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This paper is dedicated to the efficient numerical computation of solutionsto the 1D stationary Schr\"odinger equation in the highly oscillatory regime.We compute an approximate solution based on the well-known WKB-ansatz, whichrelies on an asymptotic expansion w.r.t. the small parameter $\varepsilon$.Assuming that the coefficient in the equation is analytic, we derive anexplicit error estimate for the truncated WKB series, in terms of $\varepsilon$and the truncation order $N$. For any fixed $\varepsilon$, this allows todetermine the optimal truncation order $N_{opt}$ which turns out to beproportional to $\varepsilon^{-1}$. When chosen this way, the resulting errorof the optimally truncated WKB series behaves like$\mathcal{O}(\varepsilon^{-2}\exp(-r/\varepsilon))$, with some parameter $r>0$.The theoretical results established in this paper are confirmed by severalnumerical examples.

Optimally truncated WKB approximation for the 1D stationary Schr\"odinger equation in the highly oscillatory regime