Rigorous WKB for finite-order linear recurrence relations with smooth coefficients
Rigorous WKB for finite-order linear recurrence relations with smooth coefficients
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DOI：
10.1137/s0036141093248037
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发表时间：
2006-08
影响因子：
2
通讯作者：
O. Costin;R. Costin
中科院分区：
文献类型：
--
作者：
O. Costin;R. Costin
研究了递推关系$\sum\NoLimits_{j=0}，L{a_j(k\epsilon，\epsilon)y_{k+j}=0}$，$k\在{bf Z}$(L不变)中递推关系的性态。$a_j$是每个变量中的$C^\inty$函数，对于有界区间i和$\epsilon_0>；0$，在$i\Times[0，\epsilon_0]$上。在一定的正则性假设下，我们得到了这类递归解的渐近行为。在典型情况下，存在形式为${exp(-1}F_m(k\epsilon，\epsilon))}{m=1，ldots，L}$的基本解集，其中函数$F_m$是与$a_j$在同一区域上的每个变量的$C^\inty$，特别地，证明了形式摄动级数解渐近于这些递推的真解。文中还简要讨论了它的一些应用。
The $\epsilon \to 0$ behavior of recurrence relations of the type $\sum\nolimits_{j = 0}^l {a_j (k\epsilon ,\epsilon )y_{k + j} = 0} $, $k \in {\bf Z}$ (l fixed) is studied. The $a_j $ are $C^\infty $ functions in each variable on $I \times [0,\epsilon _0 ]$ for a bounded interval I and $\epsilon _0 > 0$. 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 (\epsilon ^{ - 1} F_m (k\epsilon ,\epsilon ))} \}_{m = 1, \ldots ,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.