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

Jannis Körner, Anton Arnold, Christian Klein, Jens Markus Melenk

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 $c>0$.

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

Abstract

We discuss the numerical solution of initial value problems for in the highly oscillatory regime, i.e., with and . We analyze and implement an approximate solution based on the well-known WKB-ansatz. The resulting approximation error is of magnitude where refers to the truncation order of the underlying asymptotic series. When the optimal truncation order is chosen, the error behaves like with some .
Paper Structure (5 sections, 3 theorems, 10 equations, 2 figures)

This paper contains 5 sections, 3 theorems, 10 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. WKB APPROXIMATION
  3. ERROR ANALYSIS OF THE WKB APPROXIMATION
  4. NUMERICAL TEST
  5. ACKNOWLEDGMENTS

Key Result

Lemma 1

There exist constants $K_{1},K_{2}>0$ depending only on $G$ and $S_{0}'$ such that

Figures (2)

  • Figure 1: Left: Semilog plot of the $L^{\infty}$-norms of $S_{n}'$ as a function of $n$. Right: Log-Log plot of the $L^{\infty}$-error of the WKB approximation as a function of $\varepsilon$, for several choices of $N$.
  • Figure 2: Left: Log-Log plot of the optimal truncation order as a function of $\varepsilon$. Right: The corresponding values of the optimal error as a function of $\varepsilon$, also as a Log-Log plot. The dashed line is proportional to $\varepsilon^{-2}\exp(-\frac{6}{5\varepsilon})$.

Theorems & Definitions (4)

  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Remark 1