$n$能级散射系统在奇异极限下的指数渐近性
Exponential Asymptotics in a Singular Limit for $n$-Level Scattering Systems
摘要：
考虑了与方程$i\eps dψ(t)/dt=H(t)ψ(t)$相关的$S$ -矩阵的奇异极限$\eps\ra 0$，其中解析生成元$H(t)\in M_n(\C)$的谱是实的且对所有$t\in\R$都是非退化的.充分条件允许计算渐近公式的指数小的非对角元素的$S$ -矩阵$\eps\ra 0$ explicited和广泛的一类发电机，这些条件进行了验证的定义。这些发电机是通过发电机，其频谱表现出本征值交叉扰动，以这样一种方式，这些交叉转向避免交叉。通过复WKB方法与超渐近重整化的联合应用，证明了所导出的指数小渐近公式在指数小的相对误差范围内是有效的.最后，将这些结果应用于含时薛定谔方程的量子绝热跃迁和多通道稳态薛定谔方程的半经典散射性质的研究，从而结束了本文。这里给出的结果是以前仅为$2$-级系统所知的结果到$n$-级系统$n\geq 2$的推广.
英文摘要：
The singular limit $\eps\ra 0$ of the $S$-matrix associated with the equation $i\eps dψ(t)/dt=H(t)ψ(t)$ is considered, where the analytic generator $H(t)\in M_n(\C)$ is such that its spectrum is real and non-degenerate for all $t\in\R$. Sufficient conditions allowing to compute asymptotic formulas for the exponentially small off-diagonal elements of the $S$-matrix as $\eps\ra 0$ are explicited and a wide class of generators for which these conditions are verified is defined. These generators are obtained by means of generators whose spectrum exhibits eigenvalue crossings which are perturbed in such a way that these crossings turn to avoided crossings. The exponentially small asymptotic formulas which are derived are shown to be valid up to exponentially small relative error, by means of a joint application of the complex WKB method together with superasymptotic renormalization. The application of these results to the study of quantum adiabatic transitions in the time dependent Schrödinger equation and of the semiclassical scattering properties of the multichannel stationary Schrödinger equation closes this paper. The results presented here are a generalization to $n$-level systems, $n\geq 2$, of results previously known for $2$-level systems only.
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