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

We discuss the numerical solution of initial value problems for ${\epsilon }^2{\phi }^{\mathrm{\prime }\mathrm{\prime }}+a(x)\phi =0$ in the highly oscillatory regime, i.e., with $a(x)>0$ and $0<\epsilon \ll 1$. We analyze and implement an approximate solution based on the well-known WKB-ansatz. The resulting approximation error is of magnitude $\mathcal{O}({\epsilon }^N)$ 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 $\mathcal{O}({\epsilon }^{-2}\exp (-c{\epsilon }^{-1}))$ with some $c>0$.