A dynamical systems approach to WKB-methods: The eigenvalue problem for a single well potential
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| Format: | Preprint |
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2025
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| author | Kristiansen, Kristian Uldall Szmolyan, Peter |
| author_facet | Kristiansen, Kristian Uldall Szmolyan, Peter |
| contents | In this paper, we revisit the eigenvalue problem of the one-dimensional Schr{ö}dinger equation for smooth single well potentials. In particular, we provide a new interpretation of the Bohr-Sommerfeld quantization formula. A novel aspect of our results, which are based on recent work of the authors on the turning point problem based upon dynamical systems methods, is that we cover all eigenvalues $E\in [0,\mathcal O(1)]$ and show that the Bohr-Sommerfeld quantitization formula approximates all of these eigenvalues (in a sense that is made precise). At the same time, we provide rigorous smoothness statements of the eigenvalues as functions of $ε$. We find that whereas the small eigenvalues $E=\mathcal O(ε)$ are smooth functions of $ε$, the large ones $E=\mathcal O(1)$ are smooth functions of $nε\in[c_1,c_2],\,0<c_1<c_2<\infty$, and $0\le ε^{1/3}\ll 1$; here $n\in \mathbb N_0$ is the index of the eigenvalues. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_10707 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A dynamical systems approach to WKB-methods: The eigenvalue problem for a single well potential Kristiansen, Kristian Uldall Szmolyan, Peter Mathematical Physics Dynamical Systems Spectral Theory In this paper, we revisit the eigenvalue problem of the one-dimensional Schr{ö}dinger equation for smooth single well potentials. In particular, we provide a new interpretation of the Bohr-Sommerfeld quantization formula. A novel aspect of our results, which are based on recent work of the authors on the turning point problem based upon dynamical systems methods, is that we cover all eigenvalues $E\in [0,\mathcal O(1)]$ and show that the Bohr-Sommerfeld quantitization formula approximates all of these eigenvalues (in a sense that is made precise). At the same time, we provide rigorous smoothness statements of the eigenvalues as functions of $ε$. We find that whereas the small eigenvalues $E=\mathcal O(ε)$ are smooth functions of $ε$, the large ones $E=\mathcal O(1)$ are smooth functions of $nε\in[c_1,c_2],\,0<c_1<c_2<\infty$, and $0\le ε^{1/3}\ll 1$; here $n\in \mathbb N_0$ is the index of the eigenvalues. |
| title | A dynamical systems approach to WKB-methods: The eigenvalue problem for a single well potential |
| topic | Mathematical Physics Dynamical Systems Spectral Theory |
| url | https://arxiv.org/abs/2501.10707 |