光滑系数有穷级线性递推关系的严格WKB
Rigorous WKB for finite order linear recurrence relations with smooth coefficients
摘要：
本文研究了$\sum_{j=0}^l a_j(kε,ε)y_{k+j}=0,$ $k\in \zdd$（$l$固定）型递归关系的$ε\to 0$行为. $a_j$是$I\times [0,\e_0]$上每个变量的$C^{\infty}$函数，用于有界区间$I$和$\e_0>0$。在一定的正则性假设下，我们找到了此类递推式解的渐进行为。在典型的情况下，存在一个基本的一组解决方案的形式$\{\exp(\epi F_m(kε,ε))\}_{m=1... l}$的功能$F_m$是$C^{\infty}$在每个变量的同一域上的$a_j$，特别是显示正式的扰动系列解决方案是渐近的真正解决方案，这些递归。并简要讨论了一些应用。
英文摘要：
We study the $ε\to 0$ behavior of recurrence relations of the type $\sum_{j=0}^l a_j(kε,ε)y_{k+j}=0,$ $k\in \zdd$ ($l$ fixed). The $a_j$ are $C^{\infty}$ functions in each variable on $I\times [0,\e_0]$ for a bounded interval $I$ and $\e_0>0$. Under certain regularity assumptions we find the asymptotic behavior of the solutions of such recurrences. In typical cases there exists a fundamental set of solutions in the form $\{\exp(\epi F_m(kε,ε))\}_{m=1... l}$ where the functions $F_m$ are $C^{\infty}$ in each variable on the same domain as the $a_j$, showing in particular that the formal perturbation-series solutions are asymptotic to true solutions of these recurrences. Some applications are also briefly discussed.
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