  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.
