Optimally truncated WKB approximation for the highly oscillatory stationary 1D Schrödinger equation

Abstract: We discuss the numerical solution of initial value problems for $\varepsilon<sup>2\,\varphi&#39;&#39;+a(x)\,\varphi=0$ in the highly oscillatory regime, i.e., with $a(x)&gt;0$ and $0&lt;\varepsilon\ll 1$. We analyze and implement an approximate solution based on the well-known WKB-ansatz. The resulting approximation error is of magnitude $\mathcal{O}(\varepsilon<sup>{N})$ where $N$ refers to the truncation order of the underlying asymptotic series. When the optimal truncation order $N_{opt}$ is chosen, the error behaves like $$\mathcal{O}(\varepsilon<sup>{-2}\exp(-c\varepsilon<sup>{-1}))$$ with some $c&gt;0$.