Rigorous WKB for finite order linear recurrence relations with smooth coefficients

Details Rigorous WKB for finite order linear recurrence relations with smooth coefficients Rigorous WKB for finite order linear recurrence relations with smooth coefficients

2006-08-16

arxiv.org/abs/math/0608413v1

SIAM.J.Math.Anal. v27 no1 (1996)

Mathematics

Classical Analysis and ODEs

Scientific paper

We study the $\epsilon \to 0$ behavior of recurrence relations of the type $\sum_{j=0}^l a_j(k\epsilon,\epsilon)y_{k+j}=0,$ $k\in \zdd$ ($l$ 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 $\{\exp(\epi F_m(k\epsilon,\epsilon))\}_{m=1... l}$ 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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