个有界的子集内，传输系数的缩放极限具有一个通用形式，依赖于一个参数，该参数可由有效势计算。结果是，如果微观结构电位的振荡尺度λ足够小，那么在虚轴上的上半平面上，在距离0的ε 2阶处存在透射系数的极点（因此也存在解析系数的极点）。由此可以得出Schrödinger算子H q ε =−∂x 2 + q ε (x)具有一个l2束缚态，其负能量位于距离连续谱边缘O （ε 4）处。最后,我们使用这个详细的信息来证明局部能量衰减估计:| (1 + | · | ) −−3 e i t H问e P L cψ0 |∞≤* c t−1 / 2 (1 + e 4 R (L∫∫Λeff) 2 t)−1 | |·|(1 + 3)ψ0 | L 1, P c表示投影H问ε连续光谱的一部分。©2013 Wiley期刊公司
We investigate scattering, localization, and dispersive time decay properties for the one‐dimensional Schrödinger equation with a rapidly oscillating and spatially localized potential q ε = q ( x , x / ε ) , where q ( x , y ) is periodic and mean zero with respect to y. Such potentials model a microstructured medium. Homogenization theory fails to capture the correct low‐energy (k small) behavior of scattering quantities, e.g., the transmission coefficient t q ε ( k ) as ∊ tends to zero. We derive an effective potential well σeffε ( x ) = − ε 2 Λ eff ( x ) such that t q ε ( k ) − t σeffε ( k ) is small, uniformly for k ∈ R as well as in any bounded subset of a suitable complex strip. Within such a bounded subset, the scaled limit of the transmission coefficient has a universal form, depending on a single parameter, which is computable from the effective potential. A consequence is that if ϵ, the scale of oscillation of the microstructure potential, is sufficiently small, then there is a pole of the transmission coefficient (and hence of the resolvent) in the upper half‐plane on the imaginary axis at a distance of order ε 2 from 0 . It follows that the Schrödinger operator H q ε = − ∂ x 2 + q ε ( x ) has an L 2 bound state with negative energy situated a distance O ( ε 4 ) from the edge of the continuous spectrum. Finally, we use this detailed information to prove the local energy time decay estimate: | ( 1 + | · | ) − 3 e − i t H q e P c ψ 0 | L ∞ ≤ * C t − 1 / 2 ( 1 + e 4 ( ∫ l ∫ R Λ eff ) 2 t ) − 1 | ( 1 + | · | 3 ) ψ 0 | L 1 , where P c denotes the projection onto the continuous spectral part of H q ε . © 2013 Wiley Periodicals, Inc.