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

TL;DR

This paper develops an optimal truncated WKB approximation method for solving the highly oscillatory stationary 1D Schrödinger equation, achieving exponentially small errors with optimal truncation.

Contribution

It introduces a method to optimally truncate the WKB series, significantly reducing approximation errors in highly oscillatory regimes.

Findings

Error magnitude of the approximation is D7(\u0000D7)"],[

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Abstract

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