Optimally truncated WKB approximation for the highly oscillatory stationary 1D Schrödinger equation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912108605603840 |
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| author | Körner, Jannis Arnold, Anton Klein, Christian Melenk, Jens Markus |
| author_facet | Körner, Jannis Arnold, Anton Klein, Christian Melenk, Jens Markus |
| contents | We discuss the numerical solution of initial value problems for $\varepsilon^2\,φ''+a(x)\,φ=0$ 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 $c>0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_00955 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimally truncated WKB approximation for the highly oscillatory stationary 1D Schrödinger equation Körner, Jannis Arnold, Anton Klein, Christian Melenk, Jens Markus Numerical Analysis We discuss the numerical solution of initial value problems for $\varepsilon^2\,φ''+a(x)\,φ=0$ 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 $c>0$. |
| title | Optimally truncated WKB approximation for the highly oscillatory stationary 1D Schrödinger equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2310.00955 |