This paper is concerned with a 1D Schrödinger scattering problem involving both oscillatory and evanescent regimes, separated by jump discontinuities in the potential function, to avoid “turning points”. We derive a non-overlapping domain decomposition method to split the original problem into sub-problems on these regions, both for the continuous and afterwards for the discrete problem. Further, a hybrid WKB-based numerical method is designed for its efficient and accurate solution in the semi-classical limit: a WKB-marching method for the oscillatory regions and a FEM with WKB-basis functions in the evanescent regions. We provide a complete error analysis of this hybrid method and illustrate our convergence results by numerical tests.

中文翻译：

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半经典极限中的稳态薛定谔方程：振荡和渐逝区的数值耦合

本文涉及一维薛定谔散射问题，该问题涉及振荡和渐逝状态，由势函数中的跳跃不连续性分隔，以避免“转折点”。我们推导出一种非重叠域分解方法，将原始问题分解为这些区域上的子问题，包括连续问题和离散问题之后的子问题。此外，设计了一种基于混合 WKB 的数值方法，以便在半经典极限中高效准确地求解：振荡区域的 WKB 行进方法和渐逝区域中具有 WKB 基函数的 FEM。我们提供了这种混合方法的完整误差分析，并通过数值测试说明了我们的收敛结果。