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

By O. Costin, R. Costin. 2006

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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.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/95cc0edd5a08d0c7ff61d802827b4c215dbcd0eb