Scattering resonances for highly oscillatory potentials
Scattering resonances for highly oscillatory potentials
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高振荡电势的散射共振
DOI：
10.24033/asens.2368
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
A. Drouot
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文献类型：
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作者：
A. Drouot
我们研究紧支撑势$ V_\varepsilon = W ( x, x/\varepsilon ) $的共振，其中$ W : \mathbb{R}^d \times \mathbb{R}^d / ( 2\pi \mathbb{Z}) ^d \to \mathbb{C} $, $ d $奇数。这意味着$ V_\varepsilon $是一个缓慢变化的电势$ W_0 ( x) $和一个以$1/\varepsilon$频率振荡的电势之和。对于$ W_0 \equiv 0 $，我们证明在$\text{Im} \lambda = -A \ln(\varepsilon^{-1})$线以上不存在共振，除了可能存在模量$\sim \varepsilon^2$的简单共振，当$ d=1$。我们通过构造一个一维的例子来证明这个结果是最优的。在$ W_0 \neq 0 $的情况下，我们证明了固定条的共振允许$\varepsilon$的幂次扩展。该参数提供了一种计算展开系数的方法。特别地，我们产生了一个一致收敛到$W_0$为$\varepsilon \rightarrow 0$的有效势，其共振接近于$V_\varepsilon$模$O(\varepsilon^4)$的共振。
We study resonances of compactly supported potentials $ V_\varepsilon = W ( x, x/\varepsilon ) $ where $ W : \mathbb{R}^d \times \mathbb{R}^d / ( 2\pi \mathbb{Z}) ^d \to \mathbb{C} $, $ d $ odd. That means that $ V_\varepsilon $ is a sum of a slowly varying potential, $ W_0 ( x) $, and one oscillating at frequency $1/\varepsilon$. For $ W_0 \equiv 0 $ we prove that there are no resonances above the line $\text{Im} \lambda = -A \ln(\varepsilon^{-1})$, except possibly a simple resonance of modulus $\sim \varepsilon^2$, when $ d=1$. We show that this result is optimal by constructing a one-dimensional example. In the case when $ W_0 \neq 0 $ we prove that resonances in fixed strips admit an expansion in powers of $\varepsilon$. The argument provides a method for computing the coefficients of the expansion. In particular we produce an effective potential converging uniformly to $W_0$ as $\varepsilon \rightarrow 0$ and whose resonances approach resonances of $V_\varepsilon$ modulo $O(\varepsilon^4)$.