This paper introduces an efficient high-order numerical method for solving the 1D stationary Schr\"odinger equation in the highly oscillatory regime. Building upon the ideas from [2], we first analytically transform the given equation into a smoother (i.e. less oscillatory) equation. By developing sufficiently accurate quadratures for several (iterated) oscillatory integrals occurring in the Picard approximation of the solution, we obtain a one-step method that is third order w.r.t. the step size. The accuracy and efficiency of the method are illustrated through several numerical examples.

翻译：本文提出了一种高效的高阶数值方法，用于求解高度振荡区域中的一维定态薛定谔方程。基于文献[2]的思想，我们首先通过解析变换将给定方程转化为更平滑（即振荡性更弱）的方程。通过为解皮卡近似中出现的若干（迭代）振荡积分构造足够精确的求积公式，我们获得了一个关于步长为三阶的单步方法。多个数值算例验证了该方法的精度与效率。